Patent Publication Number: US-10329988-B2

Title: Apparatus for measuring ammonia concentration, system for measuring ammonia concentration, system for treating exhaust gas, and method for measuring ammonia concentration

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to an apparatus for measuring ammonia concentration, a system for measuring ammonia concentration, a system for treating an exhaust gas, and a method for measuring ammonia concentration. 
     2. Description of the Related Art 
     Hitherto, ammonia sensors to detect ammonia concentrations in target gases such as exhaust gases of automobiles have been known. For example, Patent Literature 1 describes a multi-gas sensor including an ammonia sensing section having a pair of electrodes arranged on a solid electrolyte body. Formula (2) based on a mixed-potential formula is known as the characteristics of the electromotive force (EMF) of a mixed potential cell including solid electrolyte body and a pair of electrodes (for example, Non-Patent Literature 1). 
                   [     Math   .           ⁢   1     ]                           EMF   ∝       RT     2   ⁢   F       ⁢     (         2   3     ⁢   ln   ⁢           ⁢     p     NH   ⁢           ⁢   3         -       1   2     ⁢   ln   ⁢           ⁢     p     O   ⁢           ⁢   2         -     ln   ⁢           ⁢     p     H   ⁢           ⁢   2   ⁢   O           )               (   2   )               
(where
 
     EMF: the electromotive force of the mixed potential cell 
     R: a gas constant [J/(K·mol)] 
     T: the temperature of the mixed potential cell [K] 
     F: the Faraday constant [C/mol] 
     p NH3 : ammonia concentration in a target gas 
     p O2 : oxygen concentration in the target gas 
     p H2O : H 2 O concentration in the target gas) 
     CITATION LIST 
     Patent Literature 
     
         
         [PTL 1] Japanese Patent No. 5204160 
       
    
     Non Patent Literature 
     
         
         [NPL 1] D. Schonauer et al., Sensors and Actuators B vol. 140 (2009), p. 585-590 
       
    
     SUMMARY OF THE INVENTION 
     However, the inventors have conducted studies and have found that in actual sensor elements, the relationship among an electromotive force EMF, ammonia concentration p NH3 , oxygen concentration p O2 , H 2 O concentration p H2O  does not obey formula (2), in some cases. Thus, when the ammonia concentration p NH3  is calculated from formula (2) in a mixed potential-type ammonia sensor, the ammonia concentration in the target gas is not accurately derived, in some cases. 
     The present invention has been accomplished in order to solve these problems and mainly aims to derive ammonia concentration in a target gas with higher accuracy. 
     In the present invention, the following measures are used in order to achieve the above-described main object. 
     An apparatus of the present invention for measuring ammonia concentration in a target gas with a sensor element including a mixed potential cell that includes a solid electrolyte body, a detection electrode arranged on the solid electrolyte body, and a reference electrode arranged on the solid electrolyte body, includes: 
     an electromotive force acquisition section configured to acquire information about an electromotive force of the mixed potential cell while the detection electrode is exposed to the target gas; 
     an oxygen concentration acquisition section configured to acquire information about oxygen concentration in the target gas; and 
     an ammonia concentration derivation section configured to derive ammonia concentration in the target gas from the acquired information about the electromotive force, the acquired information about the oxygen concentration, and the relationship represented by formula (1):
 
EMF=α log a ( p   NH3 )−β log b ( p   O2 )+γ log c ( p   NH3 )×log d ( p   O2 )+B  (1)
 
(where
 
EMF: an electromotive force of the mixed potential cell,
 
α, β, γ, and B: constants (provided that each of α, β, and γ≠0),
 
a, b, c, and d: any base (provided that each of a, b, c, and d≠1, and each of a, b, c, and d&gt;0),
 
p NH3 : the ammonia concentration in the target gas, and
 
p O2 : the oxygen concentration in the target gas).
 
     In the apparatus for measuring ammonia concentration, the ammonia concentration in the target gas is derived from the information about the electromotive force of the mixed potential cell of the sensor element, the information about the oxygen concentration in the target gas, and the relationship of formula (1). In this way, the use of formula (1) can derive the ammonia concentration in the target gas with higher accuracy than that in the case of using formula (2) described above. Here, the derivation of the ammonia concentration on the basis of the relationship of formula (1) may be performed by using the relationship of formula (1) and is not limited to the derivation of the ammonia concentration using formula (1) itself. For example, the ammonia concentration may be derived from a formula obtained by modifying formula (1). The relationship among the values of the variables (EMF, p NH3 , p O2 ) of formula (1) is stored in the form of a map, and the ammonia concentration may be derived from the map. The constants α, β, and B are values depending on the sensor element and can be determined by, for example, experiments in advance. 
     In this case, the target gas may have a temperature of 150° C. or higher. The target gas may have a 200° C. or higher. The target gas may have a temperature of 400° C. or lower. 
     A system of the present invention for measuring ammonia concentration includes the sensor element and the ammonia concentration measurement apparatus. Accordingly, the system for measuring ammonia concentration has the same effect as the apparatus of the present invention for measuring ammonia concentration, i.e., for example, the effect of deriving ammonia concentration in a target gas with higher accuracy. 
     In the system for measuring ammonia concentration, the detection electrode may be composed of a Au—Pt alloy as a main component. The Au—Pt alloy is suitable for a main component of the detection electrode because a mixed potential is easily established at the triple phase boundary of the solid electrolyte body and the target gas. In this case, the detection electrode may have a degree of concentration (=amount of Au present [atom %]/amount of Pt present [atom %]) of 0.3 or more, the degree of concentration being measured by at least one of X-ray photoelectron spectroscopy (XPS) and Auger electron spectroscopy (AES). A degree of concentration of 0.3 or more can more reliably establish the mixed potential. The degree of concentration may be 0.1 or more. 
     In the system of the present invention for measuring ammonia concentration, the sensor element may include a heater configured to heat the mixed potential cell to an operating temperature of 450° C. or higher and 650° C. or lower. In the system for measuring ammonia concentration, the use of an operating temperature of 450° C. or higher can appropriately activate the solid electrolyte body. In the system for measuring ammonia concentration, the use of an operating temperature of 650° C. or lower can inhibit a decrease in measurement accuracy due to the combustion of ammonia. The operating temperature may be 600° C. or lower. 
     A system of the present invention for treating an exhaust gas includes any one of the systems for measuring ammonia concentration according to the foregoing embodiments, and an exhaust gas path through which an exhaust gas serving as the target gas from an internal combustion engine flows, the sensor element being arranged in the exhaust gas path. Accordingly, the system for treating an exhaust gas has the same effect as the system for measuring ammonia concentration, i.e., for example, the effect of deriving ammonia concentration in a target gas with higher accuracy. 
     The system of the present invention for treating an exhaust gas may further include one or more oxidation catalysts arranged in the exhaust gas path, in which the sensor element may be arranged on the downstream side of the exhaust gas path in contrast to one of the one or more oxidation catalysts arranged at the upstream end. In this case, the target gas in which a component (for example, at least one of hydrocarbons and carbon monoxide) that is present in the target gas and that affects the measurement accuracy of the ammonia concentration has been oxidized by the oxidation catalysts reaches the sensor element. Thus, in the system for treating an exhaust gas, the ammonia concentration in the target gas can be derived with higher accuracy. 
     A method of the present invention for measuring ammonia concentration in a target gas with a sensor element including a mixed potential cell that includes a solid electrolyte body, a detection electrode arranged on the solid electrolyte body, and a reference electrode arranged on the solid electrolyte body, includes: 
     an electromotive force acquisition step of acquiring information about an electromotive force of the mixed potential cell while the detection electrode is exposed to the target gas; 
     an oxygen concentration acquisition step of acquiring information about oxygen concentration in the target gas; and 
     a concentration derivation step of deriving the ammonia concentration in the target gas from the acquired information about the electromotive force, the acquired information about the oxygen concentration, and a relationship represented by formula (1):
 
EMF=α log a ( p   NH3 )−β log b ( p   O2 )+γ log c ( p   NH3 )×log d ( p   O2 )+B   (1)
 
