Which of the following graph yield a straight line?

Options

A. `x//m` vs p
B. log `x//m` vs p
C. `x//m` vs log p
D. log `x//m` vs log p

Explanation

To determine which graph yields a straight line based on the Freundlich isotherm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Freundlich Isotherm Equation**:
The Freundlich isotherm is given by the equation:
\[ \frac{X}{M} = K P^{\frac{1}{n}} \]
where \(X\) is the mass of gas absorbed, \(M\) is the mass of the adsorbent, \(P\) is the pressure, and \(K\) and \(n\) are constants.

2. **Take the Logarithm of Both Sides**:
To linearize the equation, we take the logarithm of both sides:
\[ \log\left(\frac{X}{M}\right) = \log(K) + \frac{1}{n} \log(P) \]

3. **Rearrange the Equation**:
This can be rearranged to fit the form of a straight line equation \(y = mx + c\):
\[ \log\left(\frac{X}{M}\right) = \frac{1}{n} \log(P) + \log(K) \]
Here, we can identify:
- \(y = \log\left(\frac{X}{M}\right)\)
- \(m = \frac{1}{n}\) (slope)
- \(x = \log(P)\)
- \(c = \log(K)\) (y-intercept)

4. **Identify the Graph**:
From the rearranged equation, we see that if we plot \(\log\left(\frac{X}{M}\right)\) (y-axis) against \(\log(P)\) (x-axis), we will get a straight line.

5. **Conclusion**:
Therefore, the graph that yields a straight line is the plot of \(\log\left(\frac{X}{M}\right)\) versus \(\log(P)\).

The correct answer is option D: the graph between \(\log\left(\frac{X}{M}\right)\) and \(\log(P)\).