Method and apparatus for measuring physical adsorption of gases based on dielectric measurements

A method and apparatus for measuring physical adsorption of a gas by a solid adsorbent are disclosed. Use is made of a capacitance cell in controlled gas flow communication with a sample cell initially under vacuum and containing the solid adsorbent. The capacitance cell is filled with an adsorptive gas at a predetermined pressure and the dielectric constant of the gas at the predetermined pressure is measured to determine a first density value. The gas from the capacitance cell is then allowed to expand into the sample cell for adsorption by the solid adsorbent, whereby an adsorption takes place pressure falls until eqilibrium is established. The dielectric constant of the gas at equilibrium pressure is measured to determine a second density value, and the amount of gas adsorbed by the solid adsorbent is determined from the first and second density values.

The present invention relates to gas adsorption measurements. More 
particularly, the invention is concerned with a method and apparatus for 
measuring physical adsorption of a gas by a solid adsorbent. 
Physical adsorption of a gas by a solid is the condition in which the 
concentration of the gas molecules at the gas/solid interface is greater 
than the bulk concentration. This enrichment is caused by the van der 
Waals interactions at the gas/solid interface. The nature of the solid 
plays an important role in adsorption phenomena. In general, the atoms in 
the solid are distributed in a periodic crystalline structure, and the 
total force exerted by the solid atoms on a gaseous molecule depends very 
much on the position of the gas molecule on the solid surface. Certain 
sites in the solid surface are more favorable to adsorption than others. 
For a given mass of solid, the adsorption goes up with the amount of 
surface available. Maximization of surface for a given mass may be 
realized either by breaking up the solid into fine particles or, more 
realistically, by producing an extensive network of fine pores in the 
solid. The adsorption properties of a porous solid depend on the size of 
its pores. The pore sizes classified as micropores (&lt;2 nm) are mostly 
responsible for pore filling adsorption. The bigger pore sizes like 
mesopores (between 2 and 50 nm) and macropores (&gt;50 nm) are responsible 
for monolayer/multilayer adsorption and capillary condensation. 
The adsorption studies at high pressures and high temperatures have 
important industrial applications in the fields of separation and 
purification of hydrogen, light hydrocarbons and several other gases, 
storage of fuel gases in microporous solids, catalytic reactions, and 
chromatography. The measurement of adsorption at the gas/solid interface 
also serves as an essential tool in the study of solid surfaces and the 
characterization of microporous materials. 
Although numerous instruments exist to measure gas adsorption at low 
pressures, those capable of operating at high pressures are rare and 
necessitate specialized construction. The high-pressure adsorption 
measurements have been essentially based on standard volumetric and 
gravimetric principles. 
In the volumetric method, the adsorbate gas of known pressure and volume is 
allowed to expand into a cell containing the sample. If the sample dead 
space is known, the amount of adsorption can be found by the application 
of the gas laws. This method is reliable at low pressure where all gases 
in the bulk phase closely resemble the ideal gas. At high pressure, 
however, where the gas phase deviates significantly from ideality, the 
reliability of the volumetric method is very much dependent upon the 
availability and the accuracy of the compressibility data for the gas of 
interest. This is due to the fact that the adsorbed amount is calculated 
from the density of the gas which in turn is derived from the 
experimentally measured pressure and the compressibility data for the gas. 
As an example, a 3% deviation from ideality can modify the amount of 
adsorption increments by 50% to 100% because the calculated amount is a 
small difference between two large numbers. 
In the gravimetric method, the amount of gas adsorbed is directly measured 
by the change in weight of the adsorbent using a microbalance. Depending 
on the type of balance used, the great uncertainty factor in gravimetric 
measurement of adsorption at high pressures arises from the buoyancy 
correction to be applied to the observed results. This correction could be 
as large as the weighted amounts. Moreover, the gravimetric method is 
limited to 15 MPa and it is especially convenient for measurements with 
vapors at temperatures not far removed from ambient. At both high and low 
temperatures, however, it becomes difficult to control and measure the 
exact temperature of the adsorbent. 
It is therefore an object of the present invention to overcome the above 
drawbacks and to provide a method and apparatus enabling precise 
measurement of gas adsorption at high pressures. 
