16.23. A gas is in equilibrium with a solid surface onto which its molecules can adsorb. The surface exposes M adsorption sites, each of which can be either free or occupied by a single gas molecule. Adsorbed molecules do not interact with each other, but there is a favorable energetic decrease of amount -e each time a site is осcupied. (a) Find the canonical partition function for N molecules adsorbed on M sites at a temperature T. (b) Find an expression for the chemical potential of an adsorbed molecule, µads- (c) At equilibrium at constant T, show that the dependence of the fraction of occupied sites, x = N/M, is given by the so-called Langmuir adsorption %3D isotherm, cP X = 1+ cP where c is a constant. How does c vary with temperature?
16.23. A gas is in equilibrium with a solid surface onto which its molecules can adsorb. The surface exposes M adsorption sites, each of which can be either free or occupied by a single gas molecule. Adsorbed molecules do not interact with each other, but there is a favorable energetic decrease of amount -e each time a site is осcupied. (a) Find the canonical partition function for N molecules adsorbed on M sites at a temperature T. (b) Find an expression for the chemical potential of an adsorbed molecule, µads- (c) At equilibrium at constant T, show that the dependence of the fraction of occupied sites, x = N/M, is given by the so-called Langmuir adsorption %3D isotherm, cP X = 1+ cP where c is a constant. How does c vary with temperature?
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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This problem is (16.23) from a book "
![### Problem 16.23:
A gas is in equilibrium with a solid surface onto which its molecules can adsorb. The surface provides \(M\) adsorption sites, each of which can be either free or occupied by a single gas molecule. Adsorbed molecules do not interact with each other, but there is a favorable energetic decrease of amount \(-\epsilon\) each time a site is occupied.
**Questions:**
(a) Find the canonical partition function for \(N\) molecules adsorbed on \(M\) sites at a temperature \(T\).
(b) Find an expression for the chemical potential of an adsorbed molecule, \(\mu_{\text{ads}}\).
(c) At equilibrium at constant \(T\), show that the dependence of the fraction of occupied sites, \(x = N/M\), is given by the so-called Langmuir adsorption isotherm:
\[
x = \frac{cP}{1 + cP}
\]
where \(c\) is a constant. How does \(c\) vary with temperature?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fecf9aa62-dd5c-4e9f-b87b-6613ab564f53%2Fc1758cfd-8b71-4b56-8312-5dd405092fcf%2Feevu445_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 16.23:
A gas is in equilibrium with a solid surface onto which its molecules can adsorb. The surface provides \(M\) adsorption sites, each of which can be either free or occupied by a single gas molecule. Adsorbed molecules do not interact with each other, but there is a favorable energetic decrease of amount \(-\epsilon\) each time a site is occupied.
**Questions:**
(a) Find the canonical partition function for \(N\) molecules adsorbed on \(M\) sites at a temperature \(T\).
(b) Find an expression for the chemical potential of an adsorbed molecule, \(\mu_{\text{ads}}\).
(c) At equilibrium at constant \(T\), show that the dependence of the fraction of occupied sites, \(x = N/M\), is given by the so-called Langmuir adsorption isotherm:
\[
x = \frac{cP}{1 + cP}
\]
where \(c\) is a constant. How does \(c\) vary with temperature?
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