Question

The Langmuir isotherm for the adsorption of a gas on a solid surface can be expressed as $\theta = \frac{Kp}{1+Kp}$The correct statement(s) about this isotherm is/are

• A. At very low pressures, plot of $\theta$ against $p$ is a straight line passing through the origin with slope equal to $K$
• B. At very high pressures, plot of $\theta$ against $p$ is a straight line parallel to the x-axis with the value of the y-intercept equal to 1
• C. The Langmuir isotherm can also be expressed as $\frac{1}{\theta} = 1 + \frac{1}{Kp}$
• D. The Langmuir isotherm is applicable for multilayer adsorption

Hint:

Langmuir isotherm is for monolayer adsorptionAt low pressure, $\theta \propto p$ and at high pressure, $\theta$ approaches 1

Answer 1

The Correct Option is A, B, C

Step 1: Write the given Langmuir adsorption isotherm.
\[ \theta = \frac{Kp}{1+Kp} \]
Here, $\theta$ represents the fraction of surface covered by adsorbed gas molecules

Step 2: Consider the case of very low pressure.
At very low pressure, $Kp << 1$Therefore,
\[ 1+Kp \approx 1 \]
\[ \theta = \frac{Kp}{1+Kp} \approx Kp \]

Step 3: Check statement (A).
Since $\theta = Kp$, the plot of $\theta$ against $p$ is a straight line passing through origin with slope $K$Thus, statement (A) is correct

Step 4: Consider the case of very high pressure.
At very high pressure, $Kp >> 1$Therefore,
\[ 1+Kp \approx Kp \]
\[ \theta = \frac{Kp}{Kp} = 1 \]

Step 5: Check statement (B).
At very high pressure, $\theta$ becomes constant and equal to 1So the plot of $\theta$ against $p$ becomes parallel to the x-axis with y-intercept 1Thus, statement (B) is correct

Step 6: Rearrange the isotherm.
\[ \theta = \frac{Kp}{1+Kp} \]
Taking reciprocal on both sides:
\[ \frac{1}{\theta} = \frac{1+Kp}{Kp} \]
\[ \frac{1}{\theta} = 1 + \frac{1}{Kp} \]
Thus, statement (C) is correct

Step 7: Check multilayer adsorption condition.
Langmuir isotherm assumes monolayer adsorption on a homogeneous surfaceIt is not applicable for multilayer adsorptionMultilayer adsorption is better explained by BET isothermTherefore, statement (D) is incorrect
\[ \boxed{\text{(A), (B) and (C)}} \]