The following Eqs. (1)–(4) are the numerical expressions of the IAS model. The superscript 0 means the value of a single gas, and the subscripts i and j mean the gas species. In the IAS model, the equilibrium adsorption amount is calculated by integrating the increment of adsorbed gas molecules under the assumption that the spreading pressure π, which is defined as the increment in the surface tension of a surface due to the spreading of an adsorbate over a surface, is assumed to have a relationship with the increment of the amount of adsorbates [19].
$$\Pi = \frac{\pi A}{RT} = \mathop \int \limits_{0}^{{p_{i}^{0} }} \frac{{q_{i}^{0} }}{{p_{i}^{0} }}dp_{i}^{0}$$
(1)
$$\frac{1}{{q_{t} }} = \mathop \sum \limits_{j = 1}^{n} \frac{{Z_{j} }}{{q_{j}^{0} }}$$
(2)
$$Z_{i} = \frac{{q_{i} }}{{q_{t} }}$$
(3)
$$Z_{i} = \frac{{p_{i} }}{{p_{i}^{0} }}$$
(4)
The various isotherm equations of single gases can be applied to the IAS model, setting aside the convergence property of the iterative calculation for the deduction of Π in Eq. (1). In this case, the IAS model with the following Langmuir equation Eq. (5) is considered.
$$q_{i}^{0} = \frac{{q_{\infty i} K_{i} p_{i}^{0} }}{{1 + K_{i} p_{i}^{0} }}$$
(5)
By integrating Eqs. (1) and (5), the following Eqs. (6) and (7) are obtained.
$$\Pi = q_{\infty i} { \ln }\left( {1 + K_{i} p_{i}^{0} } \right)$$
(6)
$${ \exp }\left( {\frac{\Pi }{{q_{\infty i} }}} \right) = 1 + K_{i} p_{i}^{0}$$
(7)
Since the right side of Eq. (7) becomes a direct substitution in the Langmuir equation, the gas adsorption amount of a single gas expressed by Eq. (8) can be obtained by the simultaneous equations of Eqs. (5) and (7).
$$q_{i}^{0} = \frac{{q_{\infty i} \left\{ {exp\left( {\frac{\Pi }{{q_{\infty i} }}} \right) - 1} \right\}}}{{exp\left( {\frac{\Pi }{{q_{\infty i} }}} \right)}}$$
(8)
The following Eq. (9) is obtained from Eqs. (2), (3), and (8).
$$1 = \mathop \sum \limits_{j = 1}^{n} \frac{{q_{j} \exp \left( {\frac{\Pi }{{q_{\infty j} }}} \right)}}{{q_{\infty j} \left\{ {\exp \left( {\frac{\Pi }{{q_{\infty j} }}} \right) - 1} \right\}}}$$
(9)
On the assumption that the mixed gas consists of two gas components having the same saturation capacity (q∞1 = q∞2), Eq. (9) can be deformed to Eq. (10) for derivation of the reduced spreading pressure Π.
$$\Pi = q_{\infty 1} { \ln }\left( {\frac{{ - q_{\infty 1} }}{{q_{1} + q_{2} - q_{\infty 1} }}} \right)$$
(10)
For gas component 1 (i = 1), the following Eq. (11) is obtained from Eqs. (6) and (10).
$$\frac{{ - q_{\infty 1} }}{{q_{1} + q_{2} - q_{\infty 1} }} = 1 + K_{1} p_{1}^{0}$$
(11)
When the mixed gas consists of two gas components, the total adsorption amount qt equals the simple summation of q1 and q2 described as Eq. (12). Then, Eqs. (11), (12), and the following Eq. (13), which is derived from Eqs. (3) and (4), are integrated to obtain Eq. (14).
$$q_{t} = q_{1} + q_{2}$$
(12)
$$p_{1}^{0} = \left( {\frac{{q_{t} }}{{q_{1} }}} \right)p_{1}$$
(13)
$$p_{1} = \frac{{q_{1} }}{{K_{1} \left( {q_{\infty 1} - q_{t} } \right)}}$$
(14)
The same result can be obtained for gas component 2, and the following Eq. (15) is then obtained by using the pressure ratio of these gas components.
$$K_{2} p_{2} = \left( {\frac{{q_{2} }}{{q_{1} }}} \right)K_{1} p_{1}$$
(15)
The equilibrium adsorption amount given by Eq. (16) can be derived from Eqs. (12), (14), and (15).
$$q_{1} = \frac{{q_{\infty 1} K_{1} p_{1} }}{{1 + K_{1} p_{1} + K_{2} p_{2} }}$$
(16)
When the above equation is generalized, it becomes the M–B equation Eq. (17).
$$q_{i} = \frac{{q_{\infty } K_{i} p_{i} }}{{1 + \mathop \sum \nolimits_{j} K_{j} p_{j} }}$$
(17)
This means that the IAS-LM model is completely equivalent to the M–B equation when the saturation capacity q∞ is uniform, regardless of the type of adsorbate. This result has the clear advantage of using the M–B equation prior to the IAS-LM model because it has the same thermodynamic consistency as the IAS-LM model but a smaller calculation load, as its analytical formula does not require convergent calculations.
