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Use a computer to sum the rotational partition function (equation 6.30) algebraically, keeping terms through j = 6. Then calculate the average energy and the heat capacity. Plot the heat capacity for values ofkT/ϵ ranging from 0 to 3. Have you kept enough terms in Z to give accurate results within this temperature range?

Short Answer

Expert verified

The average energy is E¯=ϵj(j+1)(2j+1)e-j(j+1)/t(2j+1)e-j(j+1)/tand

The average heat capacity is C=k(2j+1)e-j(j+1)/tj2(j+1)2(2j+1)e-j(j+1)/tt(2j+1)e-j(j+1)/t2-kj(j+1)(2j+1)e-j(j+1)/t2t(2j+1)e-j(j+1)/t2

Step by step solution

01

Given information

Heat capacity for values kT/ϵranging from 0 to 3.

j = 6

02

Explanation

Rotational partition function is:

Z=(2j+1)e-j(j+1)ϵ/kT

Let,

t=kTϵ

Therefore,

Z=(2j+1)e-j(j+1)/t(1)

The average energy is:

E¯=-1ZZβ(2)

By chain rule:

Zβ=ZttβZβ=Ztβt-1

But β=1/kT, henceβ=1/tϵhence,

Zβ=t(2j+1)e-j(j+1)/tt1tϵ-1Zβ=j(j+1)(2j+1)e-j(j+1)/t1t2-1t2ϵ-1Zβ=-ϵj(j+1)(2j+1)e-j(j+1)/t

Substitute into (2)

role="math" localid="1647453692824" E¯=ϵj(j+1)(2j+1)e-j(j+1)/t(2j+1)e-j(j+1)/t(3)

The partial derivative of total energy with respect to temperature equals the heat capacity, which is:

C=E¯TC=kϵE¯t

Substitute from (3):

role="math" localid="1647454207193" C=ϵktϵj(j+1)(2j+1)e-j(j+1)/t(2j+1)e-j(j+1)/tC=k(2j+1)e-j(j+1)/tj2(j+1)2(2j+1)e-j(j+1)/tt(2j+1)e-j(j+1)/t2-kj(j+1)(2j+1)e-j(j+1)/t2t(2j+1)e-j(j+1)/t2

03

Explanation

Using python to plot a function between t and C/k. The code is given below:

The graph is:

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Most popular questions from this chapter

This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid or the internal vibrations of gas molecules) at temperature T. As in Section 2.2, the allowed energies of each oscillator are 0, hf, 2hf, and so on. (

a) Prove by long division that

11-x=1+x+x2+x3+

For what values of x does this series have a finite sum?

(b) Evaluate the partition function for a single harmonic oscillator. Use the result of part (a) to simplify your answer as much as possible.

(c) Use formula 6.25 to find an expression for the average energy of a single oscillator at temperature T. Simplify your answer as much as possible.

(d) What is the total energy of the system of N oscillators at temperature T? Your result should agree with what you found in Problem 3.25.

(e) If you haven't already done so in Problem 3.25, compute the heat capacity of this system and check t hat it has the expected limits as T0 and T.

2. Consider a classical particle moving in a one-dimensional potential well u(x), as shown The particle is in thermal equilibrium with a reservoir at temperature T, so the probabilities of its various states are determined by Boltzmann statistics.

{a) Show that the average position of the particle is given by

where each integral is over the entire xaxis.

A one-dimensional potential well. The higher the temperature, the farther the particle will stray from the equilibrium point.

(b) If the temperature is reasonably low (but still high enough for classical mechanics to apply), the particle will spend most of its time near the bottom of the potential well. In that case we can expand u(x)in a Taylor series about the equilibrium point x0: u(x)=ux0+x-x0dudxx0+12x-x02d2udx2x0

+13!x-x03d3udx3x0+

Show that the linear term must be zero, and that truncating the series after the quadratic term results in the trivial prediction x=x0.

