# Thermodynamic Characteristics of CH4 Adsorption on Shale at Different Temperatures

Authors: Pei Xue; Quansheng Liang; Shiyan Hao; Jintao Yin; Chao Gao; Lixia Zhang; Qianping Zhao
Journal: Geofluids
 (2022)

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## Abstract

In order to clarify the effect of temperature on the isosteric heat of adsorption of CH4 onto shale, by using the isothermal adsorption data of shale in the temperature range of 300~473 K and the pressure range of 0.08~14 MPa, the isosteric heat of adsorption is calculated, respectively, in the temperature range of 300~308 K, 338~373 K, and 423~473 K. The results show that the influence of temperature on the isosteric heat of adsorption of CH4 onto shale is mainly reflected in three aspects: the variation range of the isosteric heat of adsorption, the absolute adsorption capacity at which the isosteric heat of adsorption reach the maximum, and the change rate of the isosteric heat of adsorption. Specifically, as the temperature range moves to a higher range, the variation range of the isosteric heat increases significantly, and the absolute adsorption capacity decreases. The curve shape of the change rate of the isosteric heat of adsorption in different temperature ranges remained the same (decreased first and then increased), but the change rate increased gradually as the temperature range moved to a higher temperature range.


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## Body

## 1. Introduction

In recent years, the isosteric heat of adsorption of solid–gas adsorption system, as a key thermodynamic parameter for describing the industrial design such as mixed gas separation [1], gas storage [2], and adsorption refrigeration [3], has been widely used in the research field of the adsorption characteristics of unconventional natural gas (such as coalbed methane and shale gas) [4–16].The existing studies [5–16] on the isosteric heat of adsorption in the field of unconventional natural gas use the isometric plotting method to calculate the isosteric heat of adsorption, which is based on the Clausius-Clapeyron equation. In this method, the function fitting (exponential or linear function) between the natural logarithm of pressure and the adsorption capacity (lnP−nab) at different temperatures and between the natural logarithm of pressure and the reciprocal of temperature under different adsorption capacities is carried out successively through the adsorption isotherm of solid-gas adsorption system, and the isosteric heat of adsorption is calculated through the slope of lnP−1/T curve [17]. The isosteric heat of adsorption obtained by this method reflects the variation characteristics of adsorption heat with the adsorption capacity under the temperature range which the adsorption isotherm obtained in and cannot reflect the influence of temperature on isosteric heat of adsorption. Since the adsorption isotherms of the same sample at different temperatures are obviously different [18], the isometric plotting method depends on the adsorption isotherm data, and the change of the adsorption isotherm data will directly affect the calculation results of the isosteric heat of adsorption [19], it can be inferred from the angle of the isosteric heat of adsorption calculation method that the temperature will affect the isosteric heat of adsorption. However, up to now, there has not been a quantitative analysis of the effect of temperature on the isosteric heat of adsorption.To this end, the author used the isotherm adsorption data of the Bornholm shale to correct the excess adsorption capacity-pressure data into absolute adsorption capacity-fugacity data. Based on the data, the isosteric heats of adsorption of CH4 in the temperature range of 300-308 K, 338-373 K, and 423-473 K were analyzed for the same shale by using the isometric plotting method. At the same time, the change rate of isosteric heat of adsorption is defined, the variation characteristics of isosteric heat of adsorption in different temperature ranges are analyzed, and the effect of temperature on the isosteric heat of adsorption of CH4 in shale is clarified.


## 2. Isothermal Adsorption Experiment

This paper is aimed at analyzing the effect of temperature on the isosteric heat of adsorption of CH4 onto shale. Limited by the data requirements of the isometric plotting method for calculating the isosteric heat of adsorption, the adsorption isotherm of shale in a wide temperature range is needed. Therefore, the adsorption isotherm in the temperature range of 300–473 K proposed in reference [20] is adopted as the basic data for the calculation of the isosteric heat of adsorption.


### 2.1. Materials and Methods

The samples are from the shale of Well Skelbro-2 in Bornholm, Denmark (hereinafter referred to as Holm shale). The mineral composition of the experimental samples is mainly illite-montmorillonite and quartz and contains a large amount of mica (see Table1). The particle density is 2.592 g/cm3, the total organic carbon content is 6.35±0.1%, the equivalent vitrinite reflectance is 2.26%, the specific surface area is 22.8±1.6 m2/g, and the total pore volume is 0.0176±0.002 cm3/g. The samples used in the experiment were broken samples with particle sizes of 0.5–1 mm. Before the experiment, the samples were dried for 24 h at 473.15 K [17].Table 1
Mineralogical characteristics of the experimental sample [17].
Quartz (%)Plagioclase (%)Potassium feldspar (%)Limestone (%)Siderite (%)Pyrite (%)44.41.01.30.50.41.4Marcasite (%)Mica (%)Illite (%)Illite/montmorillonite (%)Kaolinite (%)Chlorite (%)0.89.55.929.90.74.2The experimental temperature range was set to be 300-473 K, and a total of 11 temperature points were set to be 300 K, 303 K, 308 K, 318 K, 338 K, 358 K, 373 K, 389 K, 423 K, 448 K, and 473 K, respectively. The equilibrium pressure range was set to be 0.08 MPa–14 MPa. The isothermal adsorption amount under 17 equilibrium pressures was measured at different equilibrium temperatures. The isothermal adsorption experiment started from the maximum set temperature of 473 K and gradually reduced the experimental temperature to 300 K. The fluctuation range of equilibrium temperature was controlled at ±0.1 K. The Hiden Isochema Intelligent Manometric Instrument (IMI) of Hiden Isochema Ltd. was selected as the experimental equipment. The maximum test temperature of the instrument was 1046.15 K, and the maximum test pressure was 20 MPa. The adsorbed gas used in the experiment was CH4 with purity of 99.995% [17]. Detailed sample characteristic parameters and experimental procedures can be found in reference [20].


### 2.2. Experimental Results

Figure1 shows the isotherm adsorption curve of Holm shale. The adsorption capacity obtained from the isothermal adsorption experiment is the excess adsorption capacity [21, 22]. Therefore, the excess adsorption amount is represented by nex in Figure 2.Figure 1
The change rate of isosteric heat of adsorption in different temperature range.
(a)
300~308 K(b)
338~373 K(c)
423~473 KFigure 2
The adsorption isothermal of CH4 onto shale (data cited from literature [20]).Since the critical temperature and critical pressure of CH4 are 190.55 K and 0.59 MPa, respectively, the temperature range of the adsorption isotherm curve of Holm shale is 300-473 K, which belongs to the supercritical adsorption category. The state changes from gas phase to supercritical fluid, and at the same time, the excess adsorption curve in the higher pressure range shows a maximum point.


## 2.1. Materials and Methods

The samples are from the shale of Well Skelbro-2 in Bornholm, Denmark (hereinafter referred to as Holm shale). The mineral composition of the experimental samples is mainly illite-montmorillonite and quartz and contains a large amount of mica (see Table1). The particle density is 2.592 g/cm3, the total organic carbon content is 6.35±0.1%, the equivalent vitrinite reflectance is 2.26%, the specific surface area is 22.8±1.6 m2/g, and the total pore volume is 0.0176±0.002 cm3/g. The samples used in the experiment were broken samples with particle sizes of 0.5–1 mm. Before the experiment, the samples were dried for 24 h at 473.15 K [17].Table 1
Mineralogical characteristics of the experimental sample [17].
Quartz (%)Plagioclase (%)Potassium feldspar (%)Limestone (%)Siderite (%)Pyrite (%)44.41.01.30.50.41.4Marcasite (%)Mica (%)Illite (%)Illite/montmorillonite (%)Kaolinite (%)Chlorite (%)0.89.55.929.90.74.2The experimental temperature range was set to be 300-473 K, and a total of 11 temperature points were set to be 300 K, 303 K, 308 K, 318 K, 338 K, 358 K, 373 K, 389 K, 423 K, 448 K, and 473 K, respectively. The equilibrium pressure range was set to be 0.08 MPa–14 MPa. The isothermal adsorption amount under 17 equilibrium pressures was measured at different equilibrium temperatures. The isothermal adsorption experiment started from the maximum set temperature of 473 K and gradually reduced the experimental temperature to 300 K. The fluctuation range of equilibrium temperature was controlled at ±0.1 K. The Hiden Isochema Intelligent Manometric Instrument (IMI) of Hiden Isochema Ltd. was selected as the experimental equipment. The maximum test temperature of the instrument was 1046.15 K, and the maximum test pressure was 20 MPa. The adsorbed gas used in the experiment was CH4 with purity of 99.995% [17]. Detailed sample characteristic parameters and experimental procedures can be found in reference [20].


## 2.2. Experimental Results

Figure1 shows the isotherm adsorption curve of Holm shale. The adsorption capacity obtained from the isothermal adsorption experiment is the excess adsorption capacity [21, 22]. Therefore, the excess adsorption amount is represented by nex in Figure 2.Figure 1
The change rate of isosteric heat of adsorption in different temperature range.
(a)
300~308 K(b)
338~373 K(c)
423~473 KFigure 2
The adsorption isothermal of CH4 onto shale (data cited from literature [20]).Since the critical temperature and critical pressure of CH4 are 190.55 K and 0.59 MPa, respectively, the temperature range of the adsorption isotherm curve of Holm shale is 300-473 K, which belongs to the supercritical adsorption category. The state changes from gas phase to supercritical fluid, and at the same time, the excess adsorption curve in the higher pressure range shows a maximum point.


## 3. Correction of Absolute Adsorption Capacity-Fugacity Curve

The adsorption amount directly obtained by the isothermal adsorption experiment is the excess adsorption amount. The excess adsorption amount represents the adsorption amount of the actual adsorption phase density minus the gas phase density, and the absolute adsorption amount represents the actual adsorption amount of the solid-gas adsorption system. Since the excess adsorption capacity cannot reflect the real adsorption capacity of the adsorption system, its value is smaller than the absolute adsorption capacity that characterizes the real adsorption capacity [21]. Using the excess adsorption capacity as the basic data, the calculated isosteric heat of adsorption is higher [19]. And in the range of low gas pressure, i. e., low adsorption amount, the isosteric heat of adsorption is less than zero, which is obviously contrary to the physical phenomenon that the adsorption process is exothermic. Therefore, from the perspective of the rationality of the calculation results, the absolute adsorption capacity should be used as the basic data for the calculation of the isosteric heat of adsorption [19, 23].At the same time, in the low pressure stage, the difference between pressure and fugacity is small, and pressure can be used to replace fugacity for thermodynamic characteristic analysis. However, with the increase of pressure, the difference between them increases. If the pressure is used as the basic data of equivalent adsorption heat calculation, the influence of gas intermolecular force in the calculation results will be too large [23]. Therefore, the absolute adsorption capacity-fugacity data is selected as the basic data for the calculation of isosteric heat of adsorption, and the excess adsorption capacity-pressure data expressed in Figure 2 is converted to the absolute adsorption capacity-fugacity data.


