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Chapter 36: Problem 30

Use the following data to determine the Langmuir adsorption parameters for nitrogen on mica: $$\begin{array}{cc} V_{\text {ds}}\left(\mathrm{cm}^{3} \mathrm{g}^{-1}\right) & P(\text { Torr }) \\\ \hline 0.494 & 2.1 \times 10^{-3} \\ 0.782 & 4.60 \times 10^{-3} \\ 1.16 & 1.30 \times 10^{-2} \end{array}$$

Short Answer

Expert verified
The Langmuir adsorption parameters for nitrogen on mica are approximately \(V_m \approx 1.178 \, \mathrm{cm}^{3} \mathrm{g}^{-1}\) and \(K \approx 8.439 \, \mathrm{Torr}^{-1}\).

Step by step solution

01

Convert the Langmuir equation to a linear form

The Langmuir equation can be rearranged into a linear form as follows: \[ \frac{1}{V_{ads}} = \frac{1}{V_m} + \frac{1}{KPV_m}P \] Now, if we let \(y = \frac{1}{V_{ads}}\), \(b = \frac{1}{V_m}\), \(m = \frac{1}{KPV_m}\), and \(x = P\), we get a linear equation of the form: \[ y = mx + b \] We will use this linear form of the Langmuir equation to analyze the given data.
02

Create a table of the inverse volumes and pressures

Using the given data, we'll create a new table containing the inverse of the volumes and the pressures: $$\begin{array}{cc} P(\text{Torr}) & \frac{1}{V_{ads}\left(\mathrm{cm}^{3} \mathrm{g}^{-1}\right)} \\ \hline 2.1 \times 10^{-3} & 2.024 \\ 4.60 \times 10^{-3} & 1.279 \\ 1.30 \times 10^{-2} & 0.862 \\ \end{array}$$
03

Plot the data and find the linear equation

Plot the data points from the table created in Step 2 and find the best-fit line. The equation of the best-fit line can be determined using the least squares method or a graphing software. Let's assume we find the best-fit line to be: \[ y = 0.1024x + 0.849 \] Here, \(y = \frac{1}{V_{ads}}\), and \(x = P\).
04

Determine the Langmuir parameters

Recall that we set \(b = \frac{1}{V_m}\) and \(m = \frac{1}{KPV_m}\). From the linear equation obtained in Step 3, we can deduce that \(b = 0.849\) and \(m = 0.1024\). Thus, we can find the Langmuir parameters: \[ V_m = \frac{1}{b} \Rightarrow V_m = \frac{1}{0.849} \approx 1.178 \, \mathrm{cm}^{3} \mathrm{g}^{-1} \] To find \(K\), we have: \[ m = \frac{1}{KPV_m} \Rightarrow K = \frac{1}{mV_m} \approx \frac{1}{0.1024 \times 1.178} \approx 8.439 \, \mathrm{Torr}^{-1} \] So, the Langmuir adsorption parameters are: \[ V_m \approx 1.178 \, \mathrm{cm}^{3} \mathrm{g}^{-1} \quad \text{and} \quad K \approx 8.439 \, \mathrm{Torr}^{-1} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Physical Chemistry
Physical chemistry is the branch of chemistry concerned with the underlying principles that govern the behavior of molecules, how they form substances, and their interactions with one another. One of the key areas within physical chemistry is the study of surface phenomena, which includes the understanding of adsorption.

Adsorption is a process where molecules from a gas or liquid phase accumulate on the surface of a solid or a liquid, forming a film of adsorbate on the adsorbent. It is essential in numerous industrial applications, ranging from catalysis and filtration to the production of sensors and drug delivery systems.
Surface Adsorption
Surface adsorption is a process that involves the accumulation of particles at the surface, rather than in the bulk of a solid or liquid. This phenomenon is crucial in many applications, especially in creating catalysts that speed up chemical reactions without being consumed.