In accordance with one aspect of the invention, there is provided a method 
of measuring physical adsorption of a gas by a solid adsorbent, wherein 
use is made of a capacitance cell in controlled gas flow communication 
with a sample cell initially under vacuum and containing the solid 
adsorbent. The method of the invention comprises the steps of: 
(a) filling the capacitance cell with an adsorptive gas at a predetermined 
pressure and measuring the dielectric constant of the gas at the 
predetermined pressure to determine a first density value; 
(b) allowing the gas from the capacitance cell to expand into the sample 
cell for adsorption by the solid adsorbent, whereby as adsorption takes 
place pressure falls until equilibrium is established; 
(c) measuring the dielectric constant of the gas at equilibrium pressure to 
determine a second density value; and 
(d) determining the amount of gas adsorbed by the solid adsorbent at the 
equilibrium pressure from the first and second density values. 
The present invention also provides, in another aspect thereof, an 
apparatus for measuring physical adsorption of a gas by a solid adsorbent, 
comprising: 
a capacitance cell having therein at least two electrode members arranged 
in opposite spaced-apart relation to one another, the electrode members 
being provided with connector means for connection to a capacitance 
bridge; 
a sample cell for containing the solid adsorbent; 
conduit means interconnecting the cells for allowing gas flow communication 
therebetween, the conduit means being provided with valve means for 
controlling the gas flow between the cells; 
vacuum means connected to the sample cell for evacuating same; and 
gas compressing means connected to the capacitance cell for filling same 
with an adsorptive gas at a predetermined pressure, whereby the adsorptive 
gas from the capacitance cell is allowed to expand into the sample cell 
for adsorption by the solid adsorbent and the dielectric constant of the 
gas is measured before and after expansion to determine the amount of gas 
adsorbed by the solid adsorbent. 
For a given gas/solid system maintained at constant temperature, the amount 
of gas adsorbed is a function of the adsorptive pressure only. The 
relation between the quantity thus adsorbed and the equilibrium pressure 
is called the adsorption isotherm. 
The method according to the invention is based on dielectric constant 
measurements and utilizes a capacitance cell of calibrated volume V.sub.c, 
connected by a valve to a sample cell of volume V.sub.s and containing the 
solid adsorbent. Initially the capacitance cell is filled with the 
adsorptive gas. Its density d.sub.1 is determined from the measurement of 
the dielectric constant .epsilon..sub.1. The sample cell is under vacuum. 
Gas from the capacitance cell is then allowed to expand into the sample 
cell. As adsorption takes place, the pressure in the system falls until 
equilibrium is established. Measurement of the dielectric constant 
.epsilon..sub.2 gives the equilibrium density d.sub.2. The amount of gas 
molecules adsorbed, N.sub.ad, is given as the difference between the 
amount of gas molecules admitted, N, and the amount, N.sub.g, required to 
fill the dead space V.sub.g around the solid adsorbent. In terms of 
measured densities, N.sub.ad is given by: 
EQU N.sub.ad =(d.sub.1 -d.sub.2)V.sub.c -d.sub.2 V.sub.g ( 1) 
Calibration of the capacitance cell volume V.sub.c can be carried out with 
the volumetric expansion technique using a standard volume cylinder which 
is evacuated. In this procedure, helium gas in the capacitance cell at an 
initial pressure P.sub.1 of about 1 MPa is allowed to expand into 
evacuated cylinder. The new equilibrium pressure P.sub.2 is a fraction q 
of P.sub.1, with the value of q fixed by V.sub.c. 
Calibration of the dead space volume V.sub.g, on the other hand, is 
advantageously carried out in a prior experiment with the apparatus 
according to the invention, by the admission in the sample cell of a 
reference gas, usually helium, which is adsorbed to a negligible extent 
(N.sub.ad .perspectiveto.0). In this procedure, helium in the capacitance 
cell at an initial density d.sub.1 determined by dielectric constant 
measurement is allowed to expand into the vacuum of the sample cell 
containing the adsorbent. The new density d.sub.2 is a fraction u of 
d.sub.1. With helium adsorption being negligible, the value of u is fixed 
by V.sub.g. 
The adsorption isotherm can be constructed point by point by maintaining a 
balance of alternate fill-ups of the capacitance cell and expansions into 
the sample cell. 
