Animated Solution for Chemistry - Surface Chemistry: CO${}_2$ gas adsorbs on charcoal following Freundlich adsorption isotherm. For a given amount of charcoal, the mass of CO${}_2$ adsorbed becomes 64 times when the pressure of CO${}_2$ is doubled. The value of $n$ in the Freundlich isotherm equation is ...... $\times 10^{-2}$. (Round off to the nearest integer)

Imagine you are wearing a gas mask in a hazardous environment. How does it protect you? The secret lies in a phenomenon called adsorption. Inside the mask, there is activated charcoal, a highly porous material with a massive surface area. When toxic gases pass through, the gas molecules stick to the surface of the charcoal. This surface-level accumulation is adsorption.

In our problem, we are looking at carbon dioxide ($C{O}_2$) gas adsorbing onto charcoal. But how do we mathematically predict how much gas will stick? This is where the Freundlich Adsorption Isotherm comes into play. It provides an empirical relationship between the quantity of gas adsorbed by a unit mass of solid adsorbent and the pressure of the gas at a constant temperature.

The Freundlich isotherm is elegantly expressed by the equation:

Here, $x$ is the mass of the gas adsorbed, $m$ is the mass of the adsorbent (charcoal), $p$ is the pressure of the gas, and $k$ and $n$ are constants that depend on the nature of the adsorbent and the gas at a particular temperature.

The problem gives us a crucial piece of information: we are dealing with a "given amount of charcoal". This means the mass of our adsorbent, $m$, is constant. When $m$ is constant, the equation simplifies beautifully. The mass of the adsorbed gas, $x$, becomes directly proportional to the pressure raised to the power of $1/n$:

We are presented with two distinct states. Let's define them clearly to build our mathematical model.

In the initial state, let the pressure be $p_1$ and the mass of the adsorbed gas be $x_1$.

In the final state, the problem states that the pressure is doubled. So, our new pressure, $p_2$, is $2p_1$. Consequently, the mass of the adsorbed gas becomes 64 times the initial mass. So, our new mass, $x_2$, is $64x_1$.

To find the unknown constant $n$, we can set up a ratio comparing the final state to the initial state:

Now, we substitute our known relationships into this ratio:

Notice how the initial variables $x_1$ and $p_1$ cancel out perfectly. This leaves us with a clean, pure exponential equation:

To solve an exponential equation like this, the most straightforward method is to express both sides with the same base. We know that 64 is a power of 2. Specifically, $2\times 2\times 2\times 2\times 2\times 2=64$. Therefore, $64=2^6$.

Substituting this back into our equation gives:

Since the bases on both sides of the equation are identical (both are 2), their exponents must be equal for the equation to hold true. This leads us to a simple linear equation:

Solving for $n$ is now trivial. We simply take the reciprocal:

Converting this fraction to a decimal gives us $n\approx \mathrm{0.1666...}$

The question asks us to round off to the nearest integer in a specific scientific notation format: $...\times 10^{-2}$. First, let's round $\mathrm{0.1666...}$ to two decimal places, which gives us $0.17$.

Now, we format $0.17$ into the requested structure:

The integer value that fills the blank in the question is 17.