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, \( p \) is pressure, and \( k, n \) are empirical constants); and
(ii) the BET isotherm is given by \[ \frac{p}{p^{\ast}-p} = \frac{\theta}{c} + \theta(c-1) \left( \frac{p}{p^{\ast}} \right) \] (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 \(p⇒ \theta\propto p\) (Freundlich with \(n=1\)); high \(p⇒ \theta\to\theta_s\) (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

To address the given question, we need to analyze how the Freundlich and BET isotherms compare with the Langmuir isotherm under specific conditions.

The Langmuir isotherm for adsorption is described by:

\[\theta = \frac{kp}{1+kp}\]

where \( \theta \) is the surface coverage, \( p \) is pressure, and \( k \) is a constant.

Now let's examine the scenarios:

1. At low surface coverage:
   Under low pressure or low surface coverage conditions, the Langmuir isotherm can be approximated as:

\[\theta \approx kp\]

This resembles the Freundlich isotherm form \( \theta = k\,p^{1/n} \) when \( n=1 \), indicating a linear relationship at low surface coverage. This makes the statement "At low surface coverage, the Langmuir isotherm reduces to the Freundlich isotherm with \( n=1 \)" correct.

2. At high surface coverage:
   As \( \theta \) approaches 1, the Langmuir isotherm becomes saturated, implying no resemblance to the Freundlich isotherm. Hence, the statement regarding reduction at high surface coverage is incorrect.
3. At very low pressure (\(p\ll p^{\ast}\)):
   The BET isotherm is:

\[\frac{p}{p^{\ast}-p} = \frac{\theta}{c} + \theta(c-1) \left( \frac{p}{p^{\ast}} \right)\]

Approximating for low \( p \), \(\frac{p}{p^{\ast}-p} \approx \frac{\theta}{c}\), resembling the Langmuir form. Hence this statement is correct.

4. At very high pressure (\(p\to p^{\ast}\)):
   The BET isotherm diverges, thus it doesn't simplify to the Langmuir isotherm pattern, making this statement incorrect.

Thus, the correct statement from the options is: "At low surface coverage, the Langmuir isotherm reduces to the Freundlich isotherm with \(n=1\)"