Question

For the non-dissociative adsorption of a gas on solid,
(i) the Freundlich isotherm is given by \(\theta = k\,p^{1/n}\) (where \(\theta\) is surface coverage, \(p\) pressure, \(k,n\) empirical constants); and
(ii) the BET isotherm is given by \(\displaystyle \frac{p}{p^{\ast}-p}=\frac{\theta}{c}+\theta(c-1)\left(\frac{p}{p^{\ast}}\right)\)
(where \(p^{\ast}\) and \(c\) are empirical constants, and \(p<p^{\ast}\)).
The correct statement(s) is(are)

• A. At low surface coverage, the Langmuir isotherm reduces to the Freundlich isotherm with \(n=1\)
• B. At high surface coverage, the Langmuir isotherm reduces to the Freundlich isotherm with \(n=\infty\)
• C. At very low pressure (\(p\ll p^{\ast}\)), the BET isotherm reduces to the Langmuir isotherm
• D. At very high pressure (\(p\to p^{\ast}\)), the BET isotherm reduces to the Langmuir isotherm

Hint:

Langmuir limits: low \(p\Rightarrow \theta\propto p\) (Freundlich with \(n=1\)); high \(p\Rightarrow \theta\to\theta_s\) (Freundlich with \(n\to\infty\), \(k=\theta_s\)).
BET reduces to Langmuir only in the very-low-pressure (monolayer) regime.

Answer 1

The Correct Option is A, B, C

Step 1: Langmuir at low coverage.
Langmuir: \(\displaystyle \theta=\frac{Kp}{1+Kp}\). For \(Kp\ll1\), \(\theta\approx Kp\), i.e., \(\theta\propto p\). This is the Freundlich form with \(n=1\) (and \(k=K\)). Hence (A) is true.
Step 2: Langmuir at high coverage.
For \(Kp\gg1\), \(\theta\to\theta_s\) (saturation, a constant). The Freundlich expression \(\theta=k\,p^{1/n}\) with \(n\to\infty\) gives \(\theta=k\,p^{0}=k\) (a constant). Choosing \(k=\theta_s\) reproduces the Langmuir high-coverage limit. Thus, in this limiting sense, (B) is true.
Step 3: BET at very low pressure.
From the given BET form, \[ \frac{p}{p^{\ast}-p}=\frac{\theta}{c}+\theta(c-1)\left(\frac{p}{p^{\ast}}\right). \] If \(p\ll p^{\ast}\), then \(\dfrac{p}{p^{\ast}-p}\approx\dfrac{p}{p^{\ast}}\) and the term \(\theta(c-1)\dfrac{p}{p^{\ast}}\) is negligible. Hence \[ \frac{p}{p^{\ast}}\approx \frac{\theta}{c}\quad\Rightarrow\quad \theta\approx \frac{c}{p^{\ast}}\,p, \] which is the Langmuir (monolayer, linear low-\(p\)) limit. Therefore (C) is true.
Step 4: BET near saturation.
As \(p\to p^{\ast}\), multilayer adsorption dominates and the BET equation does not collapse to the monolayer Langmuir form; thus (D) is false.