### 3.1. Correction of Absolute Adsorption Capacity

According to the definition of Gibbs adsorption capacity, the relationship between excess adsorption capacity and absolute adsorption capacity is [21, 23]
(1)nab=nex1−ρg/ρa,where nab is the absolute adsorption capacity (mmol/g). nex is the excess adsorption capacity (mmol/g). ρa is adsorption phase density (g/cm3). ρg is gas phase density (g/cm3). After determining the density of the gas phase and adsorption phase, Equation (1) can be used to correct the excess adsorption amount, so as to obtain the absolute adsorption curve. The vapor density of CH4 used in this paper comes from the NIST database, as shown in Figure 3.Figure 3
Gas phase density of CH4.It can be seen from Figure3 that at the same temperature, the gas density of CH4 gradually increases with the increase of pressure. For example, at 300 K, when the pressure increases from 0.08 MPa to 13.08 MPa, the gas density of CH4 increases from 0.00051523 g/cm3 to 0.10123 g/cm3. And in the same pressure, the gas density of CH4 gradually decreases with the increase of temperature. For example, when the pressure is 5.46 MPa and the temperature rises from 300 K to 473 K, the gas density of CH4 decreases from 0.03672 g/cm3 to 0.02339 g/cm3.The adsorption phase density cannot be directly measured under supercritical conditions, and theoretical estimation and equation fitting calculation are mostly used. Reference [17] compared the accuracy of the existing adsorption phase density calculation methods, and the results showed that the calculation results of the empirical formula method were more reasonable. Therefore, the Ozawa empirical formula [24] was used to calculate the adsorption phase density, which can be expressed as
(2)ρa=ρbexp−0.0025×T−Tb,where ρb is the density at boiling point (g/cm3), ρb,CH4=0.424g/cm3. Tb is boiling temperature (K), Tb,CH4=111.65K. T is absolute temperature (K). The calculated adsorption phase density curve of CH4 is shown in Figure 4. In the temperature range of 300–473 K, the adsorption phase density of CH4 gradually decreased with the increase of temperature, from 0.26467 g/cm3 at 300 K to 0.17174 g/cm3 at 473 K.Figure 4
Adsorption phase density of CH4.The corrected absolute adsorption capacity curve is shown in Figure5. It can be seen from Figure 5 that only when the temperature is 423 K, 448 K, and 473 K, the absolute adsorption amount curve has a maximum value. Theoretically, the absolute adsorption capacity curve represents the actual adsorption capacity. After the adsorption saturation is reached, the absolute adsorption capacity curve should remain stable, which is related to the adsorption phase density in the calibration process.Figure 5
The absolute adsorption capacity curve.


### 3.2. Calculation of Fugacity

Fugacityf and fugacity coefficient φ are important basic concepts in thermodynamic calculation of variable composition systems and important tools for studying phase equilibrium and chemical reaction equilibrium [23, 25]. The relationship between fugacity and fugacity coefficient can be expressed as
(3)Tc,CH4=190.55K,where φ is fugacity coefficient. f is fugacity (Pa). p is pressure (Pa). In this paper, Peng-Robinson equation [26] of state is used to obtain fugacity; the fugacity coefficient [27] can be expressed as
(4)lnφ=Z−1+lnRTZRT−pb−2aTαT4bRTlnZRT+2+1pbZRT−2−1pb,where aT为 is an intercohesion parameter and b is c-volume item. The intercohesion parameters can be expressed as
(5)aT=0.452724R2Tc2pc·αT,where Tc is critical temperature (K), Tc,CH4=190.55K. Pc is critical pressure (Pa), Pc,CH4=4.59MPa. aT can be expressed as
(6)αT=1+k1−Tr0.52,where Tr is the reduced temperature, Tr=T/Tc. When the eccentricity factor is 0<ω<0.5, k can be expressed as [23]
(7)k=0.3746+1.54226ω−0.26992ω2,where ω is an eccentric factor, ωCH4=0.008. b can be expressed as
(8)b=0.07780RTcpc.The fugacity curve of CH4 is shown in Figure 6 under the conditions of temperature range 300-473 K and pressure range 0.08-13.75 MPa. It can be seen from Figure 6 that with the increase of pressure, the fugacity gradually increases.Figure 6
Fugacity curve of CH4 (the red virtual line is characterized as f=p).In the lower pressure range, the fugacity is equivalent to the pressure, and the fugacity curve is close to thef=p line (the red dotted line in Figure 6). As the pressure increases, the fugacity is gradually smaller than the pressure, and the difference between the two increases [23].


### 3.3. Absolute Adsorption Capacity-Fugacity Curve

Using the correction method of absolute adsorption capacity and fugacity proposed in the above section, the excess adsorption capacity of Holm shale (see Figure2) is corrected, respectively, and the absolute adsorption capacity-fugacity curve is obtained, as shown in Figure 7. The absolute adsorption curve has maximum value only at 448 K and 473 K.Figure 7
The curve of the absolute amount and fugacity.


