According to Freundlich adsorption isotherm, which of the following is correct?

Options

A. `(x)/(m) prop P^(0)`
B. `(x)/(m) prop P^(1)`
C. `(x)/(m) prop P^(1//n)`
D. All the above are correct for different ranges of pressure

Answer

To solve the question regarding the Freundlich adsorption isotherm, we will analyze the relationship between the amount adsorbed (X) per unit mass of adsorbent (M) and the pressure (P) based on the provided options.

### Step-by-Step Solution:

1. **Understanding Freundlich Adsorption Isotherm**:
The Freundlich adsorption isotherm is an empirical relationship that describes how the amount of substance adsorbed on a surface varies with the pressure of the gas in equilibrium with the adsorbent. The general form of the Freundlich isotherm is:
\[ \frac{X}{M} = k P^{\frac{1}{n}} \]
where \( \frac{X}{M} \) is the amount adsorbed per unit mass, \( P \) is the pressure, \( k \) is a constant, and \( n \) is a parameter that indicates the adsorption intensity.

2. **Analyzing the Options**:
- **Option A**: \( \frac{X}{M} \propto P^0 \) (This indicates independence from pressure, valid at high pressure)
- **Option B**: \( \frac{X}{M} \propto P^1 \) (This indicates a linear relationship, valid at low pressure when \( n = 1 \))
- **Option C**: \( \frac{X}{M} \propto P^{\frac{1}{n}} \) (This is the general form of the Freundlich isotherm, valid at intermediate pressures)
- **Option D**: All the above are correct for different ranges of pressure.

3. **Identifying the Correct Answer**:
- At **low pressure**, \( \frac{X}{M} \) is directly proportional to \( P^1 \) (Option B).
- At **intermediate pressure**, \( \frac{X}{M} \) is proportional to \( P^{\frac{1}{n}} \) (Option C).
- At **high pressure**, \( \frac{X}{M} \) becomes independent of pressure, which corresponds to \( P^0 \) (Option A).
- Therefore, all options A, B, and C are correct under different conditions of pressure.

4. **Conclusion**:
The correct answer is **Option D**: All the above are correct for different ranges of pressure.