In Freundlich adsorption isotherm at moderate pressure, the extent of adsorption $$\left(\frac{x}{m}\right)$$ is directly proportional to $$P^x$$. The value of $$x$$ is:

The Freundlich adsorption isotherm provides an empirical relationship between the extent of adsorption $$\frac{x}{m}$$ and the pressure $$P$$ of the gas at constant temperature. The general equation is:

$$\frac{x}{m} = kP^{1/n}$$

where $$k$$ and $$n$$ are constants, and $$n > 1$$.

This isotherm describes three regimes depending on pressure. At low pressure, adsorption is directly proportional to pressure, so $$\frac{x}{m} \propto P$$ (here $$\frac{1}{n} = 1$$). At very high pressure, the surface reaches saturation and adsorption becomes independent of pressure, so $$\frac{x}{m} \propto P^0$$ (here $$\frac{1}{n} = 0$$).

At moderate pressure, which is the region where the Freundlich isotherm is most applicable, the extent of adsorption follows $$\frac{x}{m} = kP^{1/n}$$, where $$\frac{1}{n}$$ lies between 0 and 1 (since $$n > 1$$).

The question states that at moderate pressure, $$\frac{x}{m}$$ is directly proportional to $$P^x$$. Comparing with the Freundlich equation, the exponent $$x$$ equals $$\frac{1}{n}$$.

Therefore, the value of $$x$$ is $$\frac{1}{n}$$, which corresponds to option (2).