The Heat Capacity of Ideal Gases

Research: Science and Education

The Heat Capacity of Ideal Gases Robert L. Scott Department of Chemistry and Biochemistry, University of California, Los Angeles, Los Angeles, CA 90095-1569; *[email protected]

It is well known that, at high temperatures, the heat capacities, C, of each degree of freedom of an ideal gas (translation, rotation, and the vibrations of harmonic oscillators) reach constant values that are integral or half-integral values of NkB, where N is the number of molecules and kB is the Boltzmann constant (gas constant per molecule). For one mole of gas, NkB is just R, the gas constant per mole. Since the entropy, S, is just the integral of Cd(lnT ) and must remain finite, it follows that all heat capacities must approach zero at sufficiently low temperature T. While all substances condense to solids (except for helium, which remains as a liquid at ordinary pressures), it can be shown that even the heat capacities of gases would go to zero if condensation could be avoided. Experiments show that the vibrational heat capacity of a gas declines at low temperatures, as does the rotational heat capacity of hydrogen and its isotopes. It can be shown that the calculations of quantum statistical mechanics agree with these experiments, but it is not clear to many students (unless they carry out the detailed mathematics) why, for example, the heat capacities of ortho- and para-hydrogen are so different. We shall see how a new but exact variant of the usual statistical mechanical equation, one that shows the heat capacity as the sum of jumps from one energy level to another, can lead to better understanding. First, however, we briefly consider the classical view of heat capacity.

It is a feature of classical statistical mechanics that, when two volume elements in p,q space (“phase space”) of equal size have the same energies, there is an equal a priori probability of finding the system in one or in the other. (This is equivalent to the principle in quantum statistical mechanics of a priori probabilities of quantum states of the same energy.) When the energies are different, the probabilities must be modified by a simple Boltzmann factor. For a classical continuous distribution of a quantity x the probability of observing a value between x and x + dx is ᏼ(x)dx where ᏼ(x) is a probability density. Thus, we may write for the classical probability densities ᏼ( pi) and ᏼ(qi) ᏼ( pi ) d pi ∝ e

α p2 − i i kBT

∞

ε ( pi ) =

f

∑ αi pi 2 + i

(kinetic )

f

∑ βi qi 2

(1)

i

2

e

α p2 − i i kBT

where the summations are over the f degrees of freedom of a single molecule. (As originally set up, an expression such as eq 1 may contain cross terms of the type αijpipj and βijqiqj, but it is always possible to choose an appropriate coordinate system—the so-called “normal coordinates”—in which these cross terms vanish.) Thus, the molecular energy is separable into independent terms, one for each pi and one for each qi, and we can average each separately. •

dqi (2)

d pi

2

α i pi ∞ − e kBT d pi −∞ ∞ 2 − y 2

∫

∫ −∞ y

e

∞ − y 2

∫ −∞ e

(3)

dy

= dy

1 kB T 2

Exactly the same result, kBT2, is obtained for every other square term in the p’s and for each square term in the q’s. This leads to the classical “principle of equipartition of energy”, namely that an average energy kBT2 is associated with each variable, momentum or position, that contributes only a quadratic term to the total energy. Hence, we may expect an average kinetic energy of kBT2 for each molecular degree of freedom. The potential energy is somewhat more complex. For certain degrees of freedom (that is, translation and rotation), there is, in general, no restoring force, and hence no potential energy. For a vibrational degree of freedom, the potential energy upot may be expressed as a power series in the displacement q: upot − u0 = aq 2 + bq 3 +

(potential)

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∫ −∞ αi pi

= kBT

According to classical mechanics, a description of the microscopic state of a system involves the specification of the position, qi, and the momentum, pi, for each degree of freedom of the system. For a conservative system, it should then be possible to express the energy, U, of the system as a function of these variables. Similarly, for an ideal gas in which there are no interactions between molecules, the energy ε of a single molecule can be expressed as a function of its variables. If this molecular energy, which includes both kinetic energy and potential energy terms, can be represented as the sum of a series in which each variable, momentum or position, contributes only a quadratic term, we may write

βi qi 2 kBT

We can now use these equations to compute the average energy for a particular momentum coordinate pi:

The Classical View: Equipartition of Energy

ε =

dpi ; ᏼ(qi ) d qi ∝ e

−

…

(4)

To the extent that the cubic and higher terms in eq 4 can be ignored, the vibration is harmonic and the average potential energy for a vibrational degree of freedom is kBT2. Thus, we can see that, according to this classical picture, the average energy of a gas molecule should be kBT2 for each translational and rotational degree of freedom and kBT for each vibrational degree of freedom and that the total energy U of N noninteracting molecules is simply N times

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these. Since these energies are all linear in temperature, it follows that the heat capacity at constant volume C V = (∂U∂T )V is (32)NkB for translation, NkB or (32)NkB for rotation and NkB for each vibrational degree of freedom. (For molar heat capacities, the NkB reduces to the gas constant per mole R.) This prediction is found to be valid for monatomic gases (where the only effective degrees of freedom are translational) and approximately so for a few others. However, the general failure of this approach, coupled with similar anomalies in the heat capacities of solids, was one of the problems that led to the development of the concept of discrete energy levels and modern quantum theory.

