Physical adsorption of gases: the case for absolute adsorption as the basis for thermodynamic analysis

Abstract

We discuss the thermodynamics of physical adsorption of gases in porous solids. The measurement of the amount of gas adsorbed in a solid requires specialized volumetric and gravimetric techniques based upon the concept of the surface excess. Excess adsorption isotherms provide thermodynamic information about the gas-solid system but are difficult to interpret at high pressure because of peculiarities such as intersecting isotherms. Quantities such as pore density and heats of adsorption are undefined for excess isotherms at high pressure. These difficulties vanish when excess isotherms are converted to absolute adsorption. Using the proper definitions, the special features of adsorption can be incorporated into a rigorous framework of solution thermodynamics. Practical applications including mixed-gas equilibria, equations for adsorption isotherms, and methods for calculating thermodynamic properties are covered. The primary limitations of the absolute adsorption formalism arise from the need to estimate pore volumes and in the application to systems with larger mesopores or macropores at high bulk pressures and temperatures where the thermodynamic properties may be dominated by contributions from the bulk fluid. Under these circumstances a rigorous treatment of the thermodynamics requires consideration of the adsorption cell and its contents (bulk gas, porous solid and confined fluid).

Appendices

Appendix 1: Illustrative calculations based on a mathematical model for absolute adsorption in microporous adsorbents

Complete thermodynamic information on the adsorption of a single gas is provided either by a series of adsorption isotherms, or by a single adsorption isotherm coupled with calorimetric measurement of its differential energy. Equation (10.1) provides a quantitative fit of data for adsorption in microporous materials (see Fig. 15) and is therefore suitable for calculating energy and entropy.

Fig. 15

Points are experimental data for adsorption of ethylene on 13X molecular sieve at 25, 50, and 100 °C (Valenzuela et al. 1989). Solid lines are Eq. (10.1) with constants from Table 13

Table 13 Constants for adsorption of C₂H₄ on 13X molecular sieve

Isotherms for adsorption of pure gases are usually in the form n(P) for loading as a function of pressure. The calculation of thermodynamic properties requires a more general function n(P, T) for loading as a function of both pressure and temperature. It is apparent from Eq. (3.44) that an inverted form more useful for calculating thermodynamic properties would replace the independent variables P and T by n and T, with fugacity as the dependent variable:

$$ f = \frac{n}{K_\circ}\left[\frac{m}{m-n}\right] \exp\left[\frac{D(n)}{R}\!\!\left(\frac{1}{T} - \frac{1}{T_\circ}\right)\right]\exp \left[{C(n)}\right] $$

(10.1)

For C(n) = 0, D(n) = 0, and f = P, this reduces to the (inverted) Langmuir equation. m is the saturation capacity parameter of the Langmuir equation corresponding to pore filling.

1.1 Henry constant

K is the Henry constant:

$$ K = \lim_{P\to 0} \frac{n}{P} $$

(10.2)

and K ₀ in Eq. (10.1) is the Henry constant at reference temperature T ₀. From Eqs. (10.1) and (10.2):

$$ K = K_\circ \exp{\left[\frac{-D_\circ}{R}\left(\frac{1}{T} - \frac{1}{T_\circ}\right)\right]} $$

(10.3)

The Henry constant K decreases exponentially with increasing temperature.

1.2 Energy and entropy properties

The Langmuir equation accounts for molecules adsorbed singly in identical micropores and thus neglects entirely gas-gas interactions and variations in gas-solid interaction energy. The function C(n) in the exponential of Eq. (10.1) is a power series:

$$ C(n) = C_1 n + C_2 n^2 + C_3 n^3 + \cdots $$

(10.4)

The exponential factor e ^C(n) in the adsorption equation modifies the grand potential by adding a truncated power series to the Langmuir result, as shown in Eq. (10.12). C(n) is dimensionless and its C _i coefficients are measured at the reference temperature T ₀. This series converges rapidly; typically termination after 3 or 4 coefficients provides quantitative agreement with experiment. Application of Eq. (3.44) to Eq. (10.1) yields the power series D(n) for the differential energy:

$$ \overline{u} = D(n) = D_\circ + D_1 n + D_2 n^2 + D_3 n^3 + \cdots $$

(10.5)

D(n), like the differential energy, is negative in sign with units of J/mol. The constant $D_{\circ}$ is the differential energy at the limit of zero pressure. The assumption that the coefficients C _i and D _i in the power series are constants is equivalent to neglecting sensible heat in comparison to latent heat for the solid phase over a limited range of temperature. The energy $\overline{u}$ is relative to h ^° in the perfect-gas state at the reference temperature $T_{\circ}$.

