A plot of log `(x/m)` against log P for the adsorption of a gas on a sloid gives a straight a straight line with slope equal to :

Options

A. `1/n`
B. `n`
C. `log K`
D. `K`

Answer

To solve the problem of finding the slope of the plot of log `(x/m)` against log `P` for the adsorption of a gas on a solid, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adsorption Isotherm**: The Freundlich adsorption isotherm describes how the amount of gas adsorbed on a solid surface varies with pressure. It is given by the equation:
\[ \frac{x}{m} \propto P^{\frac{1}{n}} \]
where \( \frac{x}{m} \) is the amount of gas adsorbed per unit mass of the solid, \( P \) is the pressure, and \( n \) is a constant.

2. **Express the Relationship Mathematically**: We can express the proportionality as an equation:
\[ \frac{x}{m} = k P^{\frac{1}{n}} \]
where \( k \) is a constant.

3. **Take the Logarithm of Both Sides**: To analyze the relationship in a linear form, we take the logarithm of both sides:
\[ \log\left(\frac{x}{m}\right) = \log(k) + \frac{1}{n} \log(P) \]

4. **Identify the Linear Form**: The equation can be rearranged to match the linear form \( y = mx + c \):
- Here, \( y = \log\left(\frac{x}{m}\right) \)
- \( x = \log(P) \)
- The slope \( m \) is \( \frac{1}{n} \)
- The intercept \( c \) is \( \log(k) \)

5. **Determine the Slope**: From the linear equation, we can conclude that the slope of the plot of \( \log\left(\frac{x}{m}\right) \) against \( \log(P) \) is:
\[ \text{slope} = \frac{1}{n} \]

6. **Select the Correct Option**: Given the options:
- 1) \( \frac{1}{n} \)
- 2) \( n \)
- 3) \( \log(k) \)
- 4) \( k \)

The correct answer is option 1, \( \frac{1}{n} \).

### Final Answer:
The slope of the plot of log `(x/m)` against log `P` for the adsorption of a gas on a solid is \( \frac{1}{n} \).

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