The relation between the gas density and its dielectric constant is given 
by the Clausius-Mossotti function CM which, for dielectric constant 
.epsilon., may be developed in terms of molar density d by the series: 
EQU CM=(.epsilon.-1)[(.epsilon.+2)d].sup.-1 =A.sub..epsilon. +B.sub..epsilon. 
d+C.sub..epsilon. d.sup.2 ( 2) 
where A.sub..epsilon., B.sub..epsilon., and C.sub..epsilon. are, 
respectively, the first, the second, and the third dielectric virial 
coefficients. The first dielectric virial coefficient A.sub..epsilon. 
represents the contribution of individual molecules to CM, the second 
dielectric virial coefficient B.sub..epsilon. represents the contribution 
of pairs of molecules to CM, the third dielectric virial coefficient 
C.sub..epsilon. represents contributions due to triplets of molecules, and 
so on. Measurements of A.sub..epsilon. have long been used to derive 
accurate values of molecular polarizabilities and dipole moments, while 
measurements of the higher coefficients have yielded precise values of 
molecular multipole moments and provided information about intermolecular 
interactions. 
Depending on the pressure range of the experiment and the polarity of the 
measured gas, contributions to CM from B.sub..epsilon. and higher 
coefficients can be more or less significant. Keeping this in mind and the 
fact that determination of higher-order coefficients requires additional 
measurements, one can then derive separate approximations for polar and 
nonpolar gases. 
In the pressure range of adsorption measurements for most nonpolar gases, 
contributions to CM from B.sub..epsilon. and higher coefficients are 
negligible. Thus, for all practical purposes, Eq. (2) reduces to: 
EQU (.epsilon.-1)(.epsilon.+2).sup.-2 .apprxeq.A.sub..epsilon. d.sub.68, (3) 
from which the first-order density approximation is given by: 
EQU d.sup.(1) =(.epsilon.-1)/[(.epsilon.+2)A.sub..epsilon. ]. (4) 
If measurements of .epsilon. are done at known pressures P, the necessary 
value of A.sub..epsilon. can then be obtained easily by considering the 
expansion of the equation of state in the form: 
EQU P(RTd).sup.-1 =1+B.sub.P d+C.sub.P d.sup.2 + (5) 
where R is the universal gas constant, and B.sub.p, C.sub.p are the second 
and third pressure virial coefficients. Eliminating d between Eqs. (2) and 
(5), one obtains: 
EQU (.epsilon.-1)(.epsilon.+1).sup.-1 (RT/P) 
EQU =A.sub..epsilon. +(B.sub..epsilon. -A.sub..epsilon. B.sub.P)(P/RT)+(6) 
A plot of (.epsilon.-1)(.epsilon.+2)(RT/P) vs (P/RT) is a straight line 
with the intercept Ae and slope (B.sub..epsilon.-A.sub..epsilon. B.sub.p). 
As will be seen later, values of A.sub..epsilon. determined by this method 
can be reliable to a few parts in 10.sup.4, being dependent only on the 
accuracy of the measured quantities P, T, and .epsilon.. 
It should be noted at this point that A.sub..epsilon. of nonpolar gases is 
temperature independent, and for a given gas it needs to be measured only 
once. This feature is especially helpful when adsorption measurements are 
carried out at different temperatures. 
Contributions to CM of polar gases from Be.sub..epsilon., become noticeable 
at lower pressure and should be taken into account. In this case, Eq. (2) 
is written as: 
EQU CM=(.epsilon.-1)/[(.epsilon.+2].apprxeq.A.sub..epsilon. +B.sub..epsilon. d 
(7) 
Defining a function f as: 
EQU f=(.epsilon.-1)(.epsilon.+2).sup.-1, (8) 
EQU one gets: 
EQU d.sup.(2) =(f/A.sub..epsilon.)-(B.sub..epsilon. /A.sub..epsilon.).sup.2. 
(9) 
Although B.sub..epsilon. appears in the slope of Eq. (6) and can, in 
principle, be determined from pressure measurements by means of Eq. (5), 
the resultant value of B.sub..epsilon. remains one of limited accuracy. 
This is due to the fact that B.sub..epsilon. appears only in combination 
with A.sub..epsilon. B.sub.p, which is generally an order of magnitude 
larger. As such, a small error in the slope (B.sub..epsilon. 
-A.sub..epsilon. B.sub.p) could lead to a very large error in 
B.sub..epsilon.. 
