An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Textbook Question
Chapter 3.4, Problem 34P
(a) Find an expression for the entropy of this system in term of N and
(b) Write down a formula for L in terms of N and N1,
(c) For a one-dimensional system such as this, the length L is analogous to the volume V of a three-dimensional system. Similarly, the pressure P is replaced by the tension force F. Taking F to be positive when the rubber band is pulling inward, write down and explain the appropriate
(d) Using the thermodynamic Identity, you can flow express the tension force F in terms of a partial derivative of the entropy. From this expression, compute the tension in terms of L, T, N, and
(e) Show that when
(f) Discuss the dependence of the tension force on temperature. If you increase the temperature of a rubber band, does it tend to expand or contract? Does this behavior make sense?
(g) Suppose that you hold a relaxed rubber band in both hands and suddenly stretch it. Would you expect its temperature to increase or decrease? Explain. Test your prediction with a real rubber band (preferably a fairly heavy one with lots of stretch), using your lips or forehead as a thermometer. (Hint: The entropy you computed in part (a) is not the total entropy of the rubber band. There is additional entropy a8sociated with the vibrational energy of the molecules; this entropy depends on U but is approximately independent of L.)
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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
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b) Evaluate the partition function for a single harmonic oscillator. Use the result of
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дz
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d) What is the total energy of the system of N oscillators at temperature T?
1.7 A crystal has a basis of one atom per lattice point and a set of primitive translation
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c = 1.5(i + j+ k),
a = 3i,
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system. What is the Bravais lattice type of this crystal, and what are the Miller indices
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b= 3j.
In this and following questions, we develop a model for spontaneous
emission of a photon by a diatomic molecule AB (a model molecule),
which rotates and vibrates. In intermediate calculations, atomic units
(a.u.) will be used: unit of mass = the mass of electron, unit of charge
is the proton charge e, (e is a positive constant so that the charge of
electron is -e). The initial state of the molecule is an excited
rotational (1=1) and excited vibrational state (v=1). We consider a
molecule with the reduced mass µ = 10,000 a.u. (it is similar to the
mass of CO). After emitting a photon, the molecule will go to the 1=0,
v=0 state. The first question is about the model potential of the
molecule. It is represented by a potential of the form:
V(r)
=
C6
p12
C12
p6
"
where r is the distance between A and B in the molecule, C6 and C12
are positive constants (C6 =2 and C₁2-1). This potential has a well
meaning that the molecule is bound. The first thing to do is find
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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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