Variable temperature selectivities

[Gotzias]

When isotherms for single-component adsorption are available at three or more temperatures, the relationship between pressure ( p ) and uptake ( q ) (or coverage) for an isotherm at temperature ( T ) can be described by the Clausius-Clapeyron equation\cite{Gregg_1967,Builes2013}:
$$(ln:p){q,T}=-\frac{(Q{st})q}{RT};+;constant$$
Where ((Q{st})q) denotes the isosteric heat of adsorption at a specific coverage ( q ). We define (n) equally spaced (q_i) values within the interval ([q_1, q_n]) and calculate the heat of adsorption ((Q{st})q_i) for each coverage level in this series. In this context, (q_n) is the maximum coverage of the highest temperature isotherm, while (q_1) represents the greatest of the minimum (non-zero) coverages from the available isotherms. With the temperatures of the available isotherms being (T_1), (T_2), and (T_3), we apply least squares to fit straight lines through the points: A((T_1^{-1}), (\ln p_{(q_i, T_1)})), B((T_2^{-1}), (\ln p_{(q_i, T_2)})), and C((T_3^{-1}), (\ln p_{(q_i, T_3)})) for each (i) from 1 to (n). From these lines, we derive a slope (S_{q_i}) and an intercept (I_{q_i}) for each index (i).