(where
 
     EMF: an electromotive force of the mixed potential cell, 
     α, β, γ, and B: constants (provided that each of α, β, and γ≠0), 
     a, b, c, and d: any base (provided that each of a, b, c, and d≠1, and each of a, b, c, and d&gt;0), 
     p NH3 : the ammonia concentration in the target gas, and 
     p O2 : the oxygen concentration in the target gas). 
     As with the foregoing apparatus for measuring ammonia concentration, the ammonia concentration in the target gas can be derived with higher accuracy from the relationship of formula (1) by the method for measuring ammonia concentration. In the method for measuring ammonia concentration, the apparatus for measuring ammonia concentration, the system for measuring ammonia concentration, and the system for treating an exhaust gas according to various embodiments may be used, and steps of providing these functions may be added. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is an explanatory drawing of a system  2  for treating an exhaust gas of an engine  1 . 
         FIG. 2  is an explanatory drawing of a system  20  for measuring ammonia concentration. 
         FIG. 3  is a flow chart illustrating an example of an ammonia concentration derivation routine. 
         FIG. 4  is a flow chart illustrating an example of a constant derivation processing. 
         FIG. 5  is a graph depicting the relationships between ammonia concentrations p NH3  [ppm] and electromotive forces EMFs [mV] of a sensor element  1 . 
         FIG. 6  is a graph depicting the relationship between the oxygen concentration p O2  and the slope K of the sensor element  1 . 
         FIG. 7  is a graph depicting the relationship between the oxygen concentration p O2  and the intercept L of the sensor element  1 . 
         FIG. 8  is a graph depicting actually measured electromotive forces EMFs of the sensor element  1  and straight lines derived from formula (8). 
         FIG. 9  is a graph depicting the relationship between the H 2 O concentration p H2O  [%] and the electromotive force EMF [mV] of the sensor element  1 . 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Embodiments of the present invention will be described below with reference to the drawings.  FIG. 1  is an explanatory drawing of a system  2  for treating an exhaust gas of an engine  1 .  FIG. 2  is an explanatory drawing of a system  20  for measuring ammonia concentration. 
     The system  2  for treating an exhaust gas is a system for treating an exhaust gas serving as a target gas from the engine  1 . In this embodiment, the engine  1  is a diesel engine. As illustrated in  FIG. 1 , the system  2  for treating an exhaust gas includes an exhaust gas path  3  connected to the engine  1  and the system  20  for measuring ammonia concentration including a gas sensor  30  arranged in the exhaust gas path  3 . In the system  2  for treating an exhaust gas, a diesel oxidation catalyst (DOC)  4 , a diesel particulate filter (DPF)  5 , an injector  6 , a selective catalytic reduction (SCR)  7 , the gas sensor  30 , and an ammonia slip catalyst (ASC)  8  are arranged, in this order, from the upstream side toward the downstream side of the exhaust gas. The DOC  4  is one of oxidation catalysts included in the system  2  for treating an exhaust gas and converts HC and CO in the exhaust gas into water and carbon dioxide for detoxification. The DPF  5  traps PM in the exhaust gas. The injector  6  is a device configured to inject at least one of ammonia and a substance capable of forming ammonia (for example, urea) into an exhaust pipe to supply the at least one of ammonia and the substance to the SCR  7 . In this embodiment, the injector  6  injects urea, and the injected urea is hydrolyzed to form ammonia. The SCR  7  decomposes nitrogen oxides (NOx) into harmless N 2  and H 2 O by reduction using ammonia supplied from the injector  6  into the exhaust pipe. The exhaust gas passing through the SCR  7  flows through a pipe  10 . The gas sensor  30  is attached to the pipe  10 . The ASC  8  is arranged downstream of the pipe  10 . The ASC  8  is one of the oxidation catalysts included in the system  2  for treating an exhaust gas and is also referred to as a “downstream DOC” with respect to the DOC  4  (upstream DOC). That is, the system  2  for treating an exhaust gas according to this embodiment includes two oxidation catalysts: the DOC  4  and the ASC  8 . The gas sensor  30  is arranged downstream in contrast to the DOC  4  arranged at the upstream end among one or more oxidation catalysts (two oxidation catalysts in this embodiment) included in the system  2  for treating an exhaust gas. The ASC  8  decomposes excessive ammonia in the exhaust gas passing through the SCR  7  into harmless N 2  and H 2 O by oxidation. The exhaust gas passing through the ASC  8  is released into, for example, air. 
     The system  20  for measuring ammonia concentration includes the gas sensor  30  and an apparatus  70  for measuring ammonia concentration, the apparatus being electrically connected to the gas sensor  30 . The gas sensor  30  is an ammonia sensor configured to generate an electrical signal depending on the concentration of excessive ammonia contained in the target gas passing through the SCR  7  in the pipe  10 . The gas sensor  30  also functions as an oxygen sensor configured to generate an electrical signal depending on the concentration of oxygen in the target gas and serves as a multi-sensor. The apparatus  70  for measuring ammonia concentration derives ammonia concentration in the target gas from the electrical signal generated by the gas sensor  30  and transmits the resulting data to an engine ECU  9 . The engine ECU  9  controls the amount of urea injected from the injector  6  into the exhaust pipe in such a manner that the detected excessive ammonia concentration approaches zero. The system  20  for measuring ammonia concentration will be described in detail below. 
     As illustrated in  FIG. 1 , the gas sensor  30  is fixed in the pipe  10  in such a manner that the central axis of the gas sensor  30  is perpendicular to the flow of the target gas in the pipe  10 . The gas sensor  30  may be fixed in the pipe  10  in such a manner that the central axis of the gas sensor  30  is perpendicular to the flow of the target gas in the pipe  10  and is tilted at a predetermined angle (for example, 45°) with respect to the vertical direction (an up and down direction of  FIG. 1 ). As illustrated in the enlarged cross-sectional view of  FIG. 1 , the gas sensor  30  includes a sensor element  31 , a protective cover  32  that covers and protects the front end side (the lower end side in  FIG. 1 ) of the sensor element  31 , which is an end side of the sensor element  31  in the longitudinal direction, an element fixing portion  33  that encapsulates and fix the sensor element  31 , and a nut  37  fitted to the element fixing portion  33 . The one end side of the sensor element  31  is covered with a porous protective layer  48 . 
     The protective cover  32  is a cylindrical cover with a closed bottom, the cylindrical cover covering one end of the sensor element  31 . Although a single-layer cover is used in  FIG. 1 , for example, two-or-more-layer cover including an inner protective cover and an outer protective cover may be used. The protective cover  32  has holes through which the target gas is allowed to flow into the protective cover  32 . The one end of the sensor element  31  and the porous protective layer  48  are arranged in a cavity surrounded by the protective cover  32 . 
     The element fixing portion  33  includes a cylindrical main metal fitting  34 , a ceramic supporter  35  encapsulated in an inner through-hole of the main metal fitting  34 , and a compact  36  that is encapsulated in the inner through-hole of the main metal fitting  34  and that is formed of a ceramic powder composed of, for example, talc. The sensor element  31  is located on the central axis of the element fixing portion  33  and extends through the element fixing portion  33  in the longitudinal direction. The compact  36  is compressed between the main metal fitting  34  and the sensor element  31 . Thus, the compact  36  seals the through-hole in the main metal fitting  34  and fixes the sensor element  31 . 
     The nut  37  is fixed coaxially with the main metal fitting  34  and has an external thread portion on an outer periphery thereof. The external thread portion of the nut  37  is fitted with a fitting member  12  that is welded to the pipe  10  and that has an internal thread portion on an inner periphery thereof. Thus, the gas sensor  30  can be fixed to the pipe  10  while the one end side of the sensor element  31  and the protective cover  32  protrude into the pipe  10 . 
     The sensor element  31  will be described with reference to  FIG. 2 . The cross-sectional view of the sensor element  31  of  FIG. 2  illustrates a sectional view taken along the central axis of the sensor element  31  in the longitudinal direction (cross section taken in the up and down direction of  FIG. 1 ). The sensor element  31  includes a base  40  composed of an oxygen-ion-conducting solid electrolyte, a detection electrode  51  and an auxiliary electrode  52  arranged on the side of an end (the lower end of  FIG. 1  and the left end of  FIG. 2 ) of the sensor element  31  and on the upper surface of the base  40 , a reference electrode  53  arranged inside the base  40 , and a heater portion  60  that adjusts the temperature of the base  40 . 
     The base  40  has a plate-like structure in which four layers, i.e., a first substrate layer  41 , a second substrate layer  42 , a spacer layer  43 , and a solid electrolyte layer  44 , are stacked, in this order, from the bottom in  FIG. 2 , each of the layers being formed of an oxygen-ion-conducting solid electrolyte layer composed of, for example, zirconia (ZrO 2 ). A solid electrolyte used to form these four layers is a dense, gas-tight material. The periphery of a portion of the base  40  in the protective cover  32  is exposed to the target gas introduced into the protective cover  32 . A reference gas introduction cavity  46  is provided between an upper surface of the second substrate layer  42  and a lower surface of the solid electrolyte layer  44  in the base  40 , a side portion of the cavity being defined by a side surface of the spacer layer  43 . The reference gas introduction cavity  46  has an opening portion on the other end side (right end side of  FIG. 2 ) remote from the one end side of the sensor element  31 . For example, air is introduced into the reference gas introduction cavity  46 , air serving as a reference gas used to measure ammonia concentration and oxygen concentration. Each of the layers of the base  40  may be formed of a substrate containing 3% to 15% by mole yttria (Y 2 O 3 ) (yttria-stabilized zirconia (YSZ) substrate) serving as a stabilizer. 
     The detection electrode  51  is a porous electrode arranged on an upper surface of the solid electrolyte layer  44  of the base  40  in  FIG. 2 . The detection electrode  51 , the solid electrolyte layer  44 , and the reference electrode  53  form a mixed potential cell  55 . In the mixed potential cell  55 , a mixed potential (electromotive force EMF) is generated in the detection electrode  51 , depending on the concentration of a predetermined gas component in the target gas. The value of the electromotive force EMF between the detection electrode  51  and the reference electrode  53  is used to derive the ammonia concentration in the target gas. The detection electrode  51  is composed of, as a main component, a material that establishes a mixed potential depending on the ammonia concentration and that has detection sensitivity to the ammonia concentration. The detection electrode  51  may be composed of a noble metal, such as gold (Au), as a main component. The detection electrode  51  is preferably composed of a Au—Pt alloy as a main component. The term “main component” used here refers to a component contained in a largest amount present (atm %, atomic percent) with respect to the total amount of components contained. The detection electrode  51  preferably has a degree of concentration (=amount of Au present [atom %]/amount of Pt present [atom %]) of 0.3 or more, the degree of concentration being measured by at least one of X-ray photoelectron spectroscopy (XPS) and Auger electron spectroscopy (AES). A degree of concentration of 0.3 or more can more reliably establish the mixed potential. The degree of concentration of the detection electrode  51  refers to the degree of surface concentration on a surface of noble metal particles of the detection electrode  51 . The amount of Au present [atom %] is determined as the amount of Au present on the surfaces of the noble metal particles of the detection electrode  51 . Similarly, the amount of Pt present [atom %] is determined as the amount of Pt present on the surfaces of the noble metal particles of the detection electrode  51 . With regard to the surfaces of the noble metal particles, a surface (for example, an upper surface in  FIG. 2 ) of the detection electrode  51  or a fracture surface of the detection electrode  51  may be used. For example, in the case where the surface (the upper surface in  FIG. 2 ) of the detection electrode  51  is exposed, the degree of concentration can be measured on the surface; hence, the measurement may be performed by XPS. The degree of concentration may also be measured by AES. In the case where the detection electrode  51  is covered with the porous protective layer  48  as described in this embodiment, the fracture surface (fracture surface in the up and down direction of  FIG. 2 ) of the detection electrode  51  is subjected to measurement by XPS or AES to determine the degree of concentration. A higher degree of concentration results in a smaller amount of Pt present on the surface of the detection electrode  51 , thereby inhibiting the decomposition of ammonia in the target gas around the detection electrode  51  due to Pt. Thus, a higher degree of concentration results in a more improved derivation accuracy of the ammonia concentration in the system  20  for measuring ammonia concentration. Specifically, the degree of concentration is preferably 0.1 or more, more preferably 0.3 or more. The upper limit of the degree of concentration is not particularly set. For example, the detection electrode  51  may not contain Pt. The entire detection electrode  51  may be composed of Au. The detection electrode  51  has a porosity of, for example, 25% by volume to 60% by volume. 