The resulting equation can be written in various ways:

The Quantum-Mechanical View

where σε is the standard deviation of the molecular energy ε. Since the square of the standard deviation, the difference between the average of the square and the square of the average, is always positive, eq 11 guarantees that the heat capacity is always positive. The usual procedure for calculating the heat capacity as a function of temperature is to use eq 11, but it is hard to give a physical understanding of the detailed results. We now rewrite eq 11 in a different but equivalent form:

According to quantum mechanics, the noninteracting molecules in an ideal gas are found in a variety of quantum states of different energies εi. According to statistical mechanics, the canonical molecular partition function, Q, is then a sum over these states, Q =

∑ e

−

εi kBT

∑ e −βε

(5)

i

=

i

i

where it is convenient to write 1kBT as β. The probability ᏼi of finding a molecule in the ith state (now not a probability density because the energy distribution is discontinuous) is then e

ᏼi =

−βε i

e

=

Q

2

∑ e

CV N k B β2

=

(7)

A T 1 ∂ T

i

∑ e

−βεi

=

= σε

i

V

(8)

N kB β2

∑ ᏼi ∑ ᏼi εi

=

i

1072

∂ U ∂ T

= −kB β2 V

∂ U ∂ β

= − kB β 2 V


•

∂ β 2

∑ ᏼi εi ∑ ᏼi εi ∑ ᏼj εj

−

i

i

j

∑ ∑ ᏼi ᏼj εi 2 −∑ ∑ ᏼi ᏼj εi εj j

i

∑ ∑ ᏼi ᏼj (εi 2 i

j

− εi ε j

j

)

In the above equation there is nothing special about j as compared with i, so we can restore the symmetry by averaging with the exactly equivalent terms in which i and j are exchanged:

CV N kB β

(9)

∂ 2 (βA )

−

i

∑ ᏼj ∑ ᏼi εi 2

=

2

=

=

CV =

2

i

1 2

∑ ∑ ᏼi ᏼj (εi 2

1 2

j

+

∑ ∑ ᏼi ᏼj (ε j 2 − εj

j

)

)

− εi εj

j

∑ ∑ ᏼi ᏼj (εi i

− εi εj

i

or, recalling eq 6, it should come as no surprise that

where εi is the average energy per molecule. The heat capacity CV can be obtained by differentiation of the energy U or equivalently by double differentiation of the free energy A:

2

i

i

i

εi 2 − εi

2

V

U = N ∑ ᏼi ε i = N ε i

=

2

CV

∂ =

(11)

∑ ᏼi εi

−

i

=

it follows that the energy U can be written

U = N

∑ ᏼi εi 2

i

i

∂ (β A) ∂ β

i

=

A = −N kB T ln Q or β A = − N ln ∑ e

i

2

∑ e −β ε

(6)

−βεi

∑ εi e

−

i

j

−βεi

e

−β εi

2

Since the Helmholtz free energy, A, of the system is

i

=

∑ εi

2 −β εi

i

i

∑ εi e −β ε

i

−βε i

∑ e

−β εi

)

2

This double summation consists of three kinds of terms, those for which i < j, those for which i > j, and those for which i = j. The last group makes no contribution to the sum because then (εi − εj)2 = 0 and the other two are exactly alike; we can therefore remove the 12, exchange β for 1kBT and write

CV = N kB

(10) V

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∑ ∑ ᏼi ᏼj

εi − ε j

i< j

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kB T

2

(12)


tween the two levels; this appears in Figure 1 as an approach to constant values of ᏼ1 and ᏼ0 at high temperature. The system has an average energy g1ε1( g0 + g1), no further increase in the average energy is possible, and the electronic heat capacity will again be zero. Figure 2 shows the electronic heat capacities CelNkB for the various g1g0 ratios shown in Figure 1. We see that, if g1 = g0, Cel reaches a maximum value of 0.44NkB at T = 0.417θel. When g1 = 3g0, the maximum occurs at a slightly lower tem-

1.0

0.0

0.8

0.2

ᏼ0

0.4

g1/g 0 = 7/3 0.4

0.6

g1/g 0 = 3/1 g1/g 0 = 5/1

0.2

0.8

0.0

1.0 0.1

1

10

T / el Figure 1. Probability distribution of molecules between two electronic levels as a function of T/θel for various values of the degeneracy ratio g1/g0.