## 3.1. Correction of Absolute Adsorption Capacity

According to the definition of Gibbs adsorption capacity, the relationship between excess adsorption capacity and absolute adsorption capacity is [21, 23]
(1)nab=nex1−ρg/ρa,where nab is the absolute adsorption capacity (mmol/g). nex is the excess adsorption capacity (mmol/g). ρa is adsorption phase density (g/cm3). ρg is gas phase density (g/cm3). After determining the density of the gas phase and adsorption phase, Equation (1) can be used to correct the excess adsorption amount, so as to obtain the absolute adsorption curve. The vapor density of CH4 used in this paper comes from the NIST database, as shown in Figure 3.Figure 3
Gas phase density of CH4.It can be seen from Figure3 that at the same temperature, the gas density of CH4 gradually increases with the increase of pressure. For example, at 300 K, when the pressure increases from 0.08 MPa to 13.08 MPa, the gas density of CH4 increases from 0.00051523 g/cm3 to 0.10123 g/cm3. And in the same pressure, the gas density of CH4 gradually decreases with the increase of temperature. For example, when the pressure is 5.46 MPa and the temperature rises from 300 K to 473 K, the gas density of CH4 decreases from 0.03672 g/cm3 to 0.02339 g/cm3.The adsorption phase density cannot be directly measured under supercritical conditions, and theoretical estimation and equation fitting calculation are mostly used. Reference [17] compared the accuracy of the existing adsorption phase density calculation methods, and the results showed that the calculation results of the empirical formula method were more reasonable. Therefore, the Ozawa empirical formula [24] was used to calculate the adsorption phase density, which can be expressed as
(2)ρa=ρbexp−0.0025×T−Tb,where ρb is the density at boiling point (g/cm3), ρb,CH4=0.424g/cm3. Tb is boiling temperature (K), Tb,CH4=111.65K. T is absolute temperature (K). The calculated adsorption phase density curve of CH4 is shown in Figure 4. In the temperature range of 300–473 K, the adsorption phase density of CH4 gradually decreased with the increase of temperature, from 0.26467 g/cm3 at 300 K to 0.17174 g/cm3 at 473 K.Figure 4
Adsorption phase density of CH4.The corrected absolute adsorption capacity curve is shown in Figure5. It can be seen from Figure 5 that only when the temperature is 423 K, 448 K, and 473 K, the absolute adsorption amount curve has a maximum value. Theoretically, the absolute adsorption capacity curve represents the actual adsorption capacity. After the adsorption saturation is reached, the absolute adsorption capacity curve should remain stable, which is related to the adsorption phase density in the calibration process.Figure 5
The absolute adsorption capacity curve.


## 3.2. Calculation of Fugacity

Fugacityf and fugacity coefficient φ are important basic concepts in thermodynamic calculation of variable composition systems and important tools for studying phase equilibrium and chemical reaction equilibrium [23, 25]. The relationship between fugacity and fugacity coefficient can be expressed as
(3)Tc,CH4=190.55K,where φ is fugacity coefficient. f is fugacity (Pa). p is pressure (Pa). In this paper, Peng-Robinson equation [26] of state is used to obtain fugacity; the fugacity coefficient [27] can be expressed as
(4)lnφ=Z−1+lnRTZRT−pb−2aTαT4bRTlnZRT+2+1pbZRT−2−1pb,where aT为 is an intercohesion parameter and b is c-volume item. The intercohesion parameters can be expressed as
(5)aT=0.452724R2Tc2pc·αT,where Tc is critical temperature (K), Tc,CH4=190.55K. Pc is critical pressure (Pa), Pc,CH4=4.59MPa. aT can be expressed as
(6)αT=1+k1−Tr0.52,where Tr is the reduced temperature, Tr=T/Tc. When the eccentricity factor is 0<ω<0.5, k can be expressed as [23]
(7)k=0.3746+1.54226ω−0.26992ω2,where ω is an eccentric factor, ωCH4=0.008. b can be expressed as
(8)b=0.07780RTcpc.The fugacity curve of CH4 is shown in Figure 6 under the conditions of temperature range 300-473 K and pressure range 0.08-13.75 MPa. It can be seen from Figure 6 that with the increase of pressure, the fugacity gradually increases.Figure 6
Fugacity curve of CH4 (the red virtual line is characterized as f=p).In the lower pressure range, the fugacity is equivalent to the pressure, and the fugacity curve is close to thef=p line (the red dotted line in Figure 6). As the pressure increases, the fugacity is gradually smaller than the pressure, and the difference between the two increases [23].


## 3.3. Absolute Adsorption Capacity-Fugacity Curve

Using the correction method of absolute adsorption capacity and fugacity proposed in the above section, the excess adsorption capacity of Holm shale (see Figure2) is corrected, respectively, and the absolute adsorption capacity-fugacity curve is obtained, as shown in Figure 7. The absolute adsorption curve has maximum value only at 448 K and 473 K.Figure 7
The curve of the absolute amount and fugacity.


## 4. Characteristics of the Isosteric Heat of Adsorption in Shale

The isosteric heat of adsorption is defined as the heat released by the adsorption of infinitely small gas molecules when the adsorption capacity is constant [17, 28]. Using the Clausius-Clapeyron equation, under the adsorption capacity (n), the isosteric heat of adsorption qst can be expressed as [11, 17, 29, 30]
(9)−dlnpdTn=qstRT2,where p is the equilibrium adsorption pressure (Pa). Since the absolute adsorption capacity nab reflects the actual adsorption amount of the adsorption system, nab is used in the calculation of the equivalent adsorption heat [17, 30].And the experimental pressure range obviously exceeds the pressure range suitable for the ideal gas assumption, and fugacity is used to replace the pressure in the thermodynamic analysis. Therefore, formula (9) can be rewritten as
(10)−dlnfdTnab=qstRT2.