     The auxiliary electrode  52  is a porous electrode arranged on the upper surface of the solid electrolyte layer  44 , similarly to the detection electrode  51 . The auxiliary electrode  52 , the solid electrolyte layer  44 , and the reference electrode  53  form an electrochemical concentration cell  56 . In the concentration cell  56 , an electromotive force difference V, which is a potential difference depending on the difference in oxygen concentration between the auxiliary electrode  52  and the reference electrode  53 , is established. The value of the electromotive force difference V is used to derive the oxygen concentration (oxygen partial pressure) in the target gas. The auxiliary electrode  52  may be composed of a catalytically active noble metal. For example, Pt, Ir, Rh, Pd, or an alloy containing at least one thereof can be used for the auxiliary electrode  52 . In this embodiment, the auxiliary electrode  52  is composed of Pt. 
     The reference electrode  53  is a porous electrode arranged on the lower surface of the solid electrolyte layer  44 , i.e., arranged on a side of the solid electrolyte layer  44  opposite that on which the detection electrode  51  and the auxiliary electrode  52  are arranged. The reference electrode  53  is exposed in the reference gas introduction cavity  46 , and a reference gas (here, air) in the reference gas introduction cavity  46  is introduced thereinto. The potential of the reference electrode  53  is the standard for the electromotive force EMF and the electromotive force difference V. The reference electrode  53  may be composed of a catalytically active noble metal. For example, Pt, Ir, Rh, Pd, or an alloy containing at least one thereof can be used for the reference electrode  53 . In this embodiment, the reference electrode  53  is composed of Pt. 
     The porous protective layer  48  covers a surface of the sensor element  31  including the detection electrode  51  and the auxiliary electrode  52 . For example, the porous protective layer  48  serves to inhibit the occurrence of cracking in the sensor element  31  due to the adhesion of water in the target gas. The porous protective layer  48  is composed of, for example, alumina, zirconia, spinel, cordierite, titania, or magnesia as a main component. In this embodiment, the porous protective layer  48  is composed of alumina. The thickness of the porous protective layer  48  is, but not particularly limited to, for example, 20 to 1,000 μm. The porosity of the porous protective layer  48  is, but not particularly limited to, for example, 5% by volume to 60% by volume. The sensor element  31  may not include the porous protective layer  48 . 
     The heater portion  60  serves to control the temperature of the base  40  (in particular, the solid electrolyte layer  44 ) by heating and keeping it warm in order to activate the solid electrolyte of the base  40  to increase the oxygen-ion conductivity. The heater portion  60  includes a heater electrode  61 , a heater  62 , a through-hole  63 , a heater insulating layer  64 , and a lead wire  66 . The heater electrode  61  is an electrode arranged so as to be in contact with a lower surface of the first substrate layer  41 . The heater electrode  61  is connected to a heater power supply  77  of the apparatus  70  for measuring ammonia concentration. 
     The heater  62  is an electrical resistor arranged so as to be held between the first substrate layer  41  and the second substrate layer  42 . The heater  62  is connected to the heater electrode  61  through the lead wire  66  and the through-hole  63 . The heater  62  is fed from the heater power supply  77  through the heater electrode  61  to generate heat, so that the base  40  included in the sensor element  31  is heated and kept warm. The heater  62  is configured to be able to control the output with a temperature sensor (here, temperature acquisition section  78 ) in such a manner that the mixed potential cell  55  and the concentration cell  56  (in particular, the solid electrolyte layer  44 ) have a predetermined operating temperature. The operating temperature is preferably 450° C. or higher because the solid electrolyte layer  44  of the mixed potential cell  55  can be appropriately activated. The operating temperature is preferably 650° C. or lower because it is possible to inhibit a decrease in the measurement accuracy due to the combustion of ammonia. The operating temperature may be 600° C. or lower. The heater insulating layer  64  is an insulating layer that is arranged on upper and lower surfaces of the heater  62  and that is composed of an insulating material such as alumina, specifically porous alumina. 
     The apparatus  70  for measuring ammonia concentration is an apparatus for measuring the ammonia concentration in the target gas with the sensor element  31 . The apparatus  70  for measuring ammonia concentration also serves as a controller of the sensor element  31 . The apparatus  70  for measuring ammonia concentration includes a control section  72 , an electromotive force acquisition section  75 , an oxygen concentration acquisition section  76 , the heater power supply  77 , and the temperature acquisition section  78 . 
     The control section  72  controls the entire apparatus and, for example, is a microprocessor including CPU, RAM, and so forth. The control section  72  includes a memory part  73  that stores a processing program and various data sets. The electromotive force acquisition section  75  is a module that acquires information about the electromotive force EMF of the mixed potential cell  55 . In this embodiment, the electromotive force acquisition section  75  is connected to the detection electrode  51  and the reference electrode  53  of the mixed potential cell  55  and thus functions as a voltage detection circuit that measures an electromotive force EMF. The oxygen concentration acquisition section  76  is a module that acquires information about the oxygen concentration in the target gas. In this embodiment, the oxygen concentration acquisition section  76  is connected to the auxiliary electrode  52  and the reference electrode  53  of the concentration cell  56  and thus functions as a voltage detection circuit that measures the electromotive force difference V serving as information about the oxygen concentration. The electromotive force acquisition section  75  and the oxygen concentration acquisition section  76  output the electromotive force EMF and the electromotive force difference V that have been measured by them to the control section  72 . The control section  72  derives the ammonia concentration in the target gas from the electromotive force EMF and the electromotive force difference V. The heater power supply  77  is a power supply that supplies power to the heater  62 , and the output power is controlled by the control section  72 . The temperature acquisition section  78  is a module that acquires a value about the temperature of the heater  62  (here, value of resistance). The temperature acquisition section  78  acquires the value of resistance of the heater  62  by, for example, connecting the temperature acquisition section  78  to the heater electrode  61 , allowing a minute electric current to flow, and measuring a voltage. 
     Each of the detection electrode  51 , the auxiliary electrode  52 , and the reference electrode  53  is electrically connected to a corresponding one of lead wires arranged toward the other end of the sensor element  31  (right side of  FIG. 2 ) (not illustrated in  FIG. 2 ). The electromotive force acquisition section  75  and the oxygen concentration acquisition section  76  measure the electromotive force EMF and the electromotive force difference V, respectively, through the lead wires. 
     The measurement of the ammonia concentration with the system  20  for measuring ammonia concentration will be described below.  FIG. 3  is a flow chart illustrating an example of an ammonia concentration derivation routine executed by the control section  72 . The routine is stored in, for example, the memory part  73  of the control section  72 . When a command to derive ammonia concentration is fed from the engine ECU  9 , the routine is repeatedly executed, for example, with a predetermined period (for example, several milliseconds to several tens of milliseconds). The control section  72  controls, in advance, the temperature of the mixed potential cell  55  and the concentration cell  56  to a predetermined operating temperature (for example, a temperature in the range of 450° C. or higher and 650° C. or lower) by controlling the output power of the heater power supply  77  to produce heat from the heater  62 . For example, the control section  72  controls the temperature of the mixed potential cell  55  and the concentration cell  56  to a predetermined operating temperature by controlling the output power of the heater power supply  77  in such a manner that the temperature (here, resistance) of the heater  62  acquired by the temperature acquisition section  78  is a predetermined value. 
     When the ammonia concentration derivation routine is started, the control section  72  executes an electromotive force acquisition step of acquiring information about the electromotive force EMF of the mixed potential cell  55  with the electromotive force acquisition section  75  (step S 100 ). In this embodiment, the control section  72  acquires the value of the electromotive force EMF measured by the electromotive force acquisition section  75  on an as-is basis. The control section  72  executes the ammonia concentration derivation routine in a state in which, basically, an exhaust gas from the engine  1  flows through the pipe  10  and the protective cover  32 . Thus, the control section  72  acquires the electromotive force EMF of the mixed potential cell  55  while the detection electrode  51  is exposed to the target gas. Here, in the mixed potential cell  55 , electrochemical reactions, such as the oxidation of ammonia and the ionization of oxygen in the target gas, occur at the triple phase boundary of the detection electrode  51 , the solid electrolyte layer  44  and the target gas to establish a mixed potential on the detection electrode  51 . Thus, the electromotive force EMF is a value based on the ammonia concentration and the oxygen concentration in the target gas. 
     The control section  72  executes an oxygen concentration acquisition step of acquiring information about oxygen concentration in the target gas with the oxygen concentration acquisition section  76  (step S 110 ). In this embodiment, the control section  72  acquires the electromotive force difference V of the concentration cell  56  from the oxygen concentration acquisition section  76 . Here, in the concentration cell  56 , the electromotive force difference V is generated between the auxiliary electrode  52  and the reference electrode  53 , depending on the difference in oxygen concentration between the target gas and air in the reference gas introduction cavity  46 . Hydrocarbons, NH 3 , CO, NO, NO 2  in the target gas are subjected to redox by the catalysis of Pt serving as the auxiliary electrode  52 . The concentrations of these gas components in the target gas are significantly lower than the oxygen concentration in the target gas. Thus, the occurrence of the redox has little influence on the oxygen concentration in the target gas. Accordingly, the electromotive force difference V is a value based on the oxygen concentration in the target gas. By the control section  72 , any one of step S 100  and step S 110  may be first executed, or the steps may be executed in parallel. 
     Subsequently, the control section  72  executes a concentration derivation step of deriving ammonia concentration in the target gas from the information about the electromotive force EMF acquired in step S 100 , the information about the oxygen concentration acquired in step S 110 , and the relationship represented by formula (1) (step S 120 ) and terminates the routine. The relationship represented by formula (1) is stored in, for example, the memory part  73 , in advance.
 
