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Chapter 3: Q. 3.26 (page 108)

The results of either of the two preceding problems can also be applied to the vibrational motions of gas molecules. Looking only at the vibrational contribution to the heat capacity graph for H2shown in Figure 1.13, estimate the value of εfor the vibrational motion of anH2 molecule.

Short Answer

Expert verified

The value of εfor the vibrational motion of an H2molecule is0.44eV.

Step by step solution

01

Given Information

The given molecule is H2.

Heat capacity:

role="math" localid="1647281996407" C=ε2NeεkTkT2eεkT-12

02

Calculation

The heat capacity is given as:

C=ε2NeεkTkT2eεkT-12

Let t=kTε,

Hence, the above equation becomes,

C=ε2NeεkTkT2eεkT-12C=Nke1tt2e1t-12CNk=e1tt2e1t-12

The value of tat which the heat capacity is half of its maximum value is:

C=12Nk

Hence, the above equation becomes,

CNk=e1tt2e1t-1212t2e1t-12=e1tt=0.335

Now, by resubstituting the value of t, we get,

role="math" localid="1647282058861" kTε=0.335

For vibrational motion of H2gas molecules at its heat capacity, half of its maximum value, the temperature is, T=1700K.

Hence, εcan be calculated as:

ϵ=kT0.335ε=8.62×10-5×17000.335ε=0.44eV

03

Final answer

Hence, the value ofεcan be calculated as0.44eV.

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Most popular questions from this chapter

An ice cube (mass 30g)0°Cis left sitting on the kitchen table, where it gradually melts. The temperature in the kitchen is 25°C.

(a) Calculate the change in the entropy of the ice cube as it melts into water at 0°C. (Don't worry about the fact that the volume changes somewhat.)

(b) Calculate the change in the entropy of the water (from the melted ice) as its temperature rises from 0°Cto 25°C.

(c) Calculate the change in the entropy of the kitchen as it gives up heat to the melting ice/water.

(d) Calculate the net change in the entropy of the universe during this process. Is the net change positive, negative, or zero? Is this what you would expect?

Figure 3.3 shows graphs of entropy vs. energy for two objects, A and B. Both graphs are on the same scale. The energies of these two objects initially have the values indicated; the objects are then brought into thermal contact with each other. Explain what happens subsequently and why, without using the word "temperature."

Use a computer to study the entropy, temperature, and heat capacity of an Einstein solid, as follows. Let the solid contain 50 oscillators (initially), and from 0 to 100 units of energy. Make a table, analogous to Table 3.2, in which each row represents a different value for the energy. Use separate columns for the energy, multiplicity, entropy, temperature, and heat capacity. To calculate the temperature, evaluate ΔU/ΔSfor two nearby rows in the table. (Recall that U=qϵfor some constant ϵ.) The heat capacity (ΔU/ΔT)can be computed in a similar way. The first few rows of the table should look something like this:

(In this table I have computed derivatives using a "centered-difference" approximation. For example, the temperature .28is computed as 2/(7.15-0).) Make a graph of entropy vs. energy and a graph of heat capacity vs. temperature. Then change the number of oscillators to 5000 (to "dilute" the system and look at lower temperatures), and again make a graph of heat capacity vs. temperature. Discuss your prediction for the heat capacity, and compare it to the data for lead, aluminum, and diamond shown in Figure 1.14. Estimate the numerical value of εin electron-volts, for each of those real solids.

Experimental measurements of the heat capacity of aluminum at low temperatures (below about 50K) can be fit to the formula

CV=aT+bT3

where CVis the heat capacity of one mole of aluminum, and the constants aand bare approximately a=0.00135J/K2and b=2.48×10-5J/K4. From this data, find a formula for the entropy of a mole of aluminum as a function of temperature. Evaluate your formula at T=1Kand at T=10K, expressing your answers both in conventional units (J/K)and as unitless numbers (dividing by Boltzmann's constant).

Consider an ideal two-state electronic paramagnet such as DPPH, with μ=μB. In the experiment described above, the magnetic field strength was 2.06T and the minimum temperature was 2.2K. Calculate the energy, magnetization, and entropy of this system, expressing each quantity as a fraction of its maximum possible value. What would the experimenters have had to do to attain99% of the maximum possible magnetization?
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