Animated Solution for Chemistry - Surface Chemistry: For a linear plot of $\log (x/m)$ versus $\log p$ in a Freundlich adsorption isotherm, which of the following statements is correct? ($k$ and $n$ are constants)

Have you ever wondered how scientists make sense of complex, curved graphs? The secret often lies in a simple mathematical trick: logarithms! In this problem, we are dealing with the Freundlich adsorption isotherm, which describes how a gas accumulates on a solid surface.

The original equation is an exponential curve:

Here, $x/m$ is the mass of gas adsorbed per unit mass of the adsorbent, $p$ is the pressure, and $k$ and $n$ are constants. Graphing this directly gives a curve, which is hard to analyze. To find the exact values of $k$ and $n$, we need a straight line.

By taking the logarithm of both sides, we can transform this exponential relationship into a linear one. Let's see how:

Using the properties of logarithms ($\log (ab)=\log a+\log b$ and $\log (a^b)=b\log a$), we can expand the right side:

Now, let's compare our new equation with the standard equation of a straight line, $y=mx+c$.

- Our $y$-axis variable is $\log (x/m)$. - Our $x$-axis variable is $\log p$. - The $y$-intercept ($c$) is $\log k$. - The slope ($m$) of the line is $1/n$.

When we plot $\log (x/m)$ against $\log p$, we get a beautiful straight line. The point where this line cuts the $y$-axis gives us $\log k$, and the steepness (slope) of the line gives us $1/n$.

Therefore, looking at the options provided, it is clear that only $1/n$ appears as the slope. The intercept is $\log k$, not $1/n$ or $\log (1/n)$. This elegant mathematical transformation is a powerful tool in physical chemistry!