EMF=α log a ( p   NH3 )−β log b ( p   O2 )+γ log c ( p   NH3 )×γ log c ( p   NH3 )×log d ( p   O2 )+B  (1)
 
(where
 
     EMF: an electromotive force of the mixed potential cell, 
     α, β, γ, and B: constants (provided that each of α, β, and γ≠0), 
     a, b, c, and d: any base (provided that each of a, b, c, and d≠1, and each of a, b, c, and d&gt;0), 
     p NH3 : the ammonia concentration in the target gas, and 
     p O2 : the oxygen concentration in the target gas). 
     In step S 120 , the control section  72  replaces “EMF” in formula (1) by the value of the electromotive force EMF acquired in step S 100 . The control section  72  derives the oxygen concentration p O2  from the electromotive force difference V acquired in step S 110  and the relationship, which is stored in the memory part  73  in advance, between the electromotive force difference V and the oxygen concentration p O2  and replaces “p O2 ” in formula (1) by the derived value. The control section  72  derives the ammonia concentration p NH3  in formula (1). The units of the electromotive force EMF may be, for example, [mV]. The ammonia concentration p NH3  is the volume fraction of ammonia in the target gas. The oxygen concentration p O2  is the volume fraction of oxygen in the target gas. With regard to the units of p NH3  and p O2 , a value given in parts per million [ppm] may be used, a value given in percent [%] may be used, or a dimensionless value (for example, in the case of 10%, the value is 0.1) may be used. p NH3  and p O2  may be given in different units. Each of the bases a, b, c, and d may be a value of 10 or Napier&#39;s constant e. Each of the constants α, β, γ, and B has a value determined, depending on the sensor element  31  and can have different values, depending on the sensor element  31 . The constants α, β, γ, and B can be determined by, for example, experiments described below, in advance. The constants α and β may satisfy α:β≠(⅔):(½). The constants α and β may have a positive value. The constant γ may have a positive value or a negative value. The derivation of the ammonia concentration p NH3  executed by the control section  72  on the basis of the relationship of formula (1) may be performed using the relationship of formula (1) and is not limited to the derivation of the ammonia concentration using formula (1) itself. For example, formula (1) itself may be stored in the memory part  73 . Formula (1a) or (1b), described below, obtained by modifying the formula (1) may be stored. Formula (1c), described below, obtained by modifying the formula (1a) in such a manner that the left side is “p NH3 ” alone may be stored. The relationship of values of the variables (EMF, p NH3 , and p O2 ) of formula (1) is stored as a map in the memory part  73 . The control section  72  may derive the ammonia concentration p NH3  from the map.
 
EMF=(α+γ′ log d ( p   O2 ))log a ( p   NH3 )−β log b ( p   O2 )+B   (1a)
 
(where γ′=γ/log a  c)
 
EMF=α log a ( p   NH3 )−(β−γ″ log c ( p   NH3 ))log b ( p   O2 )+B   (1b)
 
(where γ″=γ/log b d)
 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     2 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     p 
                     
                       NH 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       3 
                     
                   
                   = 
                   
                     
                       a 
                       
                         
                           ( 
                           
                             EMF 
                             + 
                             
                               β 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   log 
                                   b 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     p 
                                     02 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             B 
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             α 
                             + 
                             
                               
                                 γ 
                                 ′ 
                               
                               ⁢ 
                               
                                 
                                   log 
                                   d 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     p 
                                     02 
                                   
                                   ) 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ( 
                     
                       
                         where 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           γ 
                           ′ 
                         
                       
                       = 
                       
                         γ 
                         
                           
                             log 
                             a 
                           
                           ⁢ 
                           c 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   
                     1 
                     ⁢ 
                     c 
                   
                   ) 
                 
               
             
           
         
       
     
     As described above, the control section  72  derives the ammonia concentration p NH3  in the target gas from the relationship of formula (1) in this embodiment. Thus, the ammonia concentration in the target gas can be derived with high accuracy, compared with, for example, in the case of using formula (2) described above. This will be described below. 
     As described above, formula (2) is known as the characteristics of the electromotive force EMF of the mixed potential-type ammonia sensor. However, the inventors have conducted studies and have found that in an actual sensor element (for example, the sensor element  31 ), the relationship among the electromotive force EMF, the ammonia concentration p NH3 , the oxygen concentration p O2 , the H 2 O concentration p H2O  does not obey formula (2). For example, although the relationship between the degree of the effect of the ammonia concentration p NH3  on the electromotive force EMF (NH 3  sensitivity) and the degree of the effect of the oxygen concentration p O2  on the electromotive force EMF (O 2  interference) should be NH 3  sensitivity:O 2  interference=(⅔):(½) from the coefficients of the term p NH3  and the term p O2  in the right side of formula (2), the relationship was not obtained, in some cases. According to formula (2), although the effect of the H 2 O concentration p H2O  on the electromotive force EMF (H 2 O interference) should be present, in fact, even when the H 2 O concentration p H2O  in the target gas was changed, the electromotive force EMF remained substantially unchanged. 
     With regard to the oxidation of ammonia and the ionization of oxygen in the target gas, an anodic reaction represented by equation (a) described below and a cathodic reaction represented by equation (b) described below occur at the triple phase boundary of the mixed potential cell  55 . Equations (a) and (b) can also be expressed as equations (a)′ and (b)′. In equation (a), “O O ” represents an oxygen ion (O 2− ) present in an oxygen site in the solid electrolyte layer  44 . The fourth term in the right side of equation (a) indicates that an oxygen ion is not present (not sufficient) in the oxygen site in the solid electrolyte layer  44 .
 
[Math. 3]
 
[Anodic reaction]
 
⅔NH 3 +O O →H 2 O+⅓N 2 +2 e   − +V O   ••   (a)
 
NH 3 +O 2 →H 2 O+N 2   +e   −   (a)′
 
[Cathodic reaction]
 
½O 2 +2 e   − +V O   •• →O O   (b)
 
O 2 +4 e   − →2O 2−   (b)′
 
     The anodic reaction and the cathodic reaction occur simultaneously at the triple phase boundary of one detection electrode (for example, the detection electrode  51 ) to form a local cell, thereby establishing an electromotive force EMF. This is a mixed potential cell (for example, the mixed potential cell  55 ). The electromotive force EMF at this time should theoretically obey formula (2). For example, the coefficient “⅔” of the ammonia concentration p NH3  in formula (2) is a value based on the coefficient “⅔” of NH 3  on the left side of equation (a). Similarly, the coefficient “½” of the oxygen concentration p O2  in the formula (2) and the coefficient “1” of the H 2 O concentration pH 2 O are values based on the coefficient “½” of O 2  on the left side of equation (b) and the coefficient “1” of H 2 O on the right side of equation (a), respectively. 
     For actual sensor elements, however, it was found in experiments that the relationship among the variables obeys formula (1), and not formula (2). The inventors have considered that the reason for this that p NH3 , p O2 , and p H2O  in formula (2) need not be replaced by the concentrations in the target gas and should be replaced by partial pressures at the triple phase boundary. Letting an NH 3  partial pressure, an O 2  partial pressure, and a H 2 O partial pressure at the triple phase boundary on the detection electrode be p NH3 *, p O2 *, and p H2O *, respectively, formula (A1) holds. This can also be derived from formula (2). The actual electromotive force EMF seemingly obeys formula (A1), and not formula (2). Because p NH3 *, p O2 *, and p H2O * at the triple phase boundary cannot be directly detected, a formula including p NH3 , p O2 , and p H2O  in the target gas need to be derived from formula (A1). The inventors thought that we could explain below that formula (1) including p NH3 , p O2 , and p H2O  holds on the basis of formula (A1). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     4 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   EMF 
                   ∝ 
                   
                     
                       RT 
                       
                         2 
                         ⁢ 
                         F 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             2 
                             3 
                           
                           ⁢ 
                           ln 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             p 
                             
                               NH 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               3 
                             
                             * 
                           
                         
                         - 
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                           ln 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             p 
                             
                               O 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                             * 
                           
                         
                         - 
                         
                           ln 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             p 
                             
                               H 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               O 
                             
                             * 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   A1 
                   ) 
                 
               
             
           
         
       
     
     Let us first consider a mixed potential equation from a microscopic point of view. As described above, the partial pressures, ln p NH3 *, ln p O2 *, and ln p H2O *, at the triple phase boundary on the detection electrode are not equal to the partial pressures, ln p NH3 , ln p O2 , and ln p H2O , in an atmospheric gas (target gas). This is because the following dynamic changes occur in the electrochemical reactions: molecules are adsorbed onto a surface of the detection electrode, diffused on the surface of the detection electrode to reach the triple phase boundary, and subjected to electrochemical reactions, and the resulting products are desorbed from the surface of the detection electrode, rather than the fact that the molecules directly reach the triple phase boundary from the gas phase. Let us now consider the product H 2 O formed in the anodic reaction. The formed H 2 O is seemingly adsorbed on the detection electrode and then desorbed into the gas phase. Because a large amount of H 2 O is present in the target gas, the H 2 O formed in the anodic reaction seems to be not readily desorbed from the surface of the detection electrode. It is thus considered that the H 2 O partial pressure p H2O * at the triple phase boundary during the adsorption of H 2 O is larger than the H 2 O partial pressure p H2O  in the target gas and that formula (A2) described below always holds. In the target gas (here, an exhaust gas), the H 2 O concentration is usually about 5% to about 15%, and the total pressure remains constant at 1 atm. For the sake of safety, considering that the H 2 O concentration changes in a wide range of 1% to 20%, formula (A3) described below holds.
 
 p   H2O   *&gt;p   H2O   (A2)
 
0.01 atm&lt; p   H2O &lt;0.2 atm  (A3)
 