### 4.1. Calculation of Isosteric Heat of Adsorption Based on Adsorption Isosteric Plotting Method

The theoretical method of the adsorption isometric line plotting method is to integrate formula (10) when the adsorption amount is a fixed value and obtain
(11)lnf=–qstRT+M,where M is a constant part of indefinite integral. It can be seen from equation (11) that 1/T and lnf have linear function characteristics, and the isosteric heat of adsorption qst can be obtained by using the linear slope of the function −qst/R. Representing the slope of the linear function −qst/R as A, that is, −qst/R, the isosteric heat of adsorption is
(12)qst=−RA.In the calculation process, the relationship betweenlnf and nab should be drawn first, and then, the fitting function should be determined according to the shape of the lnf−nab curve to fit the curve [17]. The fitting function of lnf−nab was used to calculate and draw the lnf−1/T curve under different adsorption capacity, and the isosteric heat of adsorption in the corresponding temperature range was calculated by using the slope of the lnf−1/T curve. From the data shown in Figure 6, the curve of lnf−nab of the Holm shale in the temperature ranges of 300-308 K, 338-373 K, and 423-473 K can be obtained, as shown in Figure 8.Figure 8
The curve of the exponential function fitting oflnf−nab.
(a)
300-308 K(b)
338-373 K(c)
423-473 KAccording to the morphological characteristics of thelnf−nab curve, the exponential function was used to fit the lnf−nab curve (see Figure 8). The exponential function expression is
(13)lnf=a−bcnab,where a, b, and c are equation fit constant. The fitting results of the lnf−nab curve function (see Table 2) show that the data of lnf−nab maintained a good exponential function relationship at each temperature, and the correlation coefficient R2 was greater than 0.98.Table 2
The result of exponential function fitting oflnf−nab.
Temperature (K)Equation fitting parameters and correlation coefficientsabcR230016.555575.948581.02×10−40.9990230316.574915.959776.39×10−50.99930816.782716.019818.24×10−50.9983633816.626145.536724.96×10−60.9940935816.280545.04727.84×10−80.9928137316.382415.044761.40×10−80.9901542316.054664.456531.00×10−140.9870544816.049044.372966.76×10−180.9839347315.824164.177041.84×10−250.98437Although the exponential function can fit thelnf−nab curve well, the fitting correlation coefficient shows a decreasing trend with the increase of temperature, that is, from 0.99902 at 300 K to 0.98437 at 473 K.From the shape of thelnf−nab curve (Figure 8), the reason is that at higher temperature (373–473 K), the data of lnf−nab gradually deviates from the fitting curve of the exponential function at the high adsorption of the curve (shown in Figure 7 by the red solid coil); that is, the positive increment of the ordinate of the data point does not meet the law of dlnf/dnab=−b⋅lnc⋅cnab.When the adsorption system has reached or is close to the saturated adsorption state, with the increase of fugacityf, the absolute adsorption capacity nab has tended to be stable, and the fitting accuracy of lnf−nab will decrease. From this point of view, when using the adsorption isotherm plotting method to calculate the isosteric heat of adsorption, the data in the adsorption isotherm indicating that the adsorption system has reached or approached the saturated adsorption state should be excluded.In the temperature range of 300~308 K, 338~373 K, and 423~473 K, the value range ofnab was set to be 0.01~0.3 mmol/g, 0.003~0.2 mmol/g, and 0.0005~0.08 mmol/g, and at the same time, twenty value points were set at equal intervals in the range of nab with different adsorption amounts, and then, the lnf−1/T curves under different absolute adsorption amounts were drawn, as shown in Figure 9.Figure 9
The cure oflnf−1/T curve at different adsorption amounts.
(a)
300-308 K(b)
338-373 K(c)
423-473 K


