Animated Solution for Chemistry - Surface Chemistry: Adsorption of a gas follows Freundlich adsorption isotherm. $x$ is the mass of the gas adsorbed on mass $m$ of the adsorbent. The plot of $\log \frac{x}{m}$ versus $\log p$ is shown in the given graph. $\frac{x}{m}$ is proportional to

When dealing with physical chemistry, we often encounter relationships that are exponential or power-based. The Freundlich adsorption isotherm is a classic example. It tells us how the mass of a gas adsorbed per unit mass of an adsorbent ($x/m$) varies with pressure ($p$). The relationship is given by:

While this equation is powerful, plotting a curve and extracting exact constants from it can be visually tricky. This is where the magic of logarithms comes in. By taking the logarithm on both sides, we can transform this curve into a beautiful, easy-to-read straight line.

Let's apply the logarithm to our isotherm equation:

Now, imagine you are looking at the standard equation of a straight line, $y=mx+c$. If we map our variables, we see that plotting $\log (x/m)$ on the y-axis and $\log p$ on the x-axis will yield a straight line.

The slope of this line corresponds exactly to the exponent $\frac{1}{n}$, and the y-intercept corresponds to $\log k$. This is a brilliant trick because calculating the slope of a straight line is incredibly straightforward!

In the problem, we are given a graph with a helpful little triangle drawn under the line. This triangle is essentially giving us the 'rise over run' or $\frac{\Delta y}{\Delta x}$.

The vertical side (change in y) is $2$, and the horizontal side (change in x) is $3$. Therefore, the slope of the line is:

Since we already established that the slope is equal to $\frac{1}{n}$, we can confidently say that $\frac{1}{n}=\frac{2}{3}$.

Finally, substituting this back into our original Freundlich equation, we get:

This reveals the direct proportionality: $\frac{x}{m}\propto p^{2/3}$. By simply reading a graph and understanding the power of logarithms, we've unlocked the physical relationship between adsorption and pressure!