Question

The adsorption of ethyl chloride on a sample of charcoal at \(0^{\circ} \mathrm{C}\) measured at several different pressures is as follows: $$\begin{array}{rc} \mathbf{P}_{C_{2} H_{5} C l}(\text { Torr }) & V_{\text {ads}}(\mathbf{m L}) \\\ \hline 20 & 3.0 \\ 50 & 3.8 \\ 100 & 4.3 \\ 200 & 4.7 \\ 300 & 4.8 \end{array}$$ Using the Langmuir isotherm, determine the fractional coverage at each pressure and \(V_{m}\)

Short answer

Using the Langmuir isotherm, we calculated the fractional coverage θ for each pressure using the equation \(θ = \frac{KP}{1+KP}\). We then plotted the variables \(\frac{V_{ads}}{V_m – V_{ads}}\) and pressures to observe a linear relationship. After performing a linear regression on the plotted datapoints, we determined the Langmuir constant K from the slope and calculated Vm by finding the x-intercept. Finally, we computed the final fractional coverage θ at each pressure using the Langmuir isotherm equation.

Step-by-step solution

Compute fractional coverage using the Langmuir isotherm equation

Since the pressures are given and we have the equation, compute the fractional coverage for each pressure P using the equation: θ = \(\frac{KP}{1+KP}\) At each pressure, the corresponding V_ads is given. We can use a linearized form of the Langmuir equation: \[\frac{V_{ads}}{V_m – V_{ads}} = \frac{1}{KP}\] Calculate the variable \(\frac{V_{ads}}{V_m – V_{ads}}\) for each pressure using the adsorption volume V_ads. 2.

Plot the variables

Plot the variables \(\frac{V_{ads}}{V_m – V_{ads}}\) on the y-axis and the pressures on the x-axis as scattered points. We should observe a linear relationship between these variables. 3.

Calculate the slope and intercept using linear regression

Perform a linear regression on the plotted datapoints to estimate the slope and intercept of the best fitting line through the origin. Use a statistical software or calculator for this purpose. 4.

Determine the K value

The inverse of the slope from the linear regression will give us the Langmuir constant (K). \(K = \frac{1}{\mathrm{slope}}\) 5.

Calculate Vm

After determining the value of K, calculate Vm by finding the value of the x-intercept using the linear regression, when \( \frac{V_{ads}}{V_m – V_{ads}} = 0 \). \(V_m\) will be the reciprocal of the intercept value. 6.

Compute final fractional coverage

Now that we have K and Vm, compute the final fractional coverage θ at each pressure using the Langmuir isotherm equation: \(θ = \frac{KP}{1+KP}\) From these steps, we have been able to determine the fractional coverage at each pressure and the Vm using the Langmuir isotherm.

Key concepts

Adsorption of Gases

Adsorption is a process where molecules of a gas accumulate on the surface of a solid or a liquid. This occurs because of the intermolecular forces between the gas molecules and the surface atoms or molecules. In physical chemistry, this phenomenon is crucial to understanding how various substances interact with surfaces and is widely studied in the context of catalysis, separation processes, and environmental science.

When we analyze the adsorption of gases like ethyl chloride on charcoal, we consider factors such as temperature, pressure, and the properties of both the gas and the adsorbent surface. The amount of gas adsorbed can be quantified volumetrically, as in the exercise example. Here, the adsorption increases with pressure but to a lesser extent as the pressure continues to increase. This non-linear relationship between pressure and adsorption volume is a characteristic of many adsorption processes and is described by the Langmuir isotherm.

Fractional Coverage

Fractional coverage (denoted as \( \theta \)) is a key concept in studying the adsorption of gases. It represents the fraction of the adsorption sites on a surface that are occupied by adsorbate molecules. In practical terms, fractional coverage tells us how much of the surface is 'covered' by the adsorbed gas.

In the exercise given, the Langmuir isotherm equation \( \theta = \frac{KP}{1+KP} \) is used to calculate the fractional coverage at various pressures, where \( K \) is the Langmuir constant and \( P \) is the pressure of the adsorbing gas. A key feature of the Langmuir model is its assumption that adsorption occurs at specific homogeneous sites on the surface, and once a site is occupied, no further adsorption can happen at that site. This aids in understanding the behavior of adsorption at both low and high pressures and informs us of how efficient the adsorbent material is at various conditions.

Physical Chemistry

Physical chemistry is the branch of chemistry that focuses on the understanding of the physical properties of molecules, the forces that act upon them, and the physical laws that govern their behavior. It blends principles of physics and chemistry to explain how chemical systems work and how they change with respect to time and energy. The study of adsorption is highlighted in physical chemistry because it involves the understanding of intermolecular forces, thermodynamics, and kinetics.

In the context of the exercise, utilizing the Langmuir isotherm requires an understanding of the physical interaction between the gas molecules and the adsorbent surface. By combining experimental data with physical chemistry theory, we find a practical application: determining how much of a gas will be adsorbed at different pressures, and consequently, we can predict how a system will behave in real-world scenarios. Through careful analysis and mathematical representation, such as the linearization of the Langmuir equation, physical chemists can elucidate the characteristics of the adsorption process.