(c) If we keep the cubic term in the Taylor series as well, the integrals in the formula for xbecome difficult. To simplify them, assume that the cubic term is small, so its exponential can be expanded in a Taylor series (leaving the quadratic term in the exponent). Keeping only the smallest temperature-dependent term, show that in this limit x differs from zo by a term proportional to kT. Express the coefficient of this term in terms of the coefficients of the Taylor series foru(x)

(d) The interaction of noble gas atoms can be modeled using the Lennard Jones potential,

u(x)=u0x0x12-2x0x6

Sketch this function, and show that the minimum of the potential well is at x=x0, with depth u0. For argon, x0=3.9Aand u0=0.010eV. Expand the Lennard-Jones potential in a Taylor series about the equilibrium point, and use the result of part ( c) to predict the linear thermal expansion coefficient of a noble gas crystal in terms of u0. Evaluate the result numerically for argon, and compare to the measured value

α=0.0007K-1(at80K)

In the low-temperature limit (kT<<), each term in the rotational partition function is much smaller than the one before. Since the first term is independent of T, cut off the sum after the second term and compute the average energy and the heat capacity in this approximation. Keep only the largest T-dependent term at each stage of the calculation. Is your result consistent with the third law of thermodynamics? Sketch the behavior of the heat capacity at all temperature, interpolating between the high-temperature and low- temperature expressions.

In this problem you will investigate the behavior of ordinary hydrogen, H2, at low temperatures. The constant εis0.0076eV. As noted in the text, only half of the terms in the rotational partition function, equation6.3, contribute for any given molecule. More precisely, the set of allowed jvalues is determined by the spin configuration of the two atomic nuclei. There are four independent spin configurations, classified as a single "singlet" state and three "triplet" states. The time required for a molecule to convert between the singlet and triplet configurations is ordinarily quite long, so the properties of the two types of molecules can be studied independently. The singlet molecules are known as parahydrogen while the triplet molecules are known as orthohydrogen.

(a) For parahydrogen, only the rotational states with even values of j are allowed.Use a computer (as in Problem6.28) to calculate the rotational partition function, average energy, and heat capacity of a parahydrogen molecule. Plot the heat capacity as a function of kT/t

(b) For orthohydrogen, only the rotational states with odd values of jare allowed. Repeat part (a) for orthohydrogen.

(c) At high temperature, where the number of accessible even-j states is essentially the same as the number of accessible odd-j states, a sample of hydrogen gas will ordinarily consist of a mixture of 1/4parahydrogen and 3/4orthohydrogen. A mixture with these proportions is called normal hydrogen. Suppose that normal hydrogen is cooled to low temperature without allowing the spin configurations of the molecules to change. Plot the rotational heat capacity of this mixture as a function of temperature. At what temperature does the rotational heat capacity fall to half its hightemperature value (i.e., to k/2per molecule)?

(d) Suppose now that some hydrogen is cooled in the presence of a catalyst that allows the nuclear spins to frequently change alignment. In this case all terms in the original partition function are allowed, but the odd-j terms should be counted three times each because of the nuclear spin degeneracy. Calculate the rotational partition function, average energy, and heat capacity of this system, and plot the heat capacity as a function of kT/t.

(e) A deuterium molecule, D2, has nine independent nuclear spin configurations, of which six are "symmetric" and three are "antisymmetric." The rule for nomenclature is that the variety with more independent states gets called "ortho-," while the other gets called "para-." For orthodeuterium only even-j rotational states are allowed, while for paradeuterium only oddj states are allowed. Suppose, then, that a sample of D2gas, consisting of a normal equilibrium mixture of 2/3ortho and 1/3para, is cooled without allowing the nuclear spin configurations to change. Calculate and plot the rotational heat capacity of this system as a function of temperature.*

Derive equation 6.92 and 6.93 for the entropy and chemical potential of an ideal gas.

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