     Let us next consider that what will become of p H2O * when p H2O  is changed while H 2 O is adsorbed on the surface of the detection electrode. With regard to H 2 O at the triple phase boundary, H 2 O adsorbed on the detection electrode is denoted by H 2 O(ad), and H 2 O in the gas phase is denoted by H 2 O(gas). The partial pressure of H 2 O adsorbed on the detection electrode is denoted by p H2O(ad) , and the partial pressure of H 2 O in the gas phase is denoted by p H2O(gas) . Thus, p H2O *=p H2O(ad) +p H2O(gas) . p H2O(ad)  includes the partial pressure of H 2 O that comes from the target gas and that is adsorbed on the detection electrode, and the partial pressure of H 2 O that is formed by the anodic reaction (the foregoing equations (a) and (a)′) and that is adsorbed on the detection electrode. p H2O(gas)  includes the partial pressure of H 2 O that is contained in the target gas and that is present at the triple phase boundary in a gas phase state, and the partial pressure of H 2 O that is formed by the anodic reaction and that is in a gas phase state. With regard to H 2 O(ad) and H 2 O(gas), formulae (A4) and (A5) described below hold, provided that an equilibrium constant K=(constant). Although p H2O * is supposed to be changed according to formulae (A4) and (A5), in fact, it behaves differently. The reason for this is presumably that p H2O  changes in the range represented by formula (A3) described above, whereas p H2O(ad)  cannot change once the adsorption of H 2 O on the detection electrode is stabilized and reaches a steady state (=1 atm). The reason p H2O(ad)  is 1 atm in the steady state is described below. Because H 2 O (ad)  adsorbed on the detection electrode is not in the gas phase, the amount of H 2 O(ad) is expressed as activity a H2O(ad) , and not as partial pressure, to be exact. When H 2 O(ad) is regarded as a solid, the activity a H2O(ad)  have a value of 1 (i.e., the activity is 1 irrespective of the amount adsorbed on the detection electrode), and an activity of 1 can be regarded as comparable to a partial pressure of 1 atm. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     5 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       H 
                       2 
                     
                     ⁢ 
                     
                       O 
                       ⁡ 
                       
                         ( 
                         ad 
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⇄ 
                     K 
                   
                   ⁢ 
                   
                     
                       H 
                       2 
                     
                     ⁢ 
                     
                       O 
                       ⁡ 
                       
                         ( 
                         gas 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   A4 
                   ) 
                 
               
             
             
               
                 
                   K 
                   = 
                   
                     
                       
                         p 
                         
                           H 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           
                             O 
                             ⁡ 
                             
                               ( 
                               ad 
                               ) 
                             
                           
                         
                       
                       
                         p 
                         
                           H 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           
                             O 
                             ⁡ 
                             
                               ( 
                               gas 
                               ) 
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           p 
                           
                             H 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             O 
                           
                           * 
                         
                         - 
                         
                           p 
                           
                             H 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                               O 
                               ⁡ 
                               
                                 ( 
                                 gas 
                                 ) 
                               
                             
                           
                         
                       
                       
                         p 
                         
                           H 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           
                             O 
                             ⁡ 
                             
                               ( 
                               gas 
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   A5 
                   ) 
                 
               
             
           
         
       
     
     Accordingly, p H2O(ad)  can be regarded as 1 atm. Although as with formula (A3), p H2O(gas)  seems to be about 0.01 to about 0.2 atm, because H 2 O(ad), which can be regarded as 1 atm, is present on the surface of the detection electrode, H 2 O in the gas phase is less likely to contribute to the reaction, the partial pressure p H2O(gas)  of H 2 O present in the gas phase at the triple phase boundary seems to have a value significantly smaller than 0.01 to 0.2 atm. Thus, p H2O(ad) &gt;&gt;p H2O(gas)  seemingly holds, and p H2O(gas)  seems to have a very small, negligible value. Accordingly, even if p H2O  changes while H 2 O is adsorbed on the surface of the detection electrode, p H2O * can be regarded as constant, as represented by formula (A6). Thus, formula (A1) can be regarded as formula (A7). That is, the H 2 O partial pressure p H2O * at the triple phase boundary can be regarded as having no effect (H 2 O interference) on the electromotive force EMF. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     6 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     p 
                     
                       H 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                       ⁢ 
                       O 
                     
                     * 
                   
                   = 
                   
                     
                       
                         p 
                         
                           H 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           
                             O 
                             ⁡ 
                             
                               ( 
                               ad 
                               ) 
                             
                           
                         
                       
                       + 
                       
                         p 
                         
                           H 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           
                             O 
                             ⁡ 
                             
                               ( 
                               gas 
                               ) 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                       = 
                       
                         
                             
                         
                         · 
                       
                       
                         · 
                         
                             
                         
                       
                     
                     ⁢ 
                     
                       
                         p 
                         
                           H 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           
                             O 
                             ⁡ 
                             
                               ( 
                               ad 
                               ) 
                             
                           
                         
                       
                       = 
                       
                         constant 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             atm 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   A6 
                   ) 
                 
               
             
             
               
                 
                   EMF 
                   ∝ 
                   
                     
                       RT 
                       
                         2 
                         ⁢ 
                         F 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             2 
                             3 
                           
                           ⁢ 
                           ln 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             p 
                             
                               NH 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               3 
                             
                             * 
                           
                         
                         - 
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                           ln 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             p 
                             
                               O 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                             * 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   A7 
                   ) 
                 
               
             
           
         
       
     
     Let us then consider a mixed potential equation from a macroscopic point of view. When the total pressure of the target gas is 1 atm, the concentration is equal to the partial pressure; thus, p NH3 , p O2 , and p H2O  will be explained below as partial pressures. Formula (A8) can be derived from formula (A3). Formula (A9) can be derived from formula (A6). From formulae (A8) and (A9), formula (A10) holds. Letting the ratio of ln p H2O * to ln p H2O  be a pressure adjustment factor δ, δ is defined by formula (A11). From formula (A10), δ satisfies −1&lt;δ&lt;1. Similarly, letting the ratio of ln p NH3 * to ln p NH3  be a pressure adjustment factor δ′, δ′ is defined by formula (A12). The pressure adjustment factors δ and δ′ are values characteristic of the sensor element, depending on, for example, the composition and the structure of the detection electrode.
 
−4.6&lt;ln  p   H2O &lt;−1.6  (A8)
 
ln  p   H2O *≈0  (A9)
 
|ln  p   H2O *|&lt;|ln  p   H2O |  (A10)
 
δ=ln  p   H2O */ln  p   H2O   (A11)
 
δ′=ln  p   NH3 */ln  p   NH3   (A12)
 
     Formula (A1) is transformed using the pressure adjustment factors δ and δ′ to derive formula (A13). Formula (A13) is obtained by substituting ln p H2O *=δ×ln p H2O  obtained from formula (A11), ln p NH3 *=δ×ln p NH3  obtained from formula (A12), and ln p O2 *=ln p O2  in formula (A1). In existing O 2  sensors and SOFCs, it is well known that the relationship between the oxygen concentration and the electromotive force obeys the Nernst equation; hence, it is clear that ln p O2 *=ln p O2  holds. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     7 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   EMF 
                   ∝ 
                   
                     
                       RT 
                       
                         2 
                         ⁢ 
                         F 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             2 
                             3 
                           
                           ⁢ 
                           
                             δ 
                             ′ 
                           
                           ⁢ 
                           ln 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             p 
                             
                               NH 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               3 
                             
                           
                         
                         - 
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                           ln 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             p 
                             
                               O 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                         
                         - 
                         
                           δln 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             p 
                             
                               H 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               O 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   A13 
                   ) 
                 
               
             
           
         
       
     
     From formulae (A6) and (A11), ln p H2O *=δ×ln p H2O =0 holds. Thus, formula (A13) can be expressed as formula (A14). Formula (A14) can be expressed as formula (A15). The constants A and B are values characteristic of the sensor element, depending on, for example, the composition and the structure of the detection electrode. In formula (A15), letting the base of the logarithm be freely selected values a and b and letting the coefficients of the terms in the right side be constants α and β, formula (A16) described below is derived. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     8 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   EMF 
                   ∝ 
                   
                     
                       RT 
                       
                         2 
                         ⁢ 
                         F 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             2 
                             3 
                           
                           ⁢ 
                           
                             δ 
                             ′ 
                           
                           ⁢ 
                           ln 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             p 
                             
                               NH 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               3 
                             
                           
                         
                         - 
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                           ln 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             p 
                             
                               O 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   A14 
                   ) 
                 
               
             
             
               
                 
                   EMF 
                   ∝ 
                   
                     
                       A 
                       ⁢ 
                       
                         RT 
                         
                           2 
                           ⁢ 
                           F 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               2 
                               3 
                             
                             ⁢ 
                             
                               δ 
                               ′ 
                             
                             ⁢ 
                             ln 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               p 
                               
                                 NH 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 3 
                               
                             
                           
                           - 
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             ln 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               p 
                               
                                 O 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       B 
                       ⁢ 
                       
                         
 
                       
                       ( 
                       
                         where 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         and 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         B 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         are 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         constants 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   A15 
                   ) 
                 
               
             
             
               
                 
                   EMF 
                   = 
                   
                     
                       α 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           log 
                           a 
                         
                         ⁡ 
                         
                           ( 
                           
                             p 
                             
                               NH 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               3 
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       β 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           log 
                           b 
                         
                         ⁡ 
                         
                           ( 
                           
                             p 
                             02 
                           
                           ) 
                         
                       
                     
                     + 
                     B 
                   
                 
               
               
                 
                   ( 
                   A16 
                   ) 
                 
               
             
           
         
       
     
