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In problem 6.20 you computed the partition function for a quantum harmonic oscillator :Zh.o.=11-e-βε, where ε=hfis the spacing between energy levels.

(a) Find an expression for the Helmholtz free energy of a system of Nharmonic oscillators.

(b) Find an expression for the entropy of this system as a function of temperature. (Don't worry, the result is fairly complicated.)

Short Answer

Expert verified

part (a): F=NkT1-e-βε

part (b):role="math" localid="1647054975548" S=-Nkln1-e-βε+NkεkTeβε-1

Step by step solution

01

Part (a): Step 1. Given information

The partition function of a single harmonic oscillator is given by

Zh.o.=11-e-βε...................(1)
02

Part (a): Step 2. Calculation

The formula to calculate the Helmholtz free energy for th single harmonic oscillator is given by

F=-kTlnZ.....................(2)

Substitute the value of Zfrom equation (1) into equation (2) to calculate the free energy for single harmonic oscillator.

F=-kTln11-e-βε=kTln1-e-βε.......................(3)

Since, Helmholtz free energy is an extensive property, multiply both sides of equation (3) by Nto obtain the required free energy for Nharmonic oscillators.

role="math" localid="1647054790522" FN=NF=NkTln1-e-βε

Here,FNis the Helmholtz free energy forNharmonic oscillator.

03

Part (b): Step 1. Calculation of entropy

The formula to calculate the entropySof the system is given by

S=-FTN..........................(4)

Substitute the formula for Helmholtz free energy from equation (3) into equation (4) and simplify to obtain the required entropy of the system.

role="math" localid="1647054991723" S=-TNkTln1-e-βε=-Nkln1-e-βε-NkT1-e-βε-1εe-βεβε=-Nkln1-e-βε+NkεkTeβε-1

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Most popular questions from this chapter

For a CO molecule, the constant ϵis approximately 0.00024eV. (This number is measured using microwave spectroscopy, that is, by measuring the microwave frequencies needed to excite the molecules into higher rotational states.) Calculate the rotational partition function for a COmolecule at room temperature (300K), first using the exact formula 6.30 and then using the approximate formula 6.31.

In this section we computed the single-particle translational partition function,Ztr, by summing over all definite-energy wave functions. An alternative approach, however, is to sum over all position and momentum vectors, as we did in Section 2.5. Because position and momentum are continuous variables, the sums are really integrals, and we need to slip a factor of 1h3to get a unitless number that actually counts the independent wavefunctions. Thus we might guess the formula

role="math" localid="1647147005946" Ztr=1h3d3rd3pe-EtrkT

where the single integral sign actually represents six integrals, three over the position components and three over the momentum components. The region of integration includes all momentum vectors, but only those position vectors that lie within a box of volume V. By evaluating the integrals explicitly, show that this expression yields the same result for the translational partition function as that obtained in the text.

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(a) Show that the entropy in this case is

S=NkInVZeZrotNvQ+72.

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In most paramagnetic materials, the individual magnetic particles have more than two independent states (orientations). The number of independent states depends on the particle's angular momentum “quantum number” j, which must be a multiple of 1/2. For j = 1/2 there are just two independent states, as discussed in the text above and in Section 3.3. More generally, the allowed values of the z component of a particle's magnetic moment are

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