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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

Suppose you have 10 atoms of weberium: 4 with energy 0 eV, 3 with energy 1 eV, 2 with energy 4 eV, and 1 with energy 6 eV.

(a) Compute the average energy of all your atoms, by adding up all their energies and dividing by 10.

(b) Compute the probability that one of your atoms chosen at random would have energy E, for each of the four values of E that occur.

(c) Compute the average energy again, using the formulaE¯=sE(s)P(s)

A lithium nucleus has four independent spin orientations, conventionally labeled by the quantum number m = -3/2, -1/2, 1/2, 3/2. In a magnetic field B, the energies of these four states are E = mμB, where μ the constant is 1.03 x 10-7 eV/T. In the Purcell-Pound experiment described in Section 3.3, the maximum magnetic field strength was 0.63 T and the temperature was 300 K. Calculate the probability of a lithium nucleus being in each of its four spin states under these conditions. Then show that, if the field is suddenly reversed, the probabilities of the four states obey the Boltzmann distribution for T =-300 K.

At room temperature, what fraction of the nitrogen molecules in the air are moving at less than300m/s?

The most common measure of the fluctuations of a set of numbers away from the average is the standard deviation, defined as follows.

(a) For each atom in the five-atom toy model of Figure 6.5, compute the deviation of the energy from the average energy, that is, Ei-E¯,fori=1to5. Call these deviations ΔEi.

(b) Compute the average of the squares of the five deviations, that is, ΔEi2¯. Then compute the square root of this quantity, which is the root-mean- square (rms) deviation, or standard deviation. Call this number σE. Does σEgive a reasonable measure of how far the individual values tend to stray from the average?

(c) Prove in general that

σE2=E2¯-(E¯)2

that is, the standard deviation squared is the average of the squares minus the square of the average. This formula usually gives the easier way of computing a standard deviation.

(d) Check the preceding formula for the five-atom toy model of Figure 6.5.

Cold interstellar molecular clouds often contain the molecule cyanogen (CN), whose first rotational excited states have an energy of 4.7x 10-4 eV (above the ground state). There are actually three such excited states, all with the same energy. In 1941, studies of the absorption spectrum of starlight that passes | through these molecular clouds showed that for every ten CN molecules that are in the ground state, approximately three others are in the three first excited states (that is, an average of one in each of these states). To account for this data, astronomers suggested that the molecules might be in thermal equilibrium with some "reservoir" with a well-defined temperature. What is that temperature?

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