On the other hand, a generalized equation including q∞i is desired in cases having different q∞i for each gas component (q∞1 ≠ q∞2). In these cases, the following approximation can be applied in the calculation.
The relationship expressed by Eq. (18) can be obtained by using the first and second terms of the Taylor expansion of the left side of Eq. (7).
$$\begin{aligned} { \exp }\left( {\frac{\Pi }{{q_{\infty i} }}} \right) & = 1 + \frac{{\frac{\Pi }{{q_{\infty i} }}}}{1!} + \frac{{\left( {\frac{\Pi }{{q_{\infty i} }}} \right)^{2} }}{2!} + \frac{{\left( {\frac{\Pi }{{q_{\infty i} }}} \right)^{3} }}{3!} + \ldots \cong 1 + \frac{\Pi }{{q_{\infty i} }} \\ 1 + \frac{\Pi }{{q_{\infty i} }} & = 1 + K_{i} p_{i}^{0} \\ \end{aligned}$$
(18)
Substitution of Eq. (13) in Eq. (18) gives the following Eq. (19).
$$\begin{aligned} & 1 + \frac{\Pi }{{q_{\infty 1} }} = 1 + \left( {\frac{{q_{t} }}{{q_{1} }}} \right)K_{1} p_{1} \\ &\Pi = \frac{{q_{t} K_{1} p_{1} q_{\infty 1} }}{{q_{1} }} \\ \end{aligned}$$
(19)
The approximation of Eq. (18) can also be applied to Eq. (9) and gives Eq. (20).
$$\Pi = \frac{{q_{\infty 1} q_{\infty 2} q_{t} }}{{q_{\infty 1} q_{\infty 2} - q_{1} q_{\infty 2} - q_{2} q_{\infty 1} }}$$
(20)
Equation (21) can be obtained from Eqs. (19) and (20).
$$K_{1} p_{1} = \frac{{q_{1} q_{\infty 2} }}{{q_{\infty 1} q_{\infty 2} - q_{1} q_{\infty 2} - q_{2} q_{\infty 1} }}$$
(21)
Because Eq. (19) is equally valid for gas component 2, the following Eq. (22) can be derived from the ratio of q1 and q2.
$$q_{2} = \frac{{q_{1} K_{2} p_{2} q_{\infty 2} }}{{K_{1} p_{1} q_{\infty 1} }}$$
(22)
The following Eq. (23) can be obtained by substitution of Eq. (21) in Eq. (22).
$$q_{1} = \frac{{q_{\infty 1} K_{1} p_{1} }}{{1 + K_{1} p_{1} + K_{2} p_{2} }}$$
(23)
When the above equation is generalized, it becomes the EX-LM equation shown in Eq. (24). In the case of having equal saturation capacities q∞, Eq. (24) is identical to the M–B equation.
$$q_{i} = \frac{{q_{\infty i} K_{i} p_{i} }}{{1 + \mathop \sum \nolimits_{j} K_{j} p_{j} }}$$
(24)
Equation (24) is a more generalized equation than Eq. (17) because it allows the difference of the saturation capacity q∞i for the component gases. According to the above relationship, the EX-LM equation that was derived by approximation of Eq. (18) is also applicable as an analytical formula of the IAS-LM model that has thermodynamic consistency for calculation of mixed gas adsorption. Similar derivations of the explicit forms of the IAS model with various single gas isotherms were attempted by Tarafder et al. [35]. However, they assumed strict uniformity of the saturation capacity q∞i for all the component gases in order to eliminate the logarithmic form derived after the integration of the right side of Eq. (1). This assumption restricts the use of their equations to limited cases. As mentioned previously, the saturation capacity q∞i is generally not constant and depends on the gas species. Therefore, the EX-LM equation can be used more generally than their equations. The fact that fewer parameters are required for the calculation is also an advantage of this model. As a result, the EX-LM equation is preferably used for calculation of the equilibrium adsorption amount of mixed gases in dynamic simulations owing to its analytical formula, which does not require convergent calculations.