The integral energy from Eq. (3.48) and the differential and integral entropies from Eqs. (3.45) and (3.49) are:

$$ u = D_{\rm int}(n) $$

(10.6)

$$ \frac{\overline s}{R} = -\ln\left[\frac{m}{P^\circ K}\right] - \ln\left[\frac{n}{m-n}\right] + \frac{D(n)}{RT_\circ} - C(n) $$

(10.7)

$$ \frac{s}{R} =- \ln\left[\frac{m}{P^\circ K}\right] -\ln\left[\frac{n}{m-n}\right] + \frac{m}{n}\ln\left[\frac{m}{m-n}\right] + \frac{D_{\rm int}(n)}{RT_\circ} - C _{\rm int}(n) $$

(10.8)

where

$$ D_{\rm int}(n) = \frac{\int D(n)dn}{n} = D_\circ + \frac{D_1}{2}n + \frac{D_2}{3}n^2 + \frac{D_3}{4}n^3 + \cdots $$

(10.9)

$$ C_{\rm int}(n) = \frac{\int C(n)dn}{n} = \quad\frac{C_1}{2}n + \frac{C_2}{3}n^2 + \frac{C_3}{4}n^3 + $$

(10.10)

C _int(n), like C(n), is dimensionless. D _int, like D(n), is a negative quantity with units of J/mol. The molar energy u is relative to the perfect-gas enthalpy h ^° and the molar entropy s is relative to the perfect-gas entropy $s^{\circ}$, both at the reference temperature $T_{\circ}$ in Eq. (10.1). The reference pressure $P^{\circ}$ = 1 bar in Eqs. (10.7) and (10.8).

1.3 Grand potential

Standard states for mixture equilibria are fixed by the reduced grand potential (ψ), which is obtained by integrating Eq. (3.16):

$$ \psi = \int\limits_0^f n \;d\ln f = \int\limits_0^n \frac{d\ln f}{d\ln n} dn\quad \hbox{(}\hbox{constant }\, T\hbox{)} $$

(10.11)

Substituting for the fugacity f from Eq. (10.1) followed by integration yields:

$$ \psi = m\ln\left(\frac{m}{m-n}\right) + \frac{D_\psi(n)}{R}\!\!\left(\frac{1}{T} - \frac{1}{T_\circ}\!\right) + C_\psi(n) $$

(10.12)

where the power series for ψ are:

$$ C_\psi(n) = \int\limits_0^n n C'(n) dn = \frac{1}{2}C_1 n^2 + \frac{2}{3} C_2 n^3 + \frac{3}{4}C_3 n^4+ \cdots $$

(10.13)

$$ D_\psi(n) = \int\limits_0^n n D'(n) dn = \frac{1}{2}D_1 n^2 + \frac{2}{3}D_2 n^3 + \frac{3}{4}D_3 n^4 + \cdots $$

(10.14)

C _ψ(n) has units of mol/kg and D _ψ(n) has units of J/kg. The domain of ψ is 0 ≤ n < m and the limit at low loading is:

$$ \lim_{n \to 0} \frac{\psi}{n} = 1 $$

1.4 Slope of adsorption isotherm

The calculation of differential energy or heat of adsorption of individual components of mixtures requires the slope of the adsorption isotherm evaluated at its standard state. From Eq. (10.1), the reciprocal of the dimensionless slope of an adsorption isotherm for a single gas is:

$$ \left[\frac{\partial\ln f}{\partial\ln n}\right]_T = \frac{m}{m-n} + \frac{D_s(n)}{R}\!\!\left(\!\frac{1}{T} - \frac{1}{T_\circ}\!\!\right) + C_s(n) $$

(10.15)