More accurate values of B.sub..epsilon. can be obtained from a second 
series of measurements based on a cyclic expansion technique using the 
apparatus according to the invention, but without the solid in the sample 
cell. Initially the system is filled with gas, the valve is closed, and 
the dielectric constant .epsilon..sub.1 is measured. The sample cell is 
then evacuated, the gas from the capacitance cell allowed to expand into 
the sample cell, the valve closed, and the dielectric constant 
.epsilon..sub.2 is measured. The process is repeated a number of times 
giving a series of measurements of f.sub.i =(i-1)(.epsilon..sub.i 
+2).sup.-1 at a set of densities d.sub.i in fixed ratios r such that: 
EQU d.sub.(i+k) =[V.sub.C /(V.sub.C +V.sub.B)].sup.k d.sub.i =r.sup.k d.sub.i. 
(10) Values of f after i and (i+k) expansions are, from Eq. (2), related 
by: 
EQU f.sub.i /f.sub.(i+k) =r.sup.-k +(r.sup.-k -1)(B.sub..epsilon. 
/A.sub..epsilon..sup.2)F.sub.i 
EQU +(r.sup.-k -r.sup.-k)[(C.sub..epsilon. 
/A.sub..epsilon..sup.3)-(B.sub..epsilon..sup.2 
/A.sub..epsilon..sup.4)]f.sub.i.sup.2. (11) The 
quantities r.sup.-k and (B.sub..epsilon. /A.sub..epsilon..sup.2) can be 
obtained by plotting the left-hand side of Eq. (11) vs. f.sub.i. 
Unlike the volumetric method which is limited by its depencency on the 
availability and reliability of the compressibility data for the gas to be 
studied, the method according to the invention is self-sufficient. That 
is, it determines the density of the adsorbate under the same experimental 
conditions as for the adsorption measurement. Moreover, when comparing the 
method of the invention with the volumetric method in the case of nonpolar 
gases, it should be pointed out that more than 99% of the contribution in 
the density comes from the first term (f/A.sub..epsilon.). The remaining 
contribution of 1% comes from the second and higher-order terms in the 
dielectric virial series. It turns out that for nonpolar gases, the 
contribution of the second-order correction to the ideal gas law is much 
more important in the pressure virial series than in the dielectric virial 
series. 
Thus, in the case of nonpolar gases, the method according to the invention 
becomes simple to apply especially when measuring adsorption as a function 
of temperature. First, B.sub..epsilon. does not have to be measured, and 
second, A.sub..epsilon. which is now temperature independent has to be 
measured only once during the course of the adsorption measurement. 
In the case of polar gases or at very high pressures, the contribution of 
B.sub..epsilon. becomes noticeable and should be measured. This is simply 
done by means of the cyclic expansion technique described above and using 
the same apparatus. The precision of this technique is limited to about 
10% which is generally acceptable for a second-order correction. 
The method of determining the density of a gas through measurement of its 
dielectric constant is also novel and constitutes another aspect of the 
invention. Thus, according to still a further aspect of the invention, 
there is provided a method of determining the density of gas, which 
comprises filling a capacitance cell with the gas at a predetermined 
pressure and measuring the dielectric constant of the gas at the 
predetermined pressure to determine the density thereof. 
If a first-order density approximation is desired, the first dielectric 
virial coefficient A.sub..epsilon. can be measured by varying the 
pressure of the gas in the capacitance cell and measuring the dielectric 
constant to the gas at different pressures to provide a first series of 
dielectric constant measurements from which A.sub..epsilon. is 
determined. Where a second-order density approximation is desired, the 
second dielectric virial coefficient B.sub..epsilon. can be measured by: 
(a) allowing the gas from the capacitance cell to expand into a second 
evacuated cell in controlled gas flow communication with the capacitance 
cell, and measuring the dielectric constant of the gas at equilibrium 
pressure; 
(b) evacuating the capacitance cell; 
(c) allowing the gas from the second cell to expand into the capacitance 
cell and measuring the dielectric constant of the gas at a new equilibrium 
pressure; 
(d) evacuating the second cell; and 
(e) repeating steps (a) through (d) at progressively decreasing pressures 
to provide a second series of dielectric constant measurements from which 
B.sub..epsilon. is determined. 
Further features and advantages of the invention will become more readily 
apparent from the following description of preferred embodiments with 
reference to the appended drawings, in which: 
FIG. 1 is a schematic diagram of an apparatus according to a preferred 
embodiment of the invention; 
FIG. 2 shows a plot of (.epsilon.-1)(.epsilon.+2)(RT/P) vs. (P/RT) for 
methane at 25.degree. C.; and 
FIG. 3 shows the adsorption isotherms of methane on BPL-activated carbon.