     Unlike formula (2), formula (A16) can express the fact that the relationship, p NH3  sensitivity:p O2  sensitivity=(⅔):(½), does not always hold and that substantially no H 2 O interference is present. Thus, the use of formula (A16) can derive the ammonia concentration p NH3  with high accuracy, compared with formula (2). 
     The inventors have conducted further studies and have found the following: According to formula (A16), the relationship between the ammonia concentration p NH3  and the electromotive force EMF will be a linear one with a slope of the constant α, and the constant α will not vary as the oxygen concentration p O2  is changed; however, in fact, the constant α varies depending on the oxygen concentration p O2 . For example, a higher oxygen concentration p O2  had a tendency to lead to a steeper slope of a straight line expressing the relationship between the ammonia concentration p NH3  and the electromotive force EMF, in some cases. Furthermore, it was also found that the relationship between the logarithm of the oxygen concentration p O2  and the slope of the straight line expressing the relationship between the ammonia concentration p NH3  and the electromotive force EMF is one expressed by a straight line (linear function). Thus, the relationship was reflected in formula (A16) to derive formula (1a), and formula (1a) was transformed to derive formula (1). Formula (1) corresponds to a formula obtained by the addition of the term “γ log c (p NH3 )×log d (p O2 )” to formula (A16). This term is affected by both of the ammonia concentration p NH3  and the oxygen concentration p O2  and is thus referred to as an “interaction term”. The use of formula (1) can derive the ammonia concentration p NH3  with higher accuracy than that in the case of using formula (A16). 
     An exact reason the interaction term affects the electromotive force EMF is not clear. However, the interaction term seemingly expresses the occurrence of the gas-phase combustion of ammonia and oxygen before the target gas reaches the triple phase boundary. For example, it is believed that because each of the porous protective layer  48  and the detection electrode  51  is a porous material, when the target gas passes through pores of at least one of them, ammonia molecules collide with oxygen molecules to burn. A larger amount of ammonia burned reduces larger amounts of ammonia and oxygen before ammonia and oxygen in the target gas reach the triple phase boundary; thus, ammonia and oxygen in the target gas seemingly have less effect on the electromotive force EMF. The interaction term seems to express an increase or a decrease in electromotive force EMF depending on the amount burned. 
     The constant γ in the interaction term is a value characteristic of the sensor element, depending on, for example, structures of pores (for example, the porosity and the pore size) of each of the porous protective layer and the detection electrode, and the presence or absence of the porous protective layer. In the case where the interaction term correlates with the combustion of ammonia, the constant γ of the interaction term can vary depending on the operating temperature of the mixed potential cell. For example, a higher operating temperature can result in a larger absolute value of the constant γ. 
     Formulae (A16) and (1) are not limited to the case where the target gas has a total pressure of 1 atm, and can also be applied to the case where the total pressure is about 1 atm (for example, 0.9 atm to 1.10 atm). Formulae (A16) and (1) can also be applied to the case where the total pressure of the target gas is not about 1 atm. The temperature of the target gas to which formula (1) is applied may be, but is not particularly limited to, 150° C. or higher or 200° C. or higher. The temperature of the target gas may be 400° C. or lower. 
     The constants α, β, γ, and B in formula (1) can be determined by experiments as described below, in advance.  FIG. 4  is a flow chart illustrating an example of constant derivation processing. In the constant derivation processing, the sensor element  31 , which is a target with the constants to be derived, is subjected to first electromotive force measurement processing for acquiring first electromotive force data multiple times, the first electromotive force data expressing the correspondence between the ammonia concentration p NH3  and the electromotive force EMF (step S 200 ). Specifically, the correspondence between the ammonia concentration p NH3  and the electromotive force EMF is acquired as the first electromotive force data by exposing the sensor element  31  to the target gas with the oxygen concentration p O2  and the ammonia concentration p NH3  that have been adjusted to predetermined values and measuring the electromotive force EMF. Next, first electromotive force data sets are similarly acquired by measuring the electromotive force EMF multiple times at different ammonia concentrations p NH3  in the target gas while the oxygen concentration p O2  in the target gas remains unchanged (constant). Subsequently, as with the first electromotive force measurement processing, electromotive force EMF measurement processing that is executed multiple times at different ammonia concentrations p NH3  while the oxygen concentration p O2  remains unchanged (constant) is executed multiple times at different oxygen concentrations p O2  (second to nth electromotive force measurement processing) to acquire the second to the nth electromotive force data sets (step S 210 ). n represents an integer of 2 or more. For example, only the second electromotive force measurement processing may be executed in step S 210 . 
     Next, the slope K (here, K1) and the intercept L (here, L1) of formula (3) described below are derived from the first electromotive force data sets acquired in step S 200  (step S 220 ). In the case where formula (1a) described above is regarded as a linear function of the electromotive force EMF and the logarithm of the ammonia concentration p NH3 , i.e., log a (p NH3 ), the slope K and the intercept L are defined by formulae (3) and (4), respectively, and formula (1a) can thus be expressed by formula (5). As is clear from formulae (3) and (4), each of the slope K and the intercept L is constant at a constant oxygen concentration p O2 ; hence, on the basis of the first electromotive force data sets (data sets at a constant oxygen concentration p O2 ), the slope K (K1) and the intercept L (L1) corresponding to the oxygen concentration p O2  at the time of the measurement of the data sets can be derived. Specifically, the slope and the intercept obtained when the relationship between the logarithm of the ammonia concentration p NH3 , log a (p NH3 ), and the electromotive force EMF in the first electromotive force data sets acquired by executing the first electromotive force measurement processing multiple times is approximated by a straight line (linear function) are derived as the slope K1 and the intercept L1. The approximation is performed on the basis of, for example, the method of least squares. Subsequently, as with step S 220 , the slopes K2 to Kn and the intercepts L2 to Ln are derived (step S 230 ) in the same way as the slope K1 and the intercept L1 on the basis of the second to the nth electromotive force data sets acquired in step S 210 .
 
K=α+γ′ log d ( p   O2 )  (3)
 
L=−β log b ( p   O2 )+B  (4)
 
EMF=K×log a ( p   NH3 )+L  (5)
 
