Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

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.

Short Answer

Expert verified

The graph is as follows,

Step by step solution

01

Step 1. Given Information

We are given that in the low-temperature limit (kT<<), each term in the rotational partition function is much smaller than the one before.

02

Step 2. Expanding the partition function 

Expanding the partition function by neglecting all higher order terms in the low temperature limit (kT<<).

Zrot=1+(2(1)+1)e-1(1+1)/kT=1+3e-2e/kT

The average energy of the system is given as follows:

Erot=-1ZrotZrotβ=-11+3e-2ββ(1+3e-2β)=-11+3e-2β(3e-2β(-2))=6e-2β1+3e-2β

Neglecting the term 3e-2βin the denominator of 6e-2β1+3e-2β,

Since the function 3e-2βapproaches to zero as βbecause e-=0.

Therefore, the average energy of the system is 6e-2β.

03

Step 3. Specific heat capacity of the system

The specific heat capacity of the system is given by,

C=ErotT=T(6e-2β)=6T(e-2/kT)=6(e-2/kT)-2k-1T2=122kT2e-2/kT=3k2kT2e-2/kT

Therefore, the heat capacity of the system in the low temperature limit is C=3k2kT2e-2/kT.

In the low temperature limit, the heat capacity of the system decreases exponentially to zero.

04

Step 4. Behavior ho heat capacity

The low and high temperature limit of Ckare plotted against the dimensionless parameter kT.

The graph is as follows,

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

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.

Consider a large system of Nindistinguishable, noninteracting molecules (perhaps an ideal gas or a dilute solution). Find an expression for the Helmholtz free energy of this system, in terms of Z1, the partition function for a single molecule. (Use Stirling's approximation to eliminate the N!.) Then use your result to find the chemical potential, again in terms ofZ1.

For an O2molecule, the constant is approximately 0.00018eV. Estimate the rotational partition function for an O2molecule at room temperature.

Apply the result of Problem 6.18 to obtain a formula for the standard deviation of the energy of a system of N identical harmonic oscillators (such as in an Einstein solid), in the high-temperature limit. Divide by the average energy to obtain a measure of the fractional fluctuation in energy. Evaluate this fraction numerically for N = 1, 104, and 1020. Discuss the results briefly.

Consider a hypothetical atom that has just two states: a ground state with energy zero and an excited state with energy 2 eV. Draw a graph of the partition function for this system as a function of temperature, and evaluate the partition function numerically at T = 300 K, 3000 K, 30,000 K, and 300,000 K.

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free