As illustrated in FIG. 1, the apparatus used for effecting dielectric 
constant measurements comprises a three-terminal cylindrical capacitance 
cell 10 (only two terminals shown, the third terminal consisting 
essentially of a ground to eliminate capacitance fringing effects) 
connected to a capacitance bridge 12 which is a decade transformer bridge 
(General Radio type 1615-A) having a resolution of the order of 
1.times.10.sup.-17 F. (10 aF): and an accuracy at 1 kHz of .+-.0.01%. The 
capacitance cell 10 is in gas flow communication with a cylindrical sample 
cell 14 containing a solid adsorbent 16 by means of conduit 18, the flow 
of gas between the cells being controlled by valve V.sub.1. The cell 10 is 
made of stainless steel and has a geometrical capacitance of 100.0.+-.0.1 
pF and a free volume V.sub.C of 96.+-.1 ml. The sample cell 14 of volume 
V.sub.s =146.+-.1.5 ml is also made of stainless steel. Both cells are 
connected with high-pressure tubing 20 and suitable valves V.sub.2, 
V.sub.3, V.sub.4, V.sub.5 and V.sub.6 to an external gas handling 
equipment including a pressure gauge 22, a vacuum pump 24, a tank 26 
containing an adsorptive gas and a compressor 28 for compressing same. The 
assembly is immersed in a circulating bath 30 maintained at constant 
temperature by a proportional controller (not shown). 
As shown, the capacitance cell 10 comprises a plurality of spaced-apart 
electrode members 32, 32' in the form of circular plates, each of the 
plates 32 being arranged alternatively with the plates 32' inside the cell 
with each plate 32 being connected to a conductor element 34 and each 
plate 32' connected to a conductor element 34', the conductor elements 34, 
34' being in turn connected to the capacitance bridge 12. The plates 32 
and 32' are spaced from one another to define a gap of about 2 to 5 mm, 
preferably 2 mm. The capacitance bridge 12 includes a frequency generator 
36, a transformer 38, a variable capacitor 40 and a meter 42. 
In order to determine the amount of gas adsorbed by the solid adsorbent 16 
contained in the sample cell 14, both cells 10 and 14 are first evacuated 
by opening valves V.sub.1, V.sub.2, V.sub.3 and V.sub.5, the valve V.sub.6 
being closed; valve V.sub.4 remains open during the entire experiment. 
When the system is under vacuum, valves V.sub.1, V.sub.3 and V.sub.5 are 
closed and valve V.sub.6 is opened so as to fill the capacitance cell 10 
with the adsorptive gas at a predetermined pressure. Valve V.sub.2 is 
closed and the dielectric constant of the gas is measured by means of the 
capacitance bridge 12. Valve V.sub.1 is then opened to allow the gas from 
the capacitance cell 10 to expand into the sample cell 14 for adsorption 
by the adsorbent 16. As adsorption takes place, the pressure in the system 
falls until equilibrium is established. The dielectric constant of the gas 
at equilibrium pressure is measured and the densities of the gas before 
and after expansion are determined from these dielectric constant 
measurements. The amount of gas molecules adsorbed at the equilibrium 
pressure can thereafter be calculated using Eq. (1). 
By repeating the above procedure at progressively increasing pressures 
until a pressure of about 1.2 MPa is reached, a series of dielectric 
constant measurements can be obtained from which the first dielectric 
virial coefficient A.sub..epsilon. can be determined. 
The second dielectric virial coefficient B.sub..epsilon. can also be 
determined from a second series of dielectric constant measurements using 
the same apparatus as shown in FIG. 1, but without the solid adsorbent 16 
in the sample cell 14. The capacitance cell 10 being initially filled with 
gas at a predetermined pressure and the valve V.sub.1 closed, the 
dielectric constant is measured. The sample cell is evacuated, the valve 
V.sub.1 is opened to allow the gas from the capacitance cell 10 to expand 
into the sample cell 14, and the dielectric constant of the gas at 
equilibrium pressure is measured. Valve V.sub.1 is closed and the 
capacitance cell 10 is evacuated. The valve V.sub.1 is again opened to 
allow the gas from the sample cell 14 to expand into the capacitance cell 
10, and the dielectric constant of the gas at the new equilibrium pressure 
is measured. This procedure is repeated at progressively decreasing 
pressures until a pressure of a few atmospheres (about 0.2 MPa) is 
reached. The value of B.sub..epsilon. can then be determined from a plot 
of the left-hand side of Eq. (11) vs f.sub.i. 