### 4.2. Isosteric Heat of Adsorption at Different Temperatures

A linear function is used to fit thelnf−1/T curve under different temperature ranges and absolute adsorption capacity, and the linear fitting equation can be expressed as
(14)lnf=h+Anab.Figure10 shows the linear fitting correlation coefficient curve of the lnf−1/T curve in different temperature ranges that are significantly different. Specifically, in the range of 300-308 K, the fitting correlation coefficient R2 is generally above 0.92, and the fitting effect is good. In the interval of 338~373 K and 423~473 K, the fitting correlation coefficient R2 showed in the low adsorption stage (nab<0.1585 mmol/g at 338~373 K and nab<0.0633 mmol/g at 423~473 K). In the high value range in the interval of 0.8~0.99 and in the stage of high adsorption capacity (nab≥0.1585 mmol/g at 338~373 K, nab≥0.0633 mmol/g at 423~473 K), the correlation coefficient R2 decreased rapidly with the increase of adsorption capacity and even negatively appeared at the position close to the maximum absolute adsorption capacity value (shown in red data in Figure 7). If the correlation coefficient R2=0.8 is used as the criterion for judging whether the lnf−1/T curve conforms to the linear law, the slope A of the linear fitting function of the lnf−1/T curve in the range of nab≥0.1585 mmol/g at 338~373 K and nab≥0.0674 mmol/g at 423~473 K can no longer be used for the calculation of isosteric heat of adsorption.Figure 10
Correlation coefficient oflnf−1/T.In view of the above rules of the linear fitting correlation coefficient of thelnf−1/T curve, in the subsequent calculation of the isosteric heat of adsorption using the linear fitting slope A value, this paper adopted the slope A value of the linear fitting of the lnf−1/T curve in a temperature range of 300~308 K, 338~373 K, and 423~473 K, corresponding the range of nab is 0.01~0.3 mmol/g, 0.003~0.1482 mmol/g, and 0.0047~0.0591 mmol/g, to calculate isosteric heat of adsorption. The calculation results are shown in Figure 11.Figure 11
Isosteric heat of adsorption in different temperature ranges.It can be seen from Figure11 that in different temperature ranges, with the increase of absolute adsorption amount, the isosteric heat of adsorption in each temperature range show a nonmonotonic change that first increases and then decreases. Specifically, in the lower temperature range of 300–308 K, qst increased from 17.18642 kJ/mol at nab=0.01 mmol/g to 24.55957 kJ/mol at nab=0.19316 mmol/g and then decreased to nab=0.324.0695 kJ/mol at nab=0.3 mmol/g. In the temperature range of 338~373 K, qst increased from 9.43947 kJ/mol at nab=0.003 mmol/g to 21.51892 kJ/mol at nab=0.05484 mmol/g and then decreased to 9.19229 kJ/mol at nab=0.14816 mmol/g. In the temperature range of 423~473 K, qst increased rapidly from 3.21306 kJ/mol at nab=0.0005 mmol/g to 25.56946 kJ/mol at nab=0.02142 mmol/g and then decreased to 7.95283 kJ/mol at nab=0.06326 mmol/g.As the temperature range moves to a higher range, the variation range of the isosteric heat of adsorption increases significantly. For example, in the temperature range of 300~308 K, 338~373 K, and 423~473 K, the variation range of isosteric heat of adsorption is 17.18642~24.55957 kJ/mol, 2.60997~21.51892 kJ/mol, and 2.26209~25.56946 kJ/mol, respectively. The change amplitudes are 7.37315 kJ/mol, 18.90895 kJ/mol, and 23.30737 kJ/mol in turn.And, as the temperature range moves to higher range, the absolute adsorption capacity (nab,qst−max) when the maximum of the isosteric heat of adsorption appears gradually decreases. For example, the temperature range at 300~308 K,nab,qst−max=0.19316 mmol/g, and at the temperature range of 338~373 K and 423-473 K,nab,qst−maxare 0.05484 mmol/g and 0.02142 mmol/g, respectively.In order to quantitatively reflect the change rate of the isosteric heat of adsorptionqst with the absolute adsorption amount nab in different temperature ranges, the change rate of the isosteric heat of adsorption vqst−nab is defined, and its expression is
(15)vqst−nab=∆qst∆nab=qsti+1−qstinabi+1−nabi,where vqst−nab is the change rate of isosteric heat of adsorption (kJ·g·mmol-2). According to the heat of isosteric curves in different temperature ranges (see Figure 10), combined with formula (15), the change rate of the isosteric heat of adsorption in different temperature ranges can be obtained (see Figure 1).It can be seen from Figure1 that the change rate of isosteric heat of adsorption in different temperature ranges shows a change law of first decreasing and then increasing and has a higher value in the stage of low adsorption capacity. For example, in the temperature range of 330~308 K, the change rate (vqst−nab) decreases from 0.1201 kJ·g·mmol-2 whennabis 0.02526 mmol/g to 0.002 kJ·g·mmol-2; whennabis 0.19316 mmol/g, then the rate of change increased to 0.0059 kJ·g·mmol-2 asnabincreased to 0.3 mmol/g.At the same time, the change rate of isosteric heat in the low temperature range is lower than that in the high temperature range. For example, when the temperature range is 300~308 K, 338~373 K, and 423~473 K,vqst−nab is 0.0002~0.1201 kJ·g·mmol-2, 0.0081~0.5277 kJ·g·mmol-2, and 0.0765~2.4293 kJ·g·mmol-2, respectively.The variation trend of isosteric heat of adsorption is affected by the heterogeneity of adsorption sites and the interaction force between gas molecules [30–34]. When adsorbate molecules are adsorbed on a heterogeneous surface, they will preferentially adsorb to the strong adsorption sites with higher surface energy and then gradually adsorb to the adsorption sites with lower surface energy. This results in a decrease in the isosteric heat of adsorption curve with increasing adsorption capacity. At the same time, with the increase of the adsorption capacity, the intermolecular force of the adsorbate increases, causing the isosteric heat of adsorption to increase with the increase of the adsorption amount [17, 22, 23].As shown in Figure11, the isosteric heat of adsorption curve at different temperatures increases first and then decreases with the increase of absolute adsorption capacity. When the isosteric heat of adsorption curve reaches the maximum, the heterogeneity of the adsorbent surface and the effect of the intermolecular interaction of the adsorbate on the isosteric heat of adsorption reach a balance, and then, the heterogeneity of the adsorbent surface plays a dominant role, and the isosteric heat of adsorption decreases with the increase of the adsorption amount.In addition, since the maximum of the isosteric heat of adsorption represents a balance between the heterogeneity of the shale surface and the effect of the intermolecular force of CH4 on the isosteric heat of adsorption. Therefore, when the maximum of the isosteric heat of adsorption in the higher temperature range moves to the low adsorption capacity, it reflects the weakening of the intermolecular force of the adsorbate and the enhancement of the heterogeneity of the adsorbent surface with the increase of temperature.