     Subsequently, the constants α and γ′ are derived from formula (3) described above (step S 240 ) on the basis of the correspondences between the slopes K (K1 to Kn) derived in steps S 220  and S 230  and the oxygen concentrations p O2  at the time of the derivation of the slopes K1 to Kn (for example, in the cases of the slope K1, the oxygen concentration p O2  at the time of the first electromotive force data measurement). As is clear from formula (3), the slope K is a linear function of the logarithm of the oxygen concentration p O2 , i.e., log d (p O2 ). When the relationship between the log d (p O2 ) and the slope K is approximated by a straight line (linear function), the slope is derived as the constant γ′, and the intercept is derived as the constant α. By deriving the constant γ′, the constant γ can be derived (see the proviso of formula (1a)). When base a=base c, constant γ′=constant γ. 
     As with the step S 240 , the constants β and B are derived from formula (4) on the basis of the correspondences between the intercepts L (L1 to Ln) derived in steps S 220  and S 230  and the oxygen concentrations p O2  at the time of the derivation of the intercepts L1 to Ln (for example, in the cases of the intercept L1, the oxygen concentration p O2  at the time of the first electromotive force data measurement) (step S 250 ). As is clear from formula (4), the intercept L is a linear function of the logarithm of the oxygen concentration p O2 , i.e., log b (p O2 ). When the relationship between the log b (p O2 ) and the intercept L is approximated by a straight line (linear function), the slope is derived as the constant β, the intercept is derived as the constant B. Thereby, the constants α, β, γ, and B are derived, and this processing is completed. 
     Each of the first to the nth electromotive force data sets described above is measured in a state in which the mixed potential cell  55  is heated with the heater  62  to a predetermined fixed operating temperature. Comparisons between formula (1) and formula (A15) reveal that the constants α and β vary depending on the temperature T of the mixed potential cell  55 , i.e., the operating temperature of the sensor element  31  in use. Thus, in the case where one sensor element  31  can be used at different operating temperatures, the constants α and β in formula (1) are derived at each of the different operating temperatures and stored in, for example, the memory part  73 , in advance. When the control section  72  executes the ammonia concentration derivation processing, the constants α and β corresponding to the operating temperature of the sensor element  31  are used. The constant γ can vary depending on the operating temperature of the mixed potential cell  55  as described above; thus, a value corresponding to the operating temperature of the mixed potential cell  55  is derived and stored in, for example, the memory part  73 , in advance. The constant B can also vary depending on the operating temperature of the sensor element  31  in use; thus, the constant B may be derived at each of the different operating temperatures and stored in, for example, the memory part  73 , in advance. 
     A method other than the foregoing constant derivation processing may be employed as long as the constants α, β, γ, and B can be derived. For example, in step S 200 , the electromotive force EMF may be measured multiple times at different oxygen concentrations p O2  in the target gas while the ammonia concentration p NH3  is constant (the same is true for step S 210 ). The constant derivation processing is not limited to the case where the electromotive force EMF is measured multiple times while one of the ammonia concentration p NH3  and the oxygen concentration p O2  is constant and the other is changed. The electromotive forces EMFs may be measured with at least four target gases in which at least one of the ammonia concentration p NH3  and the oxygen concentration p O2  is different when the sensor element  31  is exposed to each of the target gases. If at least four correspondences (electromotive force data sets) among the electromotive forces EMFs, the oxygen concentrations p O2 , and the ammonia concentrations p NH3  are thus acquired, simultaneous equations are solved by replacing variables in formula (1) by the values of the electromotive force data sets, so that the constants α, β, γ, and B can be derived. Like the foregoing constant derivation processing, however, the constants are preferably derived from the electromotive force data sets as many as possible. 
     Let us now clarify the correspondence between the constituent elements of this embodiment and constituent elements of the present invention. The solid electrolyte layer  44  of this embodiment corresponds to a solid electrolyte body of the present invention. The detection electrode  51  corresponds to a detection electrode. The reference electrode  53  corresponds to a reference electrode. The mixed potential cell  55  corresponds to a mixed potential cell. The electromotive force acquisition section  75  corresponds to a electromotive force acquisition section. The oxygen concentration acquisition section  76  corresponds to an oxygen concentration acquisition section. The control section  72  corresponds to an ammonia concentration derivation section. In this embodiment, an example of a method for measuring ammonia concentration of the present invention is also described by explaining the operation of the apparatus  70  for measuring ammonia concentration. 
     According to the system  2  for treating an exhaust gas described above in detail, in the apparatus  70  for measuring ammonia concentration, the use of the relationship of formula (1) can derive the ammonia concentration in the target gas with higher accuracy than that in the case of using formula (2) described above. 
     The target gas may have a temperature of 150° C. or higher. The derivation of the ammonia concentration p NH3  using the relationship of formula (1) is suitable even for a high-temperature target gas. 
     Because the detection electrode  51  is composed of the Au—Pt alloy as a main component, the mixed potential is easily established at the triple phase boundary of the solid electrolyte layer  44  and the target gas. The detection electrode  51  has a degree of concentration of 0.3 or more, which is measured by at least one of XPS and AES, and thus enables the mixed potential to be more reliably established. 
     Because the operating temperature of the mixed potential cell  55  is 450° C. or higher, the solid electrolyte layer  44  can be appropriately activated. Because the operating temperature of the mixed potential cell  55  is 650° C. or lower, a decrease in measurement accuracy due to the combustion of ammonia can be inhibited. 
     The system  2  for treating an exhaust gas includes the one or more oxidation catalysts (DOC  4  and ASC  8 ) arranged in the exhaust gas path  3 , and the sensor element  31  is arranged on the downstream side of the exhaust gas path  3  in contrast to the DOC  4 , which is one of the one or more oxidation catalysts, arranged at the upstream end. Thus, the target gas in which a component (for example, at least one of hydrocarbons and carbon monoxide) that is present in the target gas and that affects the measurement accuracy of the ammonia concentration has been oxidized by the oxidation catalysts reaches the sensor element  31 . Accordingly, in the system  2  for treating an exhaust gas, the ammonia concentration in the target gas can be derived with higher accuracy. 
     The present invention is not limited to the above-described embodiment, and can be carried out by various modes as long as they belong to the technical scope of the invention. 
     For example, in the foregoing embodiment, although the detection electrode  51  and the reference electrode  53  are arranged on the solid electrolyte layer  44 , the solid electrolyte layer  44  is not necessarily used, and they may be arranged on a solid electrolyte body. For example, the detection electrode  51  and the reference electrode  53  may be arranged on upper and lower surfaces of a solid electrolyte body including solid electrolyte layers stacked. In the foregoing embodiment, although the reference electrode  53  serves as both of the reference electrode of the mixed potential cell  55  and the reference electrode of the concentration cell  56 , this structure is not necessarily used, and the mixed potential cell  55  and the concentration cell  56  may include different reference electrodes. 
     In the foregoing embodiment, although the sensor element  31  includes the concentration cell  56  and thus can measure the oxygen concentration, this structure is not necessarily used. The sensor element  31  may not include the concentration cell  56  (specifically, the auxiliary electrode  52 ). In this case, the apparatus  70  for measuring ammonia concentration may acquire information about the oxygen concentration from other than the sensor element  31 . For example, the apparatus  70  for measuring ammonia concentration may acquire information about the oxygen concentration from another sensor that is arranged in the exhaust gas path  3  and that can detect information about the oxygen concentration (for example, an oxygen sensor, an A/F sensor, or a NOx sensor). The apparatus  70  for measuring ammonia concentration may acquire information about the oxygen concentration from another device (such as the engine ECU  9 ) other than sensors. In the case where the apparatus  70  for measuring ammonia concentration acquires information about the oxygen concentration from another sensor arranged at a position of the exhaust gas path  3 , the position being different from that of the sensor element  31 , the apparatus  70  for measuring ammonia concentration preferably derives the ammonia concentration in consideration of a measurement time lag (time lag C) the difference in position between the sensor element  31  and due to the another sensor attached. Specifically, letting the length of time that the target gas flow from the position of one, located upstream, of the sensor element  31  and the another sensor to the position of the other in the exhaust gas path  3  be the time lag C, the apparatus  70  for measuring ammonia concentration preferably derives the ammonia concentration in consideration of the time lag C. For example, in the case where the another sensor is located on the upstream side of the sensor element  31 , the control section  72  allows the memory part  73  to store the values of oxygen concentration acquired from the another sensor every predetermined period during the time lag C. Every time the electromotive force EMF is acquired from the sensor element  31 , the control section  72  reads the oldest value of oxygen concentration at that time (=value acquired in the past by the time lag C) from the memory part  73  and derives the ammonia concentration from the acquired electromotive force EMF, the value of the oxygen concentration read, and formula (1). In this way, the apparatus  70  for measuring ammonia concentration can derive the ammonia concentration with higher accuracy by considering the time lag C. 
     Although the engine  1  is a diesel engine in the foregoing embodiment, a gasoline engine may be used. 
     In the foregoing embodiment, although the apparatus  70  for measuring ammonia concentration is an apparatus different from the engine ECU  9 , the apparatus  70  for measuring ammonia concentration may be part of the engine ECU  9 . 
     EXAMPLES 
     Examples in which a method for measuring ammonia concentration was specifically performed will be described below as Examples. The present invention is not limited to Examples described below. 
     Production of Sensor Element  1   
     A sensor element to be used for the measurement of ammonia concentration with an apparatus for measuring ammonia concentration was produced. Four ceramic green sheets containing a ceramic component composed of a zirconia solid electrolyte containing 3% by mole yttria serving as a stabilizer were prepared as the layers of the base  40 . For example, sheet holes used for positioning during printing and stacking and through-holes required were formed in the green sheets. A space to be formed into the reference gas introduction cavity  46  was formed in the green sheet to be formed into the spacer layer  43  by, for example, punching, in advance. Various patterns were formed by pattern printing on the ceramic green sheets corresponding to the first substrate layer  41 , the second substrate layer  42 , the spacer layer  43 , and the solid electrolyte layer  44 , and the resulting ceramic green sheets were subjected to drying treatment. Specifically, for example, patterns for the detection electrode  51  composed of the Au—Pt alloy, the auxiliary electrode  52  and the reference electrode  53  composed of Pt, lead wires, and the heater portion  60  were formed. The pattern printing was performed by applying pattern-forming pastes to the green sheets using a known screen printing technique, each of the pattern-forming pastes being prepared to provide characteristics required for a corresponding one of the target objects. After the pattern printing and the drying were completed, printing and drying treatment of a bonding paste to stack and bond the green sheets corresponding to the layers together were performed. Compression bonding treatment was performed in which the green sheets including the bonding paste were stacked in a predetermined order while the green sheets were positioned with the sheet holes, and the resulting stack were subjected to compression bonding under predetermined temperature and pressure conditions to form a laminate. Laminated pieces having the same size as the sensor element  31  were cut from the resulting laminate. The cut laminated pieces were fired with a tubular furnace at 1,100° C. for 2 hours in an air atmosphere, thereby providing the sensor elements  31  each including the detection electrode  51 , the auxiliary electrode  52 , and the reference electrode  53  that were arranged on the solid electrolyte layer  44 . The sensor elements  31  were subjected to dipping with an alumina-containing slurry and firing to form the porous protective layers  48  on surfaces of the sensor elements  31 . In this way, the sensor element  31  was produced and was referred to as a sensor element  1 . The degree of concentration on the surface of a noble metal on the fracture surface of the detection electrode  51  in the sensor element  1  was measured by AES and found to be 0.99. The detection electrode  51  had a porosity of 45% by volume. The porous protective layer  48  had a porosity of 40% by volume. In the following tests, the operating temperature of the sensor element  1  in use was 480° C. 
     Experiment 1: Acquisition of Electromotive Force Data Sets 
     The sensor element  1  was subjected to steps S 200  and S 210  in the constant derivation processing to acquire first to sixth electromotive force data sets (13 for each data set). The first electromotive force data was acquired by measuring the electromotive forces EMFs [mV] at a fixed oxygen concentration p O2  of 1%, a fixed H 2 O concentration p H2O  of 5%, and different ammonia concentrations p NH3 , as listed in Table 1, in a target gas. A component (base gas) other than the foregoing components in the target gas was nitrogen, and the temperature was 200° C. The target gas was allowed to flow through the pipe having a diameter of 70 mm at a flow rate of 200 L/min. The second to the sixth electromotive force data sets were measured as in the first electromotive force data, except that the oxygen concentration p O2  (fixed value) in the target gas was changed as listed in Table 1. Table 1 lists the ammonia concentrations p NH3 , the oxygen concentrations p O2 , and the electromotive forces EMFs of the first to the sixth electromotive force data sets measured.  FIG. 5  is a graph depicting the relationship between the ammonia concentration p NH3  [ppm] and the electromotive force EMF [mV] of the sensor element  1  (first to sixth electromotive force data sets). The horizontal axis of  FIG. 5  is on a logarithmic scale.  FIG. 5  indicates that the relationships between the logarithms of the ammonia concentrations p NH3  at fixed oxygen concentrations p O2  and the electromotive forces EMFs can be approximated by a straight line. The results indicated that higher oxygen concentrations p O2  resulted in higher slopes K of the straight lines and that higher oxygen concentrations p O2  resulted in lower intercepts L of the straight lines. The results also indicated that the ratio of the NH 3  sensitivity to the O 2  interference in the measured values of the electromotive force EMF was not always (⅔):(½). These results indicated that the relationship among the electromotive force EMF, the ammonia concentration p NH3 , and the oxygen concentration p O2  was not matched to formula (2). 
     