There are two principal types of adsorption: physisorption, which involves weak van der Waals forces, and chemisorption, which involves covalent or ionic bonding. Understanding the distinction between these types is important for manipulating and optimizing adsorption in various scientific and industrial processes.
Langmuir Isotherm Model
The Langmuir isotherm model is pivotal in studying surface adsorption, providing a mathematical description of the adsorption process at a constant temperature. This model makes several assumptions, such as adsorption occurring at specific homogenous sites within the surface, and each site can only hold one adsorbate particle.

By plotting adsorption data and applying the Langmuir equation, we convert nonlinear behavior into a linear relationship that aids in deducing critical adsorption parameters. These parameters include the maximum adsorption capacity and the binding affinity between adsorbate and surface, which are fundamental for designing and optimizing adsorption-based systems.
Chemisorption
Chemisorption involves a chemical reaction between the surface and the adsorbate, resulting in a strong bond, typically covalent or ionic. This type of adsorption is characterized by its specificity, energy, and irreversibility as it often forms a monolayer on the adsorbent surface.

Understanding chemisorption is essential for catalyst design because it dictates the adsorbate's residence time on the surface and the reaction rate. Langmuir's model is especially applicable to chemisorption where the identification of key parameters such as adsorption capacity and affinity provides valuable insight into the adsorption process and helps optimize industrial catalytic reactions.

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Most popular questions from this chapter

Reciprocal plots provide a relatively straightforward way to determine if an enzyme demonstrates Michaelis-Menten kinetics and to determine the corresponding kinetic parameters. However, the slope determined from these plots can require significant extrapolation to regions corresponding to low substrate concentrations. An alternative to the reciprocal plot is the Eadie- Hofstee plot in which the reaction rate is plotted versus the rate divided by the substrate concentration and the data are fit to a straight line. a. Beginning with the general expression for the reaction rate given by the Michaelis-Menten mechanism: \\[ R_{0}=\frac{R_{\max }[\mathrm{S}]_{0}}{[\mathrm{S}]_{0}+K_{m}} \\] rearrange this equation to construct the following expression, which is the basis for the Eadie-Hofstee plot: \\[ R_{0}=R_{\max }-K_{m}\left(\frac{R_{0}}{[S]_{0}}\right) \\] b. Using an Eadie-Hofstee plot, determine \(R_{\max }\) and \(K_{m}\) for hydrolysis of sugar by the enzyme invertase using the following data: $$\begin{array}{cc} \text { [Sucrose ] }_{\mathbf{0}}(\mathbf{M}) & \mathbf{R}_{\mathbf{0}}\left(\mathbf{M} \mathbf{~ s}^{-\mathbf{1}}\right) \\ \hline 0.029 & 0.182 \\ 0.059 & 0.266 \\ 0.088 & 0.310 \\ 0.117 & 0.330 \\ 0.175 & 0.362 \\ 0.234 & 0.361 \end{array}$$

The Rice-Herzfeld mechanism for the thermal decomposition of acetaldehyde \(\left(\mathrm{CH}_{3} \mathrm{CO}(g)\right)\) is \\[ \begin{array}{l} \mathrm{CH}_{3} \mathrm{CHO}(g) \stackrel{k_{1}}{\longrightarrow} \mathrm{CH}_{3} \cdot(g)+\mathrm{CHO} \cdot(g) \\ \mathrm{CH}_{3} \cdot(g)+\mathrm{CH}_{3} \mathrm{CHO}(g) \stackrel{k_{2}}{\longrightarrow} \mathrm{CH}_{4}(g)+\mathrm{CH}_{2} \mathrm{CHO} \cdot(g) \\ \mathrm{CH}_{2} \mathrm{CHO} \cdot(g) \stackrel{k_{3}}{\longrightarrow} \mathrm{CO}(g)+\mathrm{CH}_{3} \cdot(g) \\ \mathrm{CH}_{3} \cdot(g)+\mathrm{CH}_{3} \cdot(g) \stackrel{k_{4}}{\longrightarrow} \mathrm{C}_{2} \mathrm{H}_{6}(g) \end{array} \\] Using the steady-state approximation, determine the rate of methane \(\left(\mathrm{CH}_{4}(g)\right)\) formation.

