Adsorption of a gas on a surface follows Freundlich adsorption isotherm. Plot of `"log " (x)/(m)` versus log P gives a straight line with slope equal to 0.5, then :
(`(x)/(m)` is the mass of the gas adsorbed per gram of adsorbent)

Options

A. Adsorption is proportional to the pressure.
B. Adsorption is proportional to the square root of pressure.
C. Adsorption is proportional to the square of pressure.
D. Adsorption is independent of pressure.

Answer

To solve the problem, we need to analyze the information provided about the adsorption of a gas on a surface that follows the Freundlich adsorption isotherm. Here’s a step-by-step breakdown of the solution:

### Step 1: Understanding the Freundlich Adsorption Isotherm
The Freundlich adsorption isotherm can be expressed as:
\[ \frac{x}{m} = k P^{\frac{1}{n}} \]
where:
- \( \frac{x}{m} \) is the mass of the gas adsorbed per gram of adsorbent,
- \( P \) is the pressure of the gas,
- \( k \) is a constant,
- \( n \) is a constant that indicates the degree of non-linearity of the adsorption.

### Step 2: Taking Logarithms
To analyze the relationship, we take the logarithm of both sides:
\[ \log\left(\frac{x}{m}\right) = \log(k) + \frac{1}{n} \log(P) \]
This equation can be compared to the linear equation \( y = mx + c \), where:
- \( y = \log\left(\frac{x}{m}\right) \),
- \( x = \log(P) \),
- \( m = \frac{1}{n} \) (the slope),
- \( c = \log(k) \) (the y-intercept).

### Step 3: Analyzing the Slope
According to the problem, the slope of the plot of \( \log\left(\frac{x}{m}\right) \) versus \( \log(P) \) is given as 0.5. Therefore:
\[ \frac{1}{n} = 0.5 \]
From this, we can find \( n \):
\[ n = \frac{1}{0.5} = 2 \]

### Step 4: Substituting Back into the Isotherm
Now, substituting \( n = 2 \) back into the Freundlich isotherm equation:
\[ \frac{x}{m} = k P^{\frac{1}{2}} = k \sqrt{P} \]
This indicates that the mass of gas adsorbed per gram of adsorbent is proportional to the square root of the pressure.

### Step 5: Conclusion
From the analysis, we conclude that:
\[ \frac{x}{m} \propto \sqrt{P} \]
Thus, the correct statement is that adsorption is proportional to the square root of pressure.

### Final Answer
The correct option is that adsorption is proportional to the square root of pressure.

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