       
         
           
               
               
               
               
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                   
                 First 
                 Second 
                 Third 
                 Fourth 
                 Fifth 
                 Sixth  
               
               
                   
                 electromotive 
                 electromotive 
                 electromotive 
                 electromotive 
                 electromotive 
                 electromotive 
               
               
                   
                 force data 
                 force data 
                 force data 
                 force data 
                 force data 
                 force data 
               
               
                   
                 p O2  = 1% 
                 p O2  = 3% 
                 p O2  = 5% 
                 p O2  = 10% 
                 p O2  = 15% 
                 p O2  = 20% 
               
               
                 p NH3 [ppm] 
                 EMF[mV] 
                 EMF[mV] 
                 EMF[mV] 
                 EMF[mV] 
                 EMF[mV] 
                 EMF[mV] 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 25 
                 241.850 
                 210.583 
                 195.600 
                 168.633 
                 152.050 
                 133.750 
               
               
                 50 
                 266.217 
                 238.567 
                 224.333 
                 204.550 
                 187.783 
                 177.417 
               
               
                 100 
                 290.133 
                 265.967 
                 262.200 
                 238.650 
                 225.700 
                 216.350 
               
               
                 300 
                 330.183 
                 312.467 
                 304.617 
                 298.050 
                 290.250 
                 284.333 
               
               
                 500 
                 359.783 
                 346.433 
                 341.917 
                 341.667 
                 336.800 
                 332.200 
               
               
                 750 
                 374.767 
                 362.800 
                 360.567 
                 362.733 
                 358.917 
                 355.583 
               
               
                 1000 
                 373.600 
                 371.967 
                 373.867 
                 376.900 
                 369.883 
                 369.600 
               
               
                 1 
                 142.917 
                 90.817 
                 56.233 
                 21.467 
                 9.833 
                 1.667 
               
               
                 3 
                 182.983 
                 143.367 
                 119.083 
                 84.017 
                 55.233 
                 31.650 
               
               
                 5 
                 196.833 
                 159.933 
                 138.467 
                 111.200 
                 85.883 
                 66.933 
               
               
                 10 
                 216.133 
                 182.033 
                 162.683 
                 140.300 
                 118.900 
                 103.367 
               
               
                 25 
                 249.083 
                 217.117 
                 200.700 
                 182.100 
                 163.600 
                 151.983 
               
               
                 50 
                 268.500 
                 238.500 
                 222.883 
                 206.400 
                 188.733 
                 178.200 
               
               
                   
               
            
           
         
       
     
     Experiment 2: Derivation of Slope K and Intercept L 
     Subsequently, steps S 220  and S 230  in the constant derivation processing were executed to derive the slopes K (K1 to K6) and the intercepts L (L1 to L6) in the first to the sixth electromotive force data sets. For example, with regard to the approximate straight line at an oxygen concentration p O2  of 1% (first electromotive force data) illustrated in  FIG. 5 , the slope was defined as K1 (=34.67), and the intercept was defined as L1 (=137.92). The intercept L1 is a value of the electromotive force EMF when log a (p NH3 ) of the approximate straight line is zero, in other words, when p NH3  is 1 ppm. Table 2 lists the derived slopes K, the derived intercepts L, and the oxygen concentrations p O2  corresponding thereto.  FIG. 6  is a graph depicting the relationship between the oxygen concentration p O2  and the slope K listed in Table 2.  FIG. 7  is a graph depicting the relationship between the oxygen concentration p O2  and the intercept L listed in Table 2. The horizontal axis of each of  FIGS. 6 and 7  is on a logarithmic scale.  FIG. 6  indicates that the relationship between the logarithm of the oxygen concentration p O2  and the slope K can be approximated by a straight line.  FIG. 7  indicates that the relationship between the logarithm of the oxygen concentration p O2  and the intercept L can be approximated by a straight line. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                   
                 p O2 [%] 
                 Slope K 
                 Intercept L 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 First electromotive 
                 1 
                 34.670 
                 137.920 
               
               
                   
                 force data 
                   
                   
                   
               
               
                   
                 Second electromotive 
                 3 
                 40.086 
                 89.489 
               
               
                   
                 force data 
                   
                   
                   
               
               
                   
                 Third electromotive 
                 5 
                 44.462 
                 58.546 
               
               
                   
                 force data 
                   
                   
                   
               
               
                   
                 Forth electromotive 
                 10 
                 50.266 
                 19.795 
               
               
                   
                 force data 
                   
                   
                   
               
               
                   
                 Fifth electromotive 
                 15 
                 52.981 
                 −5.359 
               
               
                   
                 force data 
                   
                   
                   
               
               
                   
                 Sixth electromotive 
                 20 
                 55.316 
                 −25.029 
               
               
                   
                 force data 
               
               
                   
                   
               
            
           
         
       
     
     Experiment 3: Derivation of Constants α, β, γ, and B 
     Steps S 240  and S 250  in the constant derivation processing were executed on the basis of the data sets obtained in experiments 1 and 2 to derive the constants α, β, γ, and B of the sensor element  1 . Specifically, formula (6) described below was derived as an approximate straight line expressing the relationship between the logarithm of the oxygen concentration p O2  and the slope K illustrated in  FIG. 6 . The constant γ′=7.09 and the constant α=33.636 were derived from formula (6). Formula (7) described below was derived as an approximate straight line expressing the relationship between the logarithm of the oxygen concentration p O2  and the intercept L illustrated in  FIG. 7 . The constant β=54.69 and the constant B=143.55 were derived from formula (7). In a formula derived here, each of the bases a to d in formula (1) was Napier&#39;s constant e. Thus, γ=γ′=7.09 was derived.
 
K=7.09×ln( p   O2 )+33.636  (6)
 
L=−54.69×ln( p   O2 )+143.55  (7)
 
     From experiments 1 to 3 described above, formula (8) expressing the relationship among the variables (EMF, p NH3 , and p O2 ) in the sensor element  1  was derived. In formula (8), the units of the electromotive force EMF are [mV], the units of the ammonia concentration p NH3  are [ppm], and the units of the oxygen concentration p O2  are [%].
 
EMF=33.636×ln( p   NH3 )−54.69×ln( p   O2 )+7.09×ln( p   NH3 )×ln( p   O2 )+143.55  (8)
 
     Verification Test 
     The electromotive forces EMFs corresponding to ammonia concentrations p NH3  and oxygen concentrations p O2  were derived from formula (8) under the same conditions as those in the first to the sixth electromotive force data sets. Table 3 lists the results.  FIG. 8  is a graph illustrating six straight lines represented by a formula (formula corresponding to formula (5)) derived from formula (8) at the different oxygen concentrations p O2  used in the first to the sixth electromotive force data sets.  FIG. 8  also illustrates the points of the first to the sixth electromotive force data sets (measured values) illustrated in  FIG. 5 .  FIG. 8  and comparisons between Tables 1 and 3 indicated that the measured values of the electromotive forces EMFs were matched to the electromotive forces EMFs derived from formula (8) with good accuracy. That is, although the relationship among the electromotive force EMF, the ammonia concentration p NH3 , and the oxygen concentration p O2  based on actual measurement was not expressed by formula (2) as illustrated in  FIG. 5 , the relationship based on the measured values was able to be expressed by formula (8) derived from formula (1). These results indicated that the use of formula (1) was able to derive the ammonia concentration p NH3  with higher accuracy than that in the case of using formula (2). 
     
       
         
           
               
               
               
               
               
               
               
             
               
                 TABLE 3 
               
               
                   
               
               
                   
                 p O2  = 1% 
                 p O2  = 3% 
                 p O2  = 5% 
                 p O2  = 10% 
                 p O2  = 15% 
                 p O2  = 20% 
               
               
                 p NH3 [ppm] 
                 EMF[mV] 
                 EMF[mV] 
                 EMF[mV] 
                 EMF[mV] 
                 EMF[mV] 
                 EMF[mV] 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 1 
                 143.550 
                 83.467 
                 55.530 
                 17.622 
                 −4.553 
                 −20.287 
               
               
                 3 
                 180.503 
                 128.977 
                 105.019 
                 72.510 
                 53.493 
                 40.001 
               
               
                 5 
                 197.685 
                 150.138 
                 128.030 
                 98.031 
                 80.483 
                 68.033 
               
               
                 10 
                 221.000 
                 178.852 
                 159.254 
                 132.662 
                 117.106 
                 106.069 
               
               
                 25 
                 251.820 
                 216.809 
                 200.530 
                 178.441 
                 165.520 
                 156.352 
               
               
                 50 
                 275.135 
                 245.523 
                 231.754 
                 213.071 
                 202.143 
                 194.389 
               
               
                 100 
                 298.450 
                 274.237 
                 262.979 
                 247.702 
                 238.766 
                 232.426 
               
               
                 300 
                 335.402 
                 319.747 
                 312.468 
                 302.590 
                 296.812 
                 292.713 
               
               
                 500 
                 352.585 
                 340.908 
                 335.479 
                 328.112 
                 323.802 
                 320.745 
               
               
                 750 
                 366.223 
                 357.704 
                 353.744 
                 348.369 
                 345.225 
                 342.995 
               
               
                 1000 
                 375.899 
                 369.622 
                 366.703 
                 362.742 
                 360.425 
                 358.782 
               
               
                   
               
            
           
         
       
     
     The electromotive forces EMFs of the sensor element  1  were measured with target gases having a constant ammonia concentration p NH3  of 100 ppm, a constant oxygen concentration p O2  of 10%, and different H 2 O concentrations p H2O  of 1% to 12% as listed in Table 4. Conditions other than those described above were the same as in experiment 1.  FIG. 9  is a graph depicting the relationship between the H 2 O concentration p H2O  [%} and the electromotive force EMF [mV] of the sensor element  1 .  FIG. 9  indicated that the electromotive force EMF remains almost unchanged at different H 2 O concentrations p H2O  in the target gases (substantially no H 2 O interference). That is, the results indicated that the term of the H 2 O concentration p H2O  in formula (2) was not matched to the relationship between the electromotive force EMF and the H 2 O concentration p H2O  actually measured. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 4 
               
               
                   
                   
               
               
                   
                 p H2O [%] 
                 EMF[mV] 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                   
                 1 
                 238.65 
               
               
                   
                 3 
                 238.65 
               
               
                   
                 5 
                 238.65 
               
               
                   
                 10 
                 238.65 
               
               
                   
                 12 
                 235.65 
               
               
                   
                   
               
            
           
         
       
     
     The present application claims priority from U.S. provisional Patent Application No. 62/411,736 filed on Oct. 24, 2016 and Japanese Patent Application No. 2017-117089 filed on Jun. 14, 2017, the entire contents of which are incorporated herein by reference.