Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Q.6.25 (page 237)

The analysis of this section applies also to liner polyatomic molecules, for which no rotation about the axis of symmetry is possible. An example is CO2, with ∈=0.000049eV. Estimate the rotational partition function for a CO2molecule at room temperature. (Note that the arrangement of the atoms isOCO, and the two oxygen atoms are identical.)

Short Answer

Expert verified

The rotational partition function for a CO2molecule at room temperature is localid="1649329748565" 264.

Step by step solution

01

Step 1. Given Information 

We are given aCO2molecule at room temperature.

02

Step 2. Rotational partition function  

The arrangement of the atoms in CO2molecule is same as that of the oxygen atoms,

Zrot=kT2ε

Substituting the values, we get

Zrot=kT2ε=(8.617×10-5eV)(300K)2(0.000049eV)=263.78

Rounding off to three significant figures, the rotational partition function at room temperature is 264.

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.

Start your free trial

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

Use a computer to sum the exact rotational partition function numerically, and plot the result as a function ofkT∈. Keep enough terms in the sum to be confident that the series has converged. Show that the approximation in equation 6.31 is a bit low, and estimate by how much. Explain the discrepancy

For a COmolecule, the constant €is 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

Estimate the partition function for the hypothetical system represented in Figure 6.3. Then estimate the probability of this system being in its ground state.

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 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.*
See all solutions

Recommended explanations on Physics Textbooks

Magnetism

Read Explanation

Famous Physicists

Read Explanation

Translational Dynamics

Read Explanation

Fluids

Read Explanation

Modern Physics

Read Explanation

Scientific Method Physics

Read Explanation
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