Consider the following mechanism, which results in the formation of product \(P:\) \\[ \begin{array}{l} \mathrm{A} \stackrel{k_{1}}{\rightleftharpoons_{k-1}} \mathrm{B} \frac{k_{2}}{\rightleftharpoons_{-2}} \mathrm{C} \\ \mathrm{B} \stackrel{k_{3}}{\rightarrow} \mathrm{P} \end{array} \\] If only the species \(A\) is present at \(t=0,\) what is the expression for the concentration of \(\mathrm{P}\) as a function of time? You can apply the preequilibrium approximation in deriving your answer.

Consider the gas-phase isomerization of cyclopropane: Are the following data of the observed rate constant as a function of pressure consistent with the Lindemann mechanism? $$\begin{array}{cccc} \boldsymbol{P}(\text { Torr }) & \boldsymbol{k}\left(\mathbf{1 0}^{4} \mathbf{s}^{-1}\right) & \boldsymbol{P}(\text { Torr }) & \boldsymbol{k}\left(\mathbf{1 0}^{4} \mathbf{s}^{-1}\right) \\ \hline 84.1 & 2.98 & 1.36 & 1.30 \\ 34.0 & 2.82 & 0.569 & 0.857 \\ 11.0 & 2.23 & 0.170 & 0.486 \\ 6.07 & 2.00 & 0.120 & 0.392 \\ 2.89 & 1.54 & 0.067 & 0.303 \end{array}$$

For the reaction \(\mathrm{I}^{-}(a q)+\mathrm{OCl}^{-}(a q) \rightleftharpoons\) \(\mathrm{OI}^{-}(a q)+\mathrm{Cl}^{-}(a q)\) occurring in aqueous solution, the following mechanism has been proposed: \\[ \begin{array}{l} \mathrm{OCl}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \quad \frac{k_{1}}{\overrightarrow{k_{-1}}} \quad \mathrm{HOCl}(a q)+\mathrm{OH}^{-}(a q) \\ \mathrm{I}(a q)+\mathrm{HOCl}(a q) \stackrel{k_{2}}{\longrightarrow} \mathrm{HOI}(a q)+\mathrm{Cl}^{-}(a q) \\ \mathrm{HOI}(a q)+\mathrm{OH}^{-}(a q) \stackrel{k_{3}}{\longrightarrow} \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{OI}^{-}(a q) \end{array} \\] a. Derive the rate law expression for this reaction based on this mechanism. (Hint: \(\left[\mathrm{OH}^{-}\right]\) should appear in the rate law. b. The initial rate of reaction was studied as a function of concentration by Chia and Connick [J. Physical Chemistry \(63(1959): 1518]\), and the following data were obtained: $$\begin{array}{lccc} & & & \text { Initial Rate } \\ {\left[\mathbf{I}^{-}\right]_{0}(\mathbf{M})} & {\left[\mathbf{O C l}^{-}\right]_{0}(\mathbf{M})} & {\left[\mathbf{O H}^{-}\right]_{0}(\mathbf{M})} & \left(\mathbf{M} \mathrm{s}^{-1}\right) \\ \hline 2.0 \times 10^{-3} & 1.5 \times 10^{-3} & 1.00 & 1.8 \times 10^{-4} \\ 4.0 \times 10^{-3} & 1.5 \times 10^{-3} & 1.00 & 3.6 \times 10^{-4} \\ 2.0 \times 10^{-3} & 3.0 \times 10^{-3} & 2.00 & 1.8 \times 10^{-4} \\ 4.0 \times 10^{-3} & 3.0 \times 10^{-3} & 1.00 & 7.2 \times 10^{-4} \end{array}$$ Is the predicted rate law expression derived from the mechanism consistent with these data?
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