## 4.1. Calculation of Isosteric Heat of Adsorption Based on Adsorption Isosteric Plotting Method

The theoretical method of the adsorption isometric line plotting method is to integrate formula (10) when the adsorption amount is a fixed value and obtain
(11)lnf=–qstRT+M,where M is a constant part of indefinite integral. It can be seen from equation (11) that 1/T and lnf have linear function characteristics, and the isosteric heat of adsorption qst can be obtained by using the linear slope of the function −qst/R. Representing the slope of the linear function −qst/R as A, that is, −qst/R, the isosteric heat of adsorption is
(12)qst=−RA.In the calculation process, the relationship betweenlnf and nab should be drawn first, and then, the fitting function should be determined according to the shape of the lnf−nab curve to fit the curve [17]. The fitting function of lnf−nab was used to calculate and draw the lnf−1/T curve under different adsorption capacity, and the isosteric heat of adsorption in the corresponding temperature range was calculated by using the slope of the lnf−1/T curve. From the data shown in Figure 6, the curve of lnf−nab of the Holm shale in the temperature ranges of 300-308 K, 338-373 K, and 423-473 K can be obtained, as shown in Figure 8.Figure 8
The curve of the exponential function fitting oflnf−nab.
(a)
300-308 K(b)
338-373 K(c)
423-473 KAccording to the morphological characteristics of thelnf−nab curve, the exponential function was used to fit the lnf−nab curve (see Figure 8). The exponential function expression is
(13)lnf=a−bcnab,where a, b, and c are equation fit constant. The fitting results of the lnf−nab curve function (see Table 2) show that the data of lnf−nab maintained a good exponential function relationship at each temperature, and the correlation coefficient R2 was greater than 0.98.Table 2
The result of exponential function fitting oflnf−nab.
Temperature (K)Equation fitting parameters and correlation coefficientsabcR230016.555575.948581.02×10−40.9990230316.574915.959776.39×10−50.99930816.782716.019818.24×10−50.9983633816.626145.536724.96×10−60.9940935816.280545.04727.84×10−80.9928137316.382415.044761.40×10−80.9901542316.054664.456531.00×10−140.9870544816.049044.372966.76×10−180.9839347315.824164.177041.84×10−250.98437Although the exponential function can fit thelnf−nab curve well, the fitting correlation coefficient shows a decreasing trend with the increase of temperature, that is, from 0.99902 at 300 K to 0.98437 at 473 K.From the shape of thelnf−nab curve (Figure 8), the reason is that at higher temperature (373–473 K), the data of lnf−nab gradually deviates from the fitting curve of the exponential function at the high adsorption of the curve (shown in Figure 7 by the red solid coil); that is, the positive increment of the ordinate of the data point does not meet the law of dlnf/dnab=−b⋅lnc⋅cnab.When the adsorption system has reached or is close to the saturated adsorption state, with the increase of fugacityf, the absolute adsorption capacity nab has tended to be stable, and the fitting accuracy of lnf−nab will decrease. From this point of view, when using the adsorption isotherm plotting method to calculate the isosteric heat of adsorption, the data in the adsorption isotherm indicating that the adsorption system has reached or approached the saturated adsorption state should be excluded.In the temperature range of 300~308 K, 338~373 K, and 423~473 K, the value range ofnab was set to be 0.01~0.3 mmol/g, 0.003~0.2 mmol/g, and 0.0005~0.08 mmol/g, and at the same time, twenty value points were set at equal intervals in the range of nab with different adsorption amounts, and then, the lnf−1/T curves under different absolute adsorption amounts were drawn, as shown in Figure 9.Figure 9
The cure oflnf−1/T curve at different adsorption amounts.
(a)
300-308 K(b)
338-373 K(c)
423-473 K


