Animated Solution for Chemistry - Surface Chemistry: The mass of gas adsorbed, $x$ per unit mass of adsorbate, $m$ was measured at various pressures, $p$. A graph between $\log \frac{x}{m}$ and $\log p$ gives a straight line with slope equal to 2 and the intercept equal to 0.4771. The value of $\frac{x}{m}$ at a pressure of 4 atm is (Given, $\log 3=0.4771$)

The Freundlich adsorption isotherm is a fundamental empirical relationship that describes how the quantity of a gas adsorbed by a solid surface varies with pressure at a constant temperature. In this problem, we are given experimental data plotted as a straight line, and we need to extract the physical constants to find the extent of adsorption at a specific pressure.

The Freundlich isotherm is mathematically expressed as: where $x$ is the mass of the gas adsorbed, $m$ is the mass of the adsorbent, $p$ is the equilibrium pressure, and $K$ and $n$ are empirical constants that depend on the nature of the adsorbate and adsorbent at a given temperature.

To make this equation easier to analyze experimentally, we take the logarithm on both sides:

Notice how this logarithmic equation perfectly mirrors the equation of a straight line, $y=mx+c$. If we plot $\log \frac{x}{m}$ on the y-axis and $\log p$ on the x-axis, we get a straight line where: - The slope of the line is $\frac{1}{n}$. - The y-intercept is $\log K$.

The problem states that the slope of this graph is $2$. Therefore:

The y-intercept is given as $0.4771$. Therefore: We are also given that $\log 3=0.4771$. By comparing these two, we can immediately deduce that:

Now that we have our constants, we can find the value of $\frac{x}{m}$ at a pressure of $p=4\text{ atm}$. Let's substitute these values back into the original Freundlich equation: