An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Chapter 3.3, Problem 24P
To determine
The table of entropy, temperature and heat capacity of an Einstein solid.
The graph of entropy versus energy, heat capacity versus temperature.
The comparison of heat capacity for lead, aluminum and diamond.
The numerical value of
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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.
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?
calculate the heat capacity of an Einstein solid in the low-temperature limit. Sketch the predicted heat capacity as a function of temperature. (Note: Measurements of heat capacities of actual solids at low temperatures do not confirm the prediction that you will make in this problem.
Chapter 3 Solutions
An Introduction to Thermal Physics
Ch. 3.1 - Use Table 3.1 to compute the temperature of solid...Ch. 3.1 - Use the definition of temperature to prove the...Ch. 3.1 - Figure 3.3 shows graphs of entropy vs. energy for...Ch. 3.1 - Can a miserly system, with a concave-up...Ch. 3.1 - Prob. 5PCh. 3.1 - Prob. 6PCh. 3.1 - Prob. 7PCh. 3.2 - Prob. 8PCh. 3.2 - In solid carbon monoxide, each CO molecule has two...Ch. 3.2 - An ice cube (mass 30 g) at 0C is left sitting on...
Ch. 3.2 - In order to take a nice warm bath, you mix 50...Ch. 3.2 - Estimate the change in the entropy of the universe...Ch. 3.2 - When the sun is high in the sky, it delivers...Ch. 3.2 - Experimental measurements of the heat capacity of...Ch. 3.2 - Prob. 15PCh. 3.2 - A bit of computer memory is some physical object...Ch. 3.3 - Prob. 17PCh. 3.3 - Prob. 18PCh. 3.3 - Prob. 19PCh. 3.3 - Prob. 20PCh. 3.3 - Prob. 21PCh. 3.3 - Prob. 22PCh. 3.3 - Prob. 23PCh. 3.3 - Prob. 24PCh. 3.3 - Prob. 25PCh. 3.3 - Prob. 26PCh. 3.4 - What partial-derivative relation can you derive...Ch. 3.4 - A liter of air, initially at room temperature and...Ch. 3.4 - Sketch a qualitatively accurate graph of the...Ch. 3.4 - As shown in Figure 1.14, the heat capacity of...Ch. 3.4 - Experimental measurements of heat capacities are...Ch. 3.4 - A cylinder contains one liter of air at room...Ch. 3.4 - Prob. 33PCh. 3.4 - Polymers, like rubber, are made of very long...Ch. 3.5 - Prob. 35PCh. 3.5 - Prob. 36PCh. 3.5 - Prob. 37PCh. 3.5 - Suppose you have a mixture of gases (such as air,...Ch. 3.6 - Prob. 39P
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- 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_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_forwardWhat 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_forward
- These two questions relate to the Boltzmann Equation. A gas of neutral hydrogen atoms in local thermodynamical equilibrium has 1/3 more atoms in the energy level n = 1 than in the n = 2 state. Calculate the temperature of that gas. For a gas of neutral hydrogen atoms make a graph that shows the ratios N2/N1, N3/N1, N4/N1 as a function of T.arrow_forwardStatistical Physics This is the chemical potential of an ideal gas. The second image is the answer to 4.20 problem. Please generate a solution for this problem (to validate the given answer). Thank you!arrow_forwardIn a certain physical system, there are two energy states available to a particle: the ground state with energy E₁ = 0 eV, and the excited state with energy E₂ = 1.5 eV. The system is in thermal equilibrium at a temperature T = 300 K. Calculate the Gibbs factor (also known as the Boltzmann factor) for the excited state . Give your answer to two decimal places.arrow_forward
- define the chemical potential in terms of derivatives of the ĈE energy E and enthalpy H. For a one component system, these are u= ON and Evaluate these expressions for an ideal gas and compare to ON µ = -kT In (kT/PA³ ) from H=| P,s OF and |= 1 TP ON v.rarrow_forwardConsider 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).arrow_forwardthe ideal gas? Do you expect these assumptions to work better for a dilute or dense gas? Explain. What are the two major assumptions that are made in deriving the partition function for 90arrow_forward
- e. Consider one unit cell and assume the length of the side of the cube is “a”. Remember that “a” is the distance between the centers of two adjacent atoms. How long is “a”, the edge of a unit cell, in terms of radius, r, of an atom? Write also your answer in the summary table.Answer: __________f. Based on the earlier questions, a simple cubic cell has the equivalent of only 1 atom. Recall the volume of sphere with radius, r, is expressed as V = 4/3 πr3. With this information, find the total volume of all the spheres in this unit cell, expressed in terms of r. (Hint: To do this, take the total number of atoms and multiply it by the volume of one atom, with radius, r)Answer: __________arrow_forwardProblem 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_forwardHi, could I get some help with this macro-connection physics problem involving the Ideal Gas Law? The set up is: What is the average volume in nm3 (cubic nanometers) taken up by molecules of an ideal gas at room temperature (taken as 300 K), and 1 atm of pressure or 101325 N/m2 to 4 digits of precision if kB = 1.38e-23 J/K and 1 nm = 10-9 m? Thank you.arrow_forward
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