An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Chapter 6.2, Problem 16P
To determine
The average value of energy for any state with a reservoir.
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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.
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.
Consider a classical ideal gas of N diatomic heterogeneous molecules at temperature T. The charac-
teristic rotational energy parameter is € = 1 and the natural frequency of vibrations is wo. Consider
the temperature region where T≫er/kB, but T is of the order of ħwo/kB. Ignore contributions from
all other internal modes. Calculate the canonical partition function, the average energy, and the heat
capacity at constant volume, Cv.
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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- 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_forwardA. (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_forwardIf the partition function is Z= VT and V=3 m^3, T=280 K, then the Enthalpy * :will be 1238.91 J 216931.566 J O 345.23 J O 415.77 J Oarrow_forward
- Use as T and 3 In 2 5 S(E,V, N) = Nk ln + to derive the dependence of the chemical potential u on E, V , and N for an ideal classical gas. Then use 3 E -NkT to determine µ(T, V, N). +arrow_forwardthe partition function of a monatomic ideal gas is 1 (2amkT\ 3N|2 VN Q(N, V, T) : N! h? Derive expressions for the pressure and the energy from this partition function. Also show that the ideal gas equation of state is obtained if Q is of the form f(T)V^, where f(T) is any function of temperature.arrow_forwardA simple partition function The partition function of a hypothetical system is given by In Z = aT*V. %3D where a is a constant. Evaluate the mean energy E, the pressure P, and the entropy S.arrow_forward
- 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?arrow_forwardIn z Show that PK = 1- P P(O2) for a real gas where KT is the isothermal compressibility.arrow_forwardThe partition funetion for the ensemble characterized by constant V, E, and G = µÑ is given to a very good approximation by ø(V, E, µN)=Q(N,V,E)eBHN, where G = µN is the Gibbs energy (µ is the chemical potential and N is the average number of particles). Find an expression for the characteristic thermodynamic function for this ensemble in terms of the partition function ø(V, E, µN).arrow_forward
- 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.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_forwardThe intensities of spectroscopic transitions between the vibrational states of a molecule are proportional to the square of the integral ∫ψv′xψvdx over all space. Use the relations between Hermite polynomials given in Table 7E.1 to show that the only permitted transitions are those for which v′ = v ± 1 and evaluate the integral in these cases.arrow_forward
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