1. Isotherm Selection Criteria Showing Isotherm Types (I–VII), Various Simplifications of eq 1, and the Resulting Relationships between the Specified, Mechanistically Based, Isotherm Models.

model	isotherm types and features	relationship between $\mathit{\theta }=\frac{\mathit{C}}{{\mathit{C}}_{\mathbf{SAT}}}$ and $\mathit{a}=\frac{\mathit{P}}{{\mathit{P}}_{\mathbf{V}}}$	condition	H
CMMS	hybrid isotherm types (e.g., Figure h,i)	$\theta =\frac{K_{0}a}{1-K_{\mathrm{as}}a[K_{0}a+w^2(1-K_{\mathrm{as}}a)]}$ with $w=\frac{1}{2}\left\{1-\frac{K_{1}a}{(1-K_{\mathrm{as}}a)}+\sqrt{{\left(1-\frac{K_{1}a}{1-K_{\mathrm{as}}a}\right)}^2+\frac{4K_{0}a}{1-K_{\mathrm{as}}a}}\right\}$
Henry	VII (linear: Figure g)	θ = K 0 a	a → 0
BET	II (downward concavity at low pressure: Figure b)	$\theta =\frac{K_{0}a}{[1-a][1-a+K_{0}a]}$	large K o = K 1; K as = 1	H = 1
GAB	II (downward concavity at low pressure: shape of Figure b)	$\theta =\frac{K_{0}a}{[1-K_{\mathrm{as}}a][1+(K_0-K_{\mathrm{as}})a]}$	large K o = K 1; 0 ≤ K as ≤ 1	H = 1
GAB	III (upward concavity: Figure c)	$\theta =\frac{K_{0}a}{[1-K_{\mathrm{as}}a][1+(K_0-K_{\mathrm{as}})a]}$	small K o = K 1; 0 ≤ K as ≤ 1	H = 1
Langmuir	I (downward concavity: Figure a)	$\theta =\frac{K_{0}a}{[1+K_{0}a]}$	large K o = K 1; K as = 0	H = 1
Rectangular	I (sharpened shape of Figure a)	θ = 1	K o = K 1 → ∞; K as = 0	H = 1
Ising	V (upward concavity at low pressures turns to downward concavity: Figure d,e)	$\theta =\frac{K_{0}a}{K_{0}a+\frac{1}{4}{\{1-K_{1}a+\sqrt{{(1-K_{1}a)}^2+4K_{0}a}\}}^2}$	K as = 0; K 1 > 1; K o < K 1	H > 1
Ising	III (upward concavity: shape of Figure c)	$\theta =\frac{K_{0}a}{K_{0}a+\frac{1}{4}{\{1-K_{1}a+\sqrt{{(1-K_{1}a)}^2+4K_{0}a}\}}^2}$	K as = 0; K 1 < 1; K o < K 1	H > 1
Ising	I (downward concavity: shape of Figure a)	$\theta =\frac{K_{0}a}{K_{0}a+\frac{1}{4}{\{1-K_{1}a+\sqrt{{(1-K_{1}a)}^2+4K_{0}a}\}}^2}$	K as = 0; K o > K 1	H < 1
Ising	VI (unimodal)/Stepwise V (Figure f)	$a=\frac{1}{K_1}$	K as = 0; K o ≪ K 1	H → ∞