## 4.2. Isosteric Heat of Adsorption at Different Temperatures

A linear function is used to fit thelnf−1/T curve under different temperature ranges and absolute adsorption capacity, and the linear fitting equation can be expressed as
(14)lnf=h+Anab.Figure10 shows the linear fitting correlation coefficient curve of the lnf−1/T curve in different temperature ranges that are significantly different. Specifically, in the range of 300-308 K, the fitting correlation coefficient R2 is generally above 0.92, and the fitting effect is good. In the interval of 338~373 K and 423~473 K, the fitting correlation coefficient R2 showed in the low adsorption stage (nab<0.1585 mmol/g at 338~373 K and nab<0.0633 mmol/g at 423~473 K). In the high value range in the interval of 0.8~0.99 and in the stage of high adsorption capacity (nab≥0.1585 mmol/g at 338~373 K, nab≥0.0633 mmol/g at 423~473 K), the correlation coefficient R2 decreased rapidly with the increase of adsorption capacity and even negatively appeared at the position close to the maximum absolute adsorption capacity value (shown in red data in Figure 7). If the correlation coefficient R2=0.8 is used as the criterion for judging whether the lnf−1/T curve conforms to the linear law, the slope A of the linear fitting function of the lnf−1/T curve in the range of nab≥0.1585 mmol/g at 338~373 K and nab≥0.0674 mmol/g at 423~473 K can no longer be used for the calculation of isosteric heat of adsorption.Figure 10
Correlation coefficient oflnf−1/T.In view of the above rules of the linear fitting correlation coefficient of thelnf−1/T curve, in the subsequent calculation of the isosteric heat of adsorption using the linear fitting slope A value, this paper adopted the slope A value of the linear fitting of the lnf−1/T curve in a temperature range of 300~308 K, 338~373 K, and 423~473 K, corresponding the range of nab is 0.01~0.3 mmol/g, 0.003~0.1482 mmol/g, and 0.0047~0.0591 mmol/g, to calculate isosteric heat of adsorption. The calculation results are shown in Figure 11.Figure 11
Isosteric heat of adsorption in different temperature ranges.It can be seen from Figure11 that in different temperature ranges, with the increase of absolute adsorption amount, the isosteric heat of adsorption in each temperature range show a nonmonotonic change that first increases and then decreases. Specifically, in the lower temperature range of 300–308 K, qst increased from 17.18642 kJ/mol at nab=0.01 mmol/g to 24.55957 kJ/mol at nab=0.19316 mmol/g and then decreased to nab=0.324.0695 kJ/mol at nab=0.3 mmol/g. In the temperature range of 338~373 K, qst increased from 9.43947 kJ/mol at nab=0.003 mmol/g to 21.51892 kJ/mol at nab=0.05484 mmol/g and then decreased to 9.19229 kJ/mol at nab=0.14816 mmol/g. In the temperature range of 423~473 K, qst increased rapidly from 3.21306 kJ/mol at nab=0.0005 mmol/g to 25.56946 kJ/mol at nab=0.02142 mmol/g and then decreased to 7.95283 kJ/mol at nab=0.06326 mmol/g.As the temperature range moves to a higher range, the variation range of the isosteric heat of adsorption increases significantly. For example, in the temperature range of 300~308 K, 338~373 K, and 423~473 K, the variation range of isosteric heat of adsorption is 17.18642~24.55957 kJ/mol, 2.60997~21.51892 kJ/mol, and 2.26209~25.56946 kJ/mol, respectively. The change amplitudes are 7.37315 kJ/mol, 18.90895 kJ/mol, and 23.30737 kJ/mol in turn.And, as the temperature range moves to higher range, the absolute adsorption capacity (nab,qst−max) when the maximum of the isosteric heat of adsorption appears gradually decreases. For example, the temperature range at 300~308 K,nab,qst−max=0.19316 mmol/g, and at the temperature range of 338~373 K and 423-473 K,nab,qst−maxare 0.05484 mmol/g and 0.02142 mmol/g, respectively.In order to quantitatively reflect the change rate of the isosteric heat of adsorptionqst with the absolute adsorption amount nab in different temperature ranges, the change rate of the isosteric heat of adsorption vqst−nab is defined, and its expression is
(15)vqst−nab=∆qst∆nab=qsti+1−qstinabi+1−nabi,where vqst−nab is the change rate of isosteric heat of adsorption (kJ·g·mmol-2). According to the heat of isosteric curves in different temperature ranges (see Figure 10), combined with formula (15), the change rate of the isosteric heat of adsorption in different temperature ranges can be obtained (see Figure 1).It can be seen from Figure1 that the change rate of isosteric heat of adsorption in different temperature ranges shows a change law of first decreasing and then increasing and has a higher value in the stage of low adsorption capacity. For example, in the temperature range of 330~308 K, the change rate (vqst−nab) decreases from 0.1201 kJ·g·mmol-2 whennabis 0.02526 mmol/g to 0.002 kJ·g·mmol-2; whennabis 0.19316 mmol/g, then the rate of change increased to 0.0059 kJ·g·mmol-2 asnabincreased to 0.3 mmol/g.At the same time, the change rate of isosteric heat in the low temperature range is lower than that in the high temperature range. For example, when the temperature range is 300~308 K, 338~373 K, and 423~473 K,vqst−nab is 0.0002~0.1201 kJ·g·mmol-2, 0.0081~0.5277 kJ·g·mmol-2, and 0.0765~2.4293 kJ·g·mmol-2, respectively.The variation trend of isosteric heat of adsorption is affected by the heterogeneity of adsorption sites and the interaction force between gas molecules [30–34]. When adsorbate molecules are adsorbed on a heterogeneous surface, they will preferentially adsorb to the strong adsorption sites with higher surface energy and then gradually adsorb to the adsorption sites with lower surface energy. This results in a decrease in the isosteric heat of adsorption curve with increasing adsorption capacity. At the same time, with the increase of the adsorption capacity, the intermolecular force of the adsorbate increases, causing the isosteric heat of adsorption to increase with the increase of the adsorption amount [17, 22, 23].As shown in Figure11, the isosteric heat of adsorption curve at different temperatures increases first and then decreases with the increase of absolute adsorption capacity. When the isosteric heat of adsorption curve reaches the maximum, the heterogeneity of the adsorbent surface and the effect of the intermolecular interaction of the adsorbate on the isosteric heat of adsorption reach a balance, and then, the heterogeneity of the adsorbent surface plays a dominant role, and the isosteric heat of adsorption decreases with the increase of the adsorption amount.In addition, since the maximum of the isosteric heat of adsorption represents a balance between the heterogeneity of the shale surface and the effect of the intermolecular force of CH4 on the isosteric heat of adsorption. Therefore, when the maximum of the isosteric heat of adsorption in the higher temperature range moves to the low adsorption capacity, it reflects the weakening of the intermolecular force of the adsorbate and the enhancement of the heterogeneity of the adsorbent surface with the increase of temperature.


## 5. Conclusion

The effect of temperature on the isosteric heat of adsorption of CH4 in shale is mainly reflected in three aspects: the variation range, the maximum value, and the rate of change of the isosteric heat of adsorption. The following conclusions can be made:
(1)
As the temperature range moves to a higher range, the variation range of the isosteric heat of adsorption increases significantly. For example, in the temperature range of 300~308 K, 338~373 K, and 423~473 K, the variation range is 7.37315 kJ/mol, 18.90895 kJ/mol, and 23.30737 kJ/mol, respectively(2)
As the temperature range moves to a higher range, the absolute adsorption capacity when the maximum of the isosteric heat of adsorption appears (nab,qst−max) decreases gradually. For example, within the temperature range of 300~308 K, 338~373 K, and 423~473 K, nab,qst−max are 0.19316 mmol/g, 0.05484 mmol/g, and 0.02142 mmol/g, respectively(3)
The shape of the isosteric heat change rate remains the same in different temperature ranges (decreases first and then increases), but as the temperature range moves to a higher range, the isosteric heat of adsorption change rate gradually increases



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*Source: 6167382-2022-07-28.xml*