An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Question
Chapter 6.6, Problem 44P
To determine
Expression of Helmholtz free energy and the chemical potential.
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Students have asked these similar questions
Consider a large system of N indistinguishable, noninteracting molecules (perhaps in an ideal gas or a dilute solution). Find an expression for the Helmholtz free energy of this system, in terms of Z1, the partition function for a single molecule. (Use Stirling's approximation to eliminate the N!) Then use your result to find the chemical potential, again in terms of Z1.
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?
Let Ω be a new thermodynamic potential that is a “natural” function of temperature T, volume V, and the chemical potential μ. Provide a definition of Φ in the form of a Legendre transformation and also write its total differential, or derived fundamental equation, in terms of these natural variables.
Chapter 6 Solutions
An Introduction to Thermal Physics
Ch. 6.1 - Prob. 2PCh. 6.1 - Prob. 4PCh. 6.1 - Prob. 5PCh. 6.1 - Prob. 6PCh. 6.1 - Prob. 7PCh. 6.1 - Prob. 8PCh. 6.1 - Prob. 9PCh. 6.1 - Prob. 10PCh. 6.1 - Prob. 11PCh. 6.1 - Prob. 12P
Ch. 6.1 - Prob. 13PCh. 6.1 - Prob. 14PCh. 6.2 - Prob. 15PCh. 6.2 - Prob. 16PCh. 6.2 - Prob. 17PCh. 6.2 - Prob. 18PCh. 6.2 - Prob. 19PCh. 6.2 - Prob. 20PCh. 6.2 - For an O2 molecule the constant is approximately...Ch. 6.2 - The analysis of this section applies also to...Ch. 6.3 - Prob. 31PCh. 6.4 - Calculate the most probable speed, average speed,...Ch. 6.4 - Prob. 35PCh. 6.4 - Prob. 36PCh. 6.4 - Prob. 37PCh. 6.4 - Prob. 39PCh. 6.4 - Prob. 40PCh. 6.5 - Prob. 42PCh. 6.5 - Some advanced textbooks define entropy by the...Ch. 6.6 - Prob. 44PCh. 6.7 - Prob. 45PCh. 6.7 - Equations 6.92 and 6.93 for the entropy and...Ch. 6.7 - Prob. 47PCh. 6.7 - For a diatomic gas near room temperature, the...Ch. 6.7 - Prob. 49PCh. 6.7 - Prob. 50PCh. 6.7 - Prob. 51PCh. 6.7 - Prob. 52PCh. 6.7 - Prob. 53P
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- (b) Consider the following heat system on the real line: U - U = 0, XER, 1>0 %3D u(x, 0) = | sin x), rER. i. Use the fundamental solution of the heat equation to write down a solution u to the system above as an integral. ii. Show that the solution u that you have found is bounded by 1.arrow_forwardUse the Maxwell distribution to calculate the average value of v2 for the molecules in an ideal gas. Check that your answer agrees with equation 6.41 (Attached).arrow_forward. An ideal classical gas composed of N particles, each of mass m, is enclosed in a vertical cylinder of height L placed in a uniform gravitational field (of acceleration g) and is in thermal equilibrium; ultimately, both N and L → ∞. Evaluate the partition function of the gas and derive expressions for its major thermodynamic properties. Explain why the specific heat of this system is larger than that of a corresponding system in free space.arrow_forward
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- Prove that the entropy S of an ideal gas [Sackur and Tetrode's equation] is an extensive quantity. Then show that the entropy of the gas of particles to be separated from each other is 3 S = NKB — Nku [2 - In (2/V)], and that this quantity is not extensive. Remember: by extensiveness we mean that if we scale the size of the system by a factor a (V → a V, N a N, but the particle density n = N/V remains constant), any extensive quantity a s) also scales by a factor a (here: S →arrow_forwardLet's consider a classical ideal gas whose single-particle partition function of molecules is Z ₁. statements is true? Which of the following Select one: a. If the gas molecules of N molecules cannot be separated from each other, and the partition function ZN of the system is written ZN = Z₁N, the entropy of the gas calculated from Z N is obtained, which is an extensive quantity. 1 O b. If the molecules cannot be separated from each other, the partition function of the system can be written in the form ZN = Z₁N/N!. In this case, the entropy calculated from Z N is obtained, which is an extensive quantity. 1 N O c. If the gas molecules of N molecules cannot be separated from each other, the partition function Z of the system can be written Z N = Z₁N.arrow_forwardT04.2 Atoms in a harmonic trap We consider Nparticles in one dimension in an external potential, mw2 K(x) = 2 X7. (to)Write the complete Hamiltonian function for the system. Then calculate the number of micro-states MAND) by means of the semiclassical approach. (b)Calculate the entropy in the thermodynamic limit. (c)Calculate the temperature and the work differential based on the result in part (b).arrow_forward
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