An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Chapter 2.4, Problem 15P
To determine
To Justify: Accuracy of Stirling’s approximation for N=50
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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.
2.3 (a)
A classical harmonic oscillator
p?, Kq?
H
+
2m
2
is in thermal contact with a heat bath at temperature T. Calculate the
partition function for the oscillator in the canonical ensemble and
show explicitly that
• (E) = kgT, ((E – (E))²) = k¿T²
For an ideal gas of classical non- interacting atoms in thermal equilibrium, the Cartesian
component of the velocity are statistically independent. In three dimensions, the probability
density distribution of the velocity is:
where σ² =
kBT
m
P(Vx, Vy, Vz) = (2nо²)-³/² exp
20²
1. Show that the probability density of the velocity is normalized.
2. Find an expression of the arithmetic average of the speed.
3. Find and expression of the root-mean-square value of the speed.
4. Estimate the standard deviation of the speed.
Chapter 2 Solutions
An Introduction to Thermal Physics
Ch. 2.1 - Prob. 1PCh. 2.1 - Prob. 2PCh. 2.1 - Prob. 3PCh. 2.1 - Prob. 4PCh. 2.2 - For an Einstein solid with each of the following...Ch. 2.2 - Prob. 6PCh. 2.2 - Prob. 7PCh. 2.3 - Prob. 8PCh. 2.3 - Use a computer to reproduce the table and graph in...Ch. 2.3 - Use a computer to produce a table and graph, like...
Ch. 2.3 - Use a computer to produce a table and graph, like...Ch. 2.4 - Prob. 12PCh. 2.4 - Fun with logarithms. (a) Simplify the expression...Ch. 2.4 - Write e1023 in the form 10x, for some x.Ch. 2.4 - Prob. 15PCh. 2.4 - Prob. 16PCh. 2.4 - Prob. 17PCh. 2.4 - Prob. 18PCh. 2.4 - Prob. 19PCh. 2.4 - Suppose you were to shrink Figure 2.7 until the...Ch. 2.4 - Prob. 21PCh. 2.4 - Prob. 22PCh. 2.4 - Prob. 23PCh. 2.4 - Prob. 24PCh. 2.4 - Prob. 25PCh. 2.5 - Prob. 26PCh. 2.5 - Prob. 27PCh. 2.6 - How many possible arrangements are there for a...Ch. 2.6 - Consider a system of two Einstein solids, with...Ch. 2.6 - Prob. 30PCh. 2.6 - Fill in the algebraic steps to derive the...Ch. 2.6 - Prob. 32PCh. 2.6 - Use the Sackur-Tetrode equation to calculate the...Ch. 2.6 - Prob. 34PCh. 2.6 - According to the Sackur-Tetrode equation, the...Ch. 2.6 - For either a monatomic ideal gas or a...Ch. 2.6 - Using the Same method as in the text, calculate...Ch. 2.6 - Prob. 38PCh. 2.6 - Compute the entropy of a mole of helium at room...Ch. 2.6 - For each of the following irreversible process,...Ch. 2.6 - Describe a few of your favorite, and least...Ch. 2.6 - A black hole is a region of space where gravity is...
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Similar questions
- (a) Fun fact about factorials: (N - 1)! = N! / N , since dividing by N cancels the final factor in N! and leaves just the first N-1 factors. Use this to show that the multiplicity of an Einstein solid can be expressed as: (g + N)! q! N! N q+ N Then apply Stirling's Approximation to each of the factorials, to express the multiplicity as approximately (g + N)o+N N qª NN 2nq(q+ N)' (b) When N and q are both large, we can set the entire square-root in the above multiplicity expression to 1, leaving just: (q + N)N+q N(N,q) = Using this formula, find an expression for the total entropy of the Einstein solid. (c) Use your result from part (b) to find the solid's temperature as a function of its energy. (d) Invert your answer from part (c) to find the energy as a function of temperature, then use it to find a formula for the solid's heat capacity C.arrow_forwardLet's look at the term InN!. Compare its exact value with the more accurate one of Stirling's approximation, EANt InN!=N/N-N+ [N and to a lighter form. (~N! ~ N/NN-N₁₂₁₂ (N=1₂10₂100) Calculate the relative ellors of the approximations in all casesarrow_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_forward
- A. (a) Consider a canonical ensemble having N particle, V volume and at T temperature. Write down the expression of partition function (Q(N,V,T)) of this canonical ensemble in terms of the microstate energy Ej. (b) Write down the expression for Helmhotz free energy (A) and pressure (P) in terms of Q(N,V.T). (c) Now, assume that for a system of dense gas you can write down the Q(N,V,T) as, 1 (2amk,T Q(N,V,T)= N! (V- Nb)" e Treat a and b as constants. Get the expression for pressure (P) in terms of V, a, b, N, kg and T. Rearrange that expression to get a form where in the RHS of the equation will have Nk T. Identify the equation.arrow_forwarda) Make a diagram showing how many distinct ways (how many microstates, the multiplicity) there are of putting q = 2 indistinguishable objects in N = 3 boxes. Assuming that all microstates are equally probable, what is the probability that both objects are in the left-most box? What is the correct formula for the mulitiplicity as a function of N and q? b) Make a diagram showing how many distinct ways (the multiplicity) there are of putting q = 2 distinguishable objects in N= 3 boxes. Assuming that all microstates are equally probable, what is the probability that both objects are in the left-most box? Label the two objects R and G. What is the correct formula for the mulitiplicity as a function of N and q? Below are the diagrams, started for you. Complete the diagrams. distinguishable indistinguishable RG •. !R !Garrow_forwardThe 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.arrow_forward
- Obtain relations for the characteristic lengths of a large plane wall of thickness 2L, a very long cylinder of radius ro, and a sphere of radius ro.arrow_forwardWhy might the measured molar heat capacity of Cl2 not match the prediction of the equipartition theorem as well as that of O2?arrow_forwardFor a system of fermions at room temperature, compute the probability of a single-particle state being occupied if its energy is. 1 eV less than μarrow_forward
- If the given Vander Waals equation is (P+a/V^2)(V-b)=RT. The given states that a=3.592,R=0.08,b=0.04267, and P=10 and T=320. For finding V use a four step iteration.arrow_forwardFind q, △U, and the work done for path ACB for the mono atomic ideal gas system.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
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