An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Question
Chapter 8.2, Problem 18P
To determine
The average energy in terms of partition function.
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The Einstein model for a solid assumes the system consists of 3N independent simple harmonic oscillators with frequencies &. Within these assumptions, the heat capacity at constant volume as:
Cv=3Nk() (-1)²
²
Complete the table for the molar heat capacity at various temperatures under either the Einstein model or high-temperature limit. You might like to use the Wolfram Alpha calculator to do the numerical calculations more
easily. Use k-0.695 cm /K.
High temperature limit value of molar heat capacity of metal is
T
1 K
10 K
50 K
-1
Einstein, = 100 cm Einstein, : = 500 cm
1.4021
3.8991
100 K
500 K
2.434E-4
1000 K
6.1499
2434E-4
kJ/mol.
Problem 1:
This problem concerns a collection of N identical harmonic oscillators (perhaps an
Einstein solid) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf,
and so on.
a) Prove =1+x + x² + x³ + .... Ignore Schroeder's comment about proving
1-x
the formula by long division. Prove it by first multiplying both sides of the
equation by (1 – x), and then thinking about the right-hand side of the resulting
expression.
b) Evaluate the partition function for a single harmonic oscillator. Use the result of
(a) to simplify your answer as much as possible.
c) Use E = -
дz
to find an expression for the average energy of a single oscillator.
z aB
Simplify as much as possible.
d) What is the total energy of the system of N oscillators at temperature T?
Consider N identical harmonic oscillators (as in the Einstein floor). Permissible Energies of each oscillator (E = n h f (n = 0, 1, 2 ...)) 0, hf, 2hf and so on.
A) Calculating the selection function of a single harmonic oscillator. What is the division of N oscillators?
B) Obtain the average energy of N oscillators at temperature T from the partition function.
C) Calculate this capacity and T-> 0 and At T-> infinity limits, what will the heat capacity be? Are these results consistent with the experiment? Why? What is the correct theory about this?
D) Find the Helmholtz free energy from this system.
E) Derive the expression that gives the entropy of this system for the temperature.
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- 5....Only the partition function as a function of temperature is required in order to determine the average energy in a sub-system, at fixed volume and number of particles. Is this statement true or false?arrow_forwarda) Show that for enthalpy and Gibbs free energy, H = G + TS can be written. b) Based on the exact differential of Gibbs free energyarrow_forward9.1. Consider the results for ideal gases derived from quantum mechanics. Write an expression for the function s(T, v) that includes the parameter A(T). (a) Show that the behavior of the entropy as T0 is unrealistic in a quantum sense. What approximation in the derivation is responsible for this behavior? (b) The Sackur-Tetrode equation is valid when the thermal de Broglie wave- length is much less than the average molecular separation distance, A(T) < (V/N)13, which is generally the case at high temperatures. Calculate the temperature at which this inequality is first violated for an ideal gas at atmospheric pressure and with a mass of 6.6 × 10-26 kg/molecule (which is typical of argon). xarrow_forward
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