The standard state notation is simplified by writing f ^* _i as f and n ^* _i as n. $T_{\circ}$ is the reference temperature for the determination of the C _i coefficients in Eq. (10.1). The power series for Eq. (10.15) are:

$$ C_s(n) = C_1 n + 2 C_2 n^2 + 3 C_3 n^3 + \cdots $$

(10.16)

$$ D_s(n) = D_1 n + 2 D_2 n^2 + 3 D_3 n^3 + \cdots $$

(10.17)

C _s (n) is dimensionless and D _s (n) has units of J/mol. The limit at zero loading is:

$$ \lim_{n\to 0} \left[\frac{\partial\ln f}{\partial\ln n}\right]_T = 1 $$

(10.18)

1.5 Example

The first step is to select an isotherm measured at a reference temperature ($T_{\circ}$) suitable for extracting the constants m, $K_{\circ}$, and the C _i coefficients. Equation (4.21) written in the form:

$$ \overline{u} = R \left[\frac{\partial\ln f}{\partial (1/T)}\right]_n $$

(10.19)

exploits the near-linearity of $\ln f$ versus 1/T at constant n to determine values of the differential energy, which are fit with the D _i coefficients. An example is shown in Fig. 15, which compares Eq. (10.1) with adsorption isotherms measured at 25, 50 and 100 °C. The other isotherms at 0, 75, and 125 °C are interpolations and extrapolations.

Having fit the adsorption isotherms with Eq. (10.1), the thermodynamic functions are given by explicit equations. On Fig. 16 are plotted the integral and differential functions calculated from the equations in this Appendix. The functions {S, U, F} are integrals with respect to n of the differential functions $\{\overline{s}, \overline{u}, \mu\}$. In addition, $F = U - T S, \mu = \overline{u} - T\overline{s}$, and $(F/ n) = \mu + (\Upomega/n)$.

Fig. 16

Properties for adsorption of ethylene on 13X molecular sieve at 298.15 K derived from experimental adsorption isotherms plotted on Fig. 15. Values of energy and entropy relative to perfect-gas state for gases and fully evacuated adsorbent for solid, both at 298.15 K. $\mu = \overline{u} - T\overline{s}$. $F = U - TS = \mu n + \Upomega$

The μ function is the adsorption isotherm at 298.15 K. Equation (10.1) is based upon the simplification that U and S are independent of temperature, an approximation justified for the temperature range of the experimental data on Fig. 15.

All functions are negative in sign. Entropy and free energy functions are undefined at the limit of zero loading. The energy is finite at zero loading corresponding to the energy of a single molecule interacting with the solid. The grand potential $(\Upomega)$ has a finite value (−n R T) at zero loading, where $(\Upomega/n) = -RT = -2.48$ kJ/mol.

Appendix 2: Simultaneous solution of fugacity equations

2.1 Ideal solution

The objective is the solution of the fugacity Eq. (4.16) for ψ and composition x. For an ideal binary mixture with x ₂ = (1 − x ₁), we seek the simultaneous solution of the pair of fugacity equations at temperature T:

$$ \left[ \begin{array}{l} F_1(x_1,\psi) \\ F_2(x_2,\psi) \end{array} \right] = \left[ \begin{array}{l} \ln ({f_1^* x_1}/{f_1}) \\ \ln ( {f_2^* x_2}/{f_2}) \end{array} \right] = \left[ \begin{array}{l} 0 \\ 0 \end{array} \right] $$

(11.1)

for which the Jacobian is:

$$ \left[ \begin{array}{ll} \left(\frac{\partial F_1}{\partial x_1}\right)_\psi & \left(\frac{\partial F_1}{\partial\psi}\right)_{x_1} \\ \left(\frac{\partial F_2}{\partial x_1}\right)_\psi & \left(\frac{\partial F_2}{\partial\psi}\right)_{x_1} \end{array} \right] = \left[ \begin{array}{ll} \frac{1}{x_1}& \frac{1}{n_1^*} \\ -\frac{1}{x_2} & \frac{1}{n_2^*} \end{array}\right] $$

(11.2)

The size of the Jacobian is equal to the number of gaseous components. Solution of the linear equations:

$$ \left[ \begin{array}{ll} \frac{1}{x_1} &\frac{1}{n_1^*} \\ -\frac{1}{x_2} & \frac{1}{n_2^*} \end{array} \right] \left[ \begin{array}{l} \Updelta x_1 \\ \Updelta \psi \end{array}\right] = \left[ \begin{array}{l} F_1 \\ F_2 \end{array} \right] $$

