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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 鈭�=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=kT2蔚

Substituting the values, we get

Zrot=kT2蔚=(8.617脳10-5eV)(300K)2(0.000049eV)=263.78

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

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Most popular questions from this chapter

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炉,fori=1to5. Call these deviations 螖Ei.

(b) Compute the average of the squares of the five deviations, that is, 螖Ei2炉. Then compute the square root of this quantity, which is the root-mean- square (rms) deviation, or standard deviation. Call this number 蟽E. Does 蟽Egive a reasonable measure of how far the individual values tend to stray from the average?

(c) Prove in general that

蟽E2=E2炉-(E炉)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.

A particle near earth's surface traveling faster than about 11km/shas enough kinetic energy to completely escape from the earth, despite earth's gravitational pull. Molecules in the upper atmosphere that are moving faster than this will therefore escape if they do not suffer any collisions on the way out.

(a) The temperature of earth's upper atmosphere is actually quite high, around 1000K.Calculate the probability of a nitrogen molecule at this temperature moving faster than 11km/s, and comment on the result.

(b) Repeat the calculation for a hydrogen molecule (H2)and for a helium atom, and discuss the implications.

(c) Escape speed from the moon's surface is only about 2.4km/s. Explain why the moon has no atmosphere.

Carefully plot the Maxwell speed distribution for nitrogen molecules at T=300K and atT=600K. Plot both graphs on the same axes, and label the axes with numbers.

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.

In the real world, most oscillators are not perfectly harmonic. For a quantum oscillator, this means that the spacing between energy levels is not exactly uniform. The vibrational levels of an H2 molecule, for example, are more accurately described by the approximate formula

En鈮�系1.03n-0.03n2,n=0,1,2,鈥�

where 系 is the spacing between the two lowest levels. Thus, the levels get closer together with increasing energy. (This formula is reasonably accurate only up to about n = 15; for slightly higher n it would say that En decreases with increasing n. In fact, the molecule dissociates and there are no more discrete levels beyond n 鈮�15.) Use a computer to calculate the partition function, average energy, and heat capacity of a system with this set of energy levels. Include all levels through n = 15, but check to see how the results change when you include fewer levels Plot the heat capacity as a function of kT/系. Compare to the case of a perfectly harmonic oscillator with evenly spaced levels, and also to the vibrational portion of the graph in Figure 1.13.
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