For Freundlich adsorption isotherm, `(x)/(m)=kp^1//n` , the value of n is

Options

A. 1. always greater than one
B. 2. always smaller than one
C. 3. always equal to one
D. 4. greater than one at low temperature and smaller than one at high temperature.

Explanation

To determine the value of \( n \) in the Freundlich adsorption isotherm equation given by

\[ \frac{x}{m} = k p^{\frac{1}{n}} \]

we can analyze the behavior of the adsorption process at different pressures.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation \( \frac{x}{m} = k p^{\frac{1}{n}} \) describes the relationship between the extent of adsorption (\( \frac{x}{m} \)), the pressure of the gas (\( p \)), and the constants \( k \) and \( n \). Here, \( x \) is the amount of gas adsorbed, \( m \) is the mass of the adsorbent, and \( k \) and \( n \) are constants that depend on the nature of the adsorbate and adsorbent.

2. **Behavior at Low Pressure**:
At low pressures, the extent of adsorption (\( \frac{x}{m} \)) is directly proportional to the pressure (\( p \)). This can be expressed as:
\[ \frac{x}{m} \propto p \]
In this case, we can see that when \( p \) is low, \( \frac{1}{n} \) approaches 1, which implies:
\[ n = 1 \]

3. **Behavior at High Pressure**:
At high pressures, the system reaches saturation pressure, where the amount of gas adsorbed becomes constant regardless of further increases in pressure. Thus, the extent of adsorption becomes independent of pressure:
\[ \frac{x}{m} = k p^0 \]
This indicates that \( p^{\frac{1}{n}} \) approaches a constant value, leading to:
\[ \frac{1}{n} \rightarrow 0 \quad \text{or} \quad n \rightarrow \infty \]

4. **Intermediate Pressures**:
Between low and high pressures, the value of \( n \) varies continuously. The value of \( n \) decreases from 1 (at low pressure) to a very high value (approaching infinity) as the pressure increases. Therefore, \( n \) can take values greater than 1 but less than infinity.

5. **Conclusion**:
Based on the analysis, the value of \( n \) is always greater than 1. Hence, the correct answer is that \( n \) is always greater than 1.

### Final Answer:
The value of \( n \) is always greater than 1.