(11.3)

gives successive iterations according to Newton’s method:

$$ (x_1)_{i+1}= (x_1)_i - \Updelta x_1 $$

(11.4)

$$ \psi_{i+1} = \psi_i - \Updelta\psi $$

(11.5)

If the single-gas isotherms obey Eq. (10.1), a suitable first guess is:

$$ x_i = \frac{K_i f_i}{\sum_i [K_i f_i]} $$

(11.6)

$$ \psi = \frac{\sum_i [ f_i m_i \ln(1 + K_i f_i/m_i)]}{\sum_i f_i} $$

(11.7)

K _i is the Henry constant for the ith component from Eq. (10.3).

Two pure-component functions are needed: n ^* _i (ψ) (see Appendix 3) and the composite function f ^* _i [n ^* _i (ψ)].

2.2 Activity coefficients

Insertion of activity coefficients into Eq. (11.1) gives:

$$ \left[ \begin{array}{l} F_1(x_1,\psi) \\ F_2(x_2,\psi) \end{array} \right] = \left[ \begin{array}{l} \ln ({f_1^* \gamma_1 x_1}/{f_1}) \\ \ln ( {f_2^*\gamma_2 x_2}/{f_2}) \end{array} \right] = \left[ \begin{array}{l} 0 \\ 0 \end{array} \right] $$

(11.8)

Using Eqs. (5.31), (5.32) and (5.33), the Jacobian is:

$$ \left[ \begin{array}{ll} \left(\frac{\partial F_1}{\partial x_1}\right)_\psi & \left(\frac{\partial F_1}{\partial\psi}\right)_{x_1} \\ \left(\frac{\partial F_2}{\partial x_1}\right)_\psi & \left(\frac{\partial F_2}{\partial\psi}\right)_{x_1} \end{array}\right] = \left[ \begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right] $$

(11.9)

where:

$$ \begin{aligned} a_{11} &=\frac{1}{x_1} -\frac{(A+ B T)}{R T}(1- e^{-C\psi}) 2 x_2\\ a_{21} &= -\frac{1}{x_2} + \frac{(A+ B T)}{R T}(1- e^{-C\psi}) 2 x_1 \\ a_{12} &= \frac{1}{n_1^*} + \frac{(A + B T)}{R T} C e^{-C\psi} x_2^2 \\ a_{22} &= \frac{1}{n_2^*} + \frac{(A + B T)}{R T} C e^{-C\psi} x_1^2 \\ \end{aligned} $$

Except for the more complicated form of the Jacobian, Eqs. (11.3)–(11.7) are unchanged.

Appendix 3: Inversion of grand potential function for mixture equilibria

Equation (10.12) is the function ψ(n) for single-gas adsorption but its inverse function is needed to fix the standard state. Define $\mathcal{P}(n) \equiv \psi(n)$ and find the solution of the function $\mathcal{H}(n)=0$:

$$ {\mathcal{H}}(n) = {\mathcal{P}}(n) - \psi = 0 $$

(12.1)

Solve Eq. (12.1) by Newton’s method using:

$$ n_{j+1} = n_j - \frac{{\mathcal{H}}}{{\mathcal{H'}}} $$

(12.2)

with

$$ \begin{aligned} {\mathcal{H}}' = \psi'(n) &= \frac{m}{m-n} + (C_1 n + 2 C_2 n^2 + 3 C_3 n^3 + \cdots)\\ & + \frac{1}{R}\left(\frac{1}{T} - \frac{1}{T^\circ}\right) (D_1 n + 2 D_2 n^2 + 3 D_3 n^3 + \cdots) \end{aligned} $$

(12.3)

A suitable first guess is n = m(1 − e ^−ψ/m). ψ increases monotonically with n and the iterative solution for the inverse function n(ψ) converges rapidly. The above notation for n is a simplification for n ^* _i , the standard-state loading of single gas i. ψ and its standard-state properties (f ^* _i , u ^* _i , s ^* _i ) are all explicit functions of n ^* _i .