For a linear plot of log $$\left(\frac{x}{m}\right)$$ versus log p in a Freundlich adsorption isotherm, which of the following statements is correct? (k and n are constants)

We start with the mathematical form of the Freundlich adsorption isotherm, which is customarily written as

$$\frac{x}{m}=k\,p^{\frac{1}{n}},$$

where $$\frac{x}{m}$$ is the amount of adsorbate adsorbed per unit mass of adsorbent, $$p$$ is the equilibrium pressure of the adsorbate gas, and $$k$$ and $$n$$ are empirical constants characteristic of the adsorbent-adsorbate system.

To convert the above power-law relation into a straight-line (linear) form, we take the logarithm (to the base 10) of both sides. Using the elementary logarithmic identity $$\log(a\,b)=\log a+\log b$$ and $$\log(a^{b})=b\log a,$$ we proceed as follows:

First, apply the logarithm to the left-hand side and the right-hand side:

$$\log\!\left(\frac{x}{m}\right)=\log\!\left(k\,p^{\frac{1}{n}}\right).$$

Now, split the logarithm on the right-hand side into a sum, because the argument is a product $$k \times p^{1/n}$$:

$$\log\!\left(\frac{x}{m}\right)=\log k+\log\!\left(p^{\frac{1}{n}}\right).$$

Next, use the power-rule of logarithms, $$\log\!\left(a^{b}\right)=b\,\log a,$$ on the second term:

$$\log\!\left(\frac{x}{m}\right)=\log k+\frac{1}{n}\,\log p.$$

We now have an equation of the form $$y=c+mx,$$ which is the standard straight-line equation. To make the correspondence completely explicit, we identify

$$y=\log\!\left(\frac{x}{m}\right), \qquad x=\log p,$$

so that

$$\text{slope }(m)=\frac{1}{n}, \qquad \text{intercept }(c)=\log k.$$

Observe that only the factor $$\frac{1}{n}$$ occurs as the slope, whereas the constant $$k$$ appears exclusively inside the logarithm of the intercept. No combination such as $$\log\!\left(\frac{1}{n}\right)$$ or any mixture of $$k$$ and $$\frac{1}{n}$$ shows up in the slope term.

Therefore, when a graph is drawn of $$\log\!\left(\dfrac{x}{m}\right)$$ on the vertical axis versus $$\log p$$ on the horizontal axis, the straight line obtained has $$\dfrac{1}{n}$$ as its slope.

Hence, the correct answer is Option A.