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This problem concerns a collection of N identical harmonic
Chapter 6, Problem 20P(choose chapter or problem)
This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid or the internal vibrations of gas molecules) at temperature T. As in Section 2.2, the allowed energies of each oscillator are 0, hf, 2hf, and so on.
(a) Prove by long division that
\(\frac{1}{1-x}=1+x+x^{2}+x^{3}+\cdots .\)
For what values of x does this series have a finite sum?
(b) Evaluate the partition function for a single harmonic oscillator. Use the result of part(a) to simplify your answer as much as possible.
(c) Use formula 6.25 to find an expression for the average energy of a single oscillator at temperature T. Simplify your answer as much as possible.
(d) What is the total energy of the system of N oscillators at temperature T? Your result should agree with what you found in Problem 3.25.
(e) If you haven’t already done so in Problem 3.25, compute the heat capacity of this system and check that it has the expected limits as \(T \rightarrow 0\) and \(T \rightarrow \infty\).
Questions & Answers
(1 Reviews)
QUESTION:
This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid or the internal vibrations of gas molecules) at temperature T. As in Section 2.2, the allowed energies of each oscillator are 0, hf, 2hf, and so on.
(a) Prove by long division that
\(\frac{1}{1-x}=1+x+x^{2}+x^{3}+\cdots .\)
For what values of x does this series have a finite sum?
(b) Evaluate the partition function for a single harmonic oscillator. Use the result of part(a) to simplify your answer as much as possible.
(c) Use formula 6.25 to find an expression for the average energy of a single oscillator at temperature T. Simplify your answer as much as possible.
(d) What is the total energy of the system of N oscillators at temperature T? Your result should agree with what you found in Problem 3.25.
(e) If you haven’t already done so in Problem 3.25, compute the heat capacity of this system and check that it has the expected limits as \(T \rightarrow 0\) and \(T \rightarrow \infty\).
ANSWER:Step 1 of 7
(a) Use the long division to divide 1 by \(1-x\), so we get
\(\begin{array}{cllll}
1-x & +x & +x^{2} & +\ldots & \\
\hline & 1+0 x & +0 x^{2} & +0 x^{3} & +\ldots \\
& 1-x & & & \\
x & +0 x^{2} & & & \\
x & -x^{2} & & & \\
\hline & x^{2} & +0 x^{3} & &
\end{array}\)
only when \(|x|<1\), the terms converge, because the power of x increases.
Reviews
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