The following non-limiting example further illustrates the invention. 
EXAMPLE 
Activated BPL-carbon from the Calgon Company was used as the adsorbent. The 
adsorbate was ultrahigh-purity methane supplied by Matheson with a stated 
purity of 99.97%. Degassing of the adsorbent was done under vacuum at 
200.degree. C. for 3 hr. The adsorbent was also degassed thoroughly after 
calibration of the dead space. 
Isotherm measurements were done at 25.degree. C. and pressures up to 16.5 
MPa. In order to compare the dielectric method according to the invention 
with the volumetric method under the same experimental conditions, the 
pressure was measured in addition to the dielectric constant. Methane 
being nonpolar, Eq. (4) was used as the basis for calculating amounts of 
gas adsorbed in the dielectric method. The ideal gas law was used as the 
basis for the volumetric method. The correction for gas nonideality was 
done by using the compressibility factor Z calculated from the equation of 
state of methane at the measured pressure and temperature. 
FIG. 2 shows a plot of the Clausius-Mossotti function at low pressure P 
(&lt;1.2 MPa). A.sub..epsilon. found from the intercept of the curve has a 
value of 6.5489 cm.sup.3 mole.sup.-1 with a standard deviation of 0.00035 
from the least-squares fit. The deviation corresponds to an error less 
than 2.times.10.sup.-4 in A.sub..epsilon.. 
FIG. 3 shows the adsorption isotherm obtained from dielectric measurements 
with A.sub..epsilon. determined in the same experiment and the isotherms 
obtained from pressure measurements with and without corrections for gas 
non-ideality. It can be easily seen that at low pressures, where the gas 
in the bulk phase behaves almost like an ideal gas, results from the 
dielectric method and the volumetric method are very close. However, as 
the pressure increases and the bulk phase starts deviating from ideality, 
adsorption increments given by the volumetric method before correction for 
nonideality become more and more erroneous. The error at 16.5 MPa is about 
300%. It can also be seen from FIG. 3 that the agreement between the 
dielectric results and the volumetric results becomes quite good when the 
later are corrected for nonideality. 
After completion of the first series of measurements, the adsorbent was 
removed, the sample cell was evacuated, and a second series of dielectric 
constant measurements was carried out at progressively decreasing 
densities until a pressure of a few atmospheres was reached. A binomial 
fit of the ratios f.sub.i /f.sub.(i+4) vs f.sub.i, [see Eq. (11)] gave a 
value r.sup.-4 =4.324 and a value B.sub..epsilon. =7.47.+-.0.8 cm.sup.6 
mole.sup.-2. Values of the first-order density d.sup.(1) given by Eq. (4) 
and the second-order density d.sup.(2) given by Eq. (9) are listed in 
Table I hereinbelow with values of d calculated from the equation of state 
of methane at 25.degree. C. and the measured pressure P.sub.m ; the 
pressure P.sub.c is calculated from the equation of state of methane at 
25.degree. C. and the measured density d.sup.(1), 
TABLE 1 
______________________________________ 
P.sub.m d.sup.(1) 
d.sup.(2) d P.sub.c 
(MPa) (g/l) (g/l) (g/l) (MPa) 
______________________________________ 
0.524 3.577 3.576 3.422 0.530 
1.738 11.76 11.75 11.59 1.763 
3.172 21.77 21.74 21.69 3.184 
5.400 38.43 38.33 38.34 5.412 
7.890 58.39 58.15 58.27 7.908 
11.138 85.96 85.43 85.75 11.166 
13.759 108.7 107.8 108.3 13.821 
16.386 130.4 129.2 129.9 16.452 
______________________________________ 
Table I shows that contributions of B.sub..epsilon. to the density are less 
than 0.25%, 0.5% and 1% for respective pressures of 5, 10 and 16 MPa. The 
first-order equation is thus a valid representation of the density of 
nonpolar gases. Table I also lists values of pressure calculated from the 
equation of state of methane at the measured densities d.sup.(1) and 
temperature. The differences between the measured values and the ones 
calculated are well within the precision of the transducer used for 
measuring the pressure.