An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
expand_more
expand_more
format_list_bulleted
Question
Chapter 6.6, Problem 44P
To determine
Expression of Helmholtz free energy and the chemical potential.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
Consider a monoatomic ideal gas, with Zint
1. The partition function is then
2Tm \ 3N/2
VN
zmonoatomic ideal gas
N!
h?B
Use F = -kT In Z, the Stirling approximation In N! = N In N
to derive the chemical potential of the monoatomic ideal gas as a function of T, N and V. You may want to
N and the appropriate partial derivative
compare your result with what you got in Weekly Practice 9.
(a) Take the atomic mass of Xenon to be 131 (Xenon has 8 different stable isotopes and many more
metastable ones). What is the chemical potential for pure Xe gas at 1 atm and T = 300 K? Use the ideal gas
law and give the answer in eV.
(b) Repeat the computation from part (a) if Xe is only 1% (by number density or, equivalently, partial pressure)
of a mixture of different gasses.
Note: if pure Xenon is allowed to come in contact with the gas in part (b), the net flow of Xenon atoms should
be into the mixture. This tells you that your answer to part (b) should be smaller than your answer to part (a).
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
Knowledge Booster
Similar questions
- (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_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_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
- Problem 2: Average values Prove that, for any system in equilibrium with a reservoir at temperature T, the average 1 дZ value of the energy is Ē = – z дв In Z, where ß = 1/kT. These formulas can be дв extremely useful when you have an explicit formula for the partition function.arrow_forwardPlease write down the proof of the Gay-Lussac-Joule experiment, i.e.for the ideal gas, u does not depend on v. Hint: you may use two-step process as shown in the figure.arrow_forwardStatical Mechanics (Thermal and Statical Physics) Instruction: Write ALL the solutions of this (necessary or and not direct answer). Write also the equations that are needed to solve for a certain problem. Thank you. Problem: Now, we have the number of microstates and in between E and E + ∆E in isolated system of N particles in the volume V is given by: (Please see the image attached) Where a,b, c are constants. Note: Answer also letter A-Darrow_forward
- What are the two major assumptions that are made in deriving the partition function for the ideal gas? Do you expect these assumptions to work better for a dilute or dense gas? Explain.arrow_forwardConsider an ideal gas containing N atoms in a container of volume Pressure P, and absolute temperature T1 (not to be confused with K. E. T). Use the virtual theorem to derive the equation of state for a perfect gas.arrow_forwardHow to do the integral? So that the expected value of internal energy must be as follows. Our professor said that there is Gaussian involved)...Please show steps (Thermodynamics & Statistical Mechanics)arrow_forward
- Let'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_forwardProve 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_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
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- College PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage LearningUniversity Physics (14th Edition)PhysicsISBN:9780133969290Author:Hugh D. Young, Roger A. FreedmanPublisher:PEARSONIntroduction To Quantum MechanicsPhysicsISBN:9781107189638Author:Griffiths, David J., Schroeter, Darrell F.Publisher:Cambridge University Press
- Physics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningLecture- Tutorials for Introductory AstronomyPhysicsISBN:9780321820464Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina BrissendenPublisher:Addison-WesleyCollege Physics: A Strategic Approach (4th Editio...PhysicsISBN:9780134609034Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart FieldPublisher:PEARSON
College Physics
Physics
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Cengage Learning
University Physics (14th Edition)
Physics
ISBN:9780133969290
Author:Hugh D. Young, Roger A. Freedman
Publisher:PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:9781107189638
Author:Griffiths, David J., Schroeter, Darrell F.
Publisher:Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:9781337553278
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:9780321820464
Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:Addison-Wesley
College Physics: A Strategic Approach (4th Editio...
Physics
ISBN:9780134609034
Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:PEARSON