A chemist studied the phenomenon of adsorption by putting blood charcoal in `KCL` solution. He observed difference in the behaviour with dilute `KCL` solution and with concentrated `KCL` solution. He also studied the adsorption of different gases on solid adsorbent and the effect of temperature on adsorption. He put forward a mathematical relationship relating `x//m` with equilibrium pressure.
Which of the following plot will be liner? (More than one correct)

Options

A. Plot of log `x//m` versus P
B. Plot of `m//x` versus `1//P`
C. Plot of log `(P)/(m//m)` versus P
D. Plot of log `m//x` versus log P

Explanation

To determine which plots will be linear based on the adsorption isotherm, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Adsorption Isotherm**:
The Freundlich adsorption isotherm is given by the equation:
\[ \frac{x}{m} = k P^{\frac{1}{n}} \]
where \(x\) is the mass of adsorbate, \(m\) is the mass of adsorbent, \(P\) is the equilibrium pressure, \(k\) is a constant, and \(n\) is a constant related to the adsorption intensity.

2. **Rearranging the Equation**:
To analyze the relationship, we can take logarithms of both sides:
\[ \log\left(\frac{x}{m}\right) = \log(k) + \frac{1}{n} \log(P) \]
This can be compared to the linear equation \(y = mx + c\), where:
- \(y = \log\left(\frac{x}{m}\right)\)
- \(x = \log(P)\)
- Slope \(m = \frac{1}{n}\)
- Intercept \(c = \log(k)\)

**Conclusion**: The plot of \(\log\left(\frac{x}{m}\right)\) versus \(\log(P)\) will be linear.

3. **Analyzing Other Plots**:
- **Plot of \(\frac{m}{x}\) versus \(\frac{1}{P}\)**:
From the original equation, we can express \(\frac{m}{x}\) as:
\[ \frac{m}{x} = \frac{1}{k} P^{-\frac{1}{n}} \]
This indicates that \(\frac{m}{x}\) is proportional to \(P^{-\frac{1}{n}}\), which can be rearranged to show a linear relationship when plotted against \(\frac{1}{P}\).

- **Plot of \(\log(P)\) versus \(\frac{m}{m}\)**:
Since \(m\) cancels out, we are left with \(\log(P)\) versus \(P\), which is not linear.

- **Plot of \(\log\left(\frac{m}{x}\right)\) versus \(\log(P)\)**:
This can be derived from the previous equations and will not be linear due to the negative sign introduced when taking logarithms.

4. **Final Conclusion**:
The plots that will be linear based on the analysis are:
- Plot of \(\log\left(\frac{x}{m}\right)\) versus \(\log(P)\) (Option A)
- Plot of \(\frac{m}{x}\) versus \(\frac{1}{P}\) (Option B)
- Plot of \(\log(P)\) versus \(\frac{m}{m}\) (Option C)

Thus, the correct options that yield linear plots are **A, B, and C**.