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The mixing entropy formula derived in the previous problem

An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder ISBN: 9780201380279 40

Solution for problem 38P Chapter 2

An Introduction to Thermal Physics | 1st Edition

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An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder

An Introduction to Thermal Physics | 1st Edition

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Problem 38P

The mixing entropy formula derived in the previous problem actually applies to any ideal gas, and to some dense gases, liquids, and solids as well, For the denser systems, we have to assume that the two types of molecules are the same size and that molecules of different types interact with each other in the same way as molecules of the same type (same forces, etc.). Such a system is called an ideal mixture. Explain why, for an ideal mixture, the mixing entropy is given by

where N is the total number of molecules and NA is the number of molecules of type A. Use Stirling’s approximation to show that this expression is the same as the result of the previous problem when both N and NA are large.

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Step 1 of 3

Solution 38P

 Suppose we start with a number of identical molecules. We can apply this argument  to any system in which the molecules all have similar properties and interact with each other in the same way. The entropy of this system is  which is not that easy to calculate.

 Now suppose after some time, some of the molecules  changes to a different type which has similar properties to the original.. The entropy will increase by the number of distinct ways we can choose to locate these  molecules among the  places available. The entropy due to the number of possible locations and momenta of the molecules won’t change when we replace of the molecules by a different type of molecule, since that is already accounted for by S.

We’re interested only in the extra entropy generated by introducing a second type of molecule.

The number of ways of choosing  locations from a total of  is just so the entropy of mixing is

Capture.PNG

Using Stirling’s approximation for large N and taking = ()as before ,we get

 =  

[]  (we can neglect the first term as it is negligible for large N)

This expression is the same as the result of the previous problem when both N andare large.

     

Step 2 of 3

Chapter 2, Problem 38P is Solved
Step 3 of 3

Textbook: An Introduction to Thermal Physics
Edition: 1
Author: Daniel V. Schroeder
ISBN: 9780201380279

An Introduction to Thermal Physics was written by and is associated to the ISBN: 9780201380279. The answer to “The mixing entropy formula derived in the previous problem actually applies to any ideal gas, and to some dense gases, liquids, and solids as well, For the denser systems, we have to assume that the two types of molecules are the same size and that molecules of different types interact with each other in the same way as molecules of the same type (same forces, etc.). Such a system is called an ideal mixture. Explain why, for an ideal mixture, the mixing entropy is given by where N is the total number of molecules and NA is the number of molecules of type A. Use Stirling’s approximation to show that this expression is the same as the result of the previous problem when both N and NA are large.” is broken down into a number of easy to follow steps, and 129 words. This textbook survival guide was created for the textbook: An Introduction to Thermal Physics , edition: 1. Since the solution to 38P from 2 chapter was answered, more than 647 students have viewed the full step-by-step answer. This full solution covers the following key subjects: Molecules, ideal, type, previous, types. This expansive textbook survival guide covers 10 chapters, and 454 solutions. The full step-by-step solution to problem: 38P from chapter: 2 was answered by , our top Physics solution expert on 07/05/17, 04:29AM.

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