Electronic Heat Capacity There are a few molecules with one or more unpaired electrons (for example, NO and O2) that may make an electronic contribution to the thermodynamic functions, including the heat capacity. For a substance with only two low-lying electronic levels, the ground level 0 with energy ε0 = 0 and a degeneracy g0, and an excited level 1 with energy ε1 and a degeneracy g1, eq 12 reduces to a very simple form: g 0 g1 e

2

ε1 kBT

= g 0 + g1 e

ε1 kBT ε − 1 kBT

•

1.4

g1/g 0 = 5/1

1.2

2 1.0

2

(13)

If we define a characteristic temperature θel as ε1kB, the probabilities ᏼ0 and ᏼ1 = 1 − ᏼ0 can be shown (Figure 1) as a function of a “reduced temperature” Tθel for various values of the degeneracy ratio g1g0. (The particular choice of some of these values will become apparent in subsequent sections.) At very low temperatures substantially all the molecules are in the ground level, so ᏼ0 = 1, ᏼ1 = 0, and the electronic heat capacity will be zero. At higher temperatures molecules begin to jump to the excited level and there will be a sizeable contribution to the heat capacity. At even higher temperatures, where ε1kBT is essentially zero and eε1kBT is essentially unity, the molecules are randomly distributed bewww.JCE.DivCHED.org

1.6

Cel / (N kB)

−

C el ε − ε0 = ᏼ0 ᏼ1 1 NkB kBT

g1/g 0 = 1/1

0.6

ᏼ1

This equation is new, and—to the best of my knowledge—has not been previously derived. As we shall see, this equation explains and provides insight into the detailed variation of the heat capacity with temperature. Each term in the double summation of eq 12 represents the contribution to CV of the jump from a particular state i to another higher state j. That this depends on the probability of occupancy of the jth state as well as that of the ith state may seem surprising at first, but this is required by the principle of microscopic reversibility; otherwise one would measure different heat capacities with increasing temperatures and decreasing temperatures. Moreover, it is obvious that one can collect terms for quantum states of the same energy; thus, eq 12 applies equally well to energy levels ( g degenerate states with the same energy) with the appropriate level probabilities. It may not be immediately obvious that eq 12 leads to a limiting value of zero for the heat capacity as the temperature approaches zero, but the probability of the upper state ᏼj, which has the exponential form of eq 6, decreases much faster than ∆εkBT increases. We shall see that the heat capacity approaches close to its high-temperature (classical) limit even though only a few jumps are important. Moreover we shall explode the myth that the high-temperature limit is approached only when the energy spacings ∆ε are very small by comparison with kBT. Equation 12 will prove very helpful in giving physical interpretation to the variation of heat capacity with temperature. However, before discussing the general problem of heat capacities at temperatures where quantization is important, we first consider some very simple cases where there are only two levels of importance.

g1/g 0 = 3/1 0.8

0.6

g1/g 0 = 7/3

0.4

g1/g 0 = 1/1 0.2

0.0 0.1

1

10

T / el Figure 2. The electronic heat capacity CelNkB as a function of Tθel for the two-level systems shown in Figure 1.

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perature (0.35θel), and is very much higher (Cel = 1.02NkB); this is because the greater degeneracy of the upper level increases the probability of exciting molecules to that level when the temperature is right. This is confirmed by a consideration of the curves in Figure 1. Not only does the probability ᏼ1 rise at an earlier temperature for larger values of the ratio g1g0, but the rises are steeper. [The curves in Figure 1 and all the subsequent figures are shown with the logarithm of

1.0

j4 j=3

0.8

j=2 j=1

ᏼ

0.6

0.4

the reduced temperature for two reasons: to expand the scale and because the entropy is then proportional to the area under such a heat capacity curve S = ∫0∞ Cd(lnT ), or, as shown in the figures, SNkB = ∫0∞ (CNkB)d(lnT )] For nitric oxide ( g0 = g1 = 2, θel = 173 K), the maximum at 72 K cannot be observed because this temperature is far below the boiling point at 122 K, but the heat capacity is anomalously high and decreases with temperature between 130 and 300 K in essentially complete agreement with theoretical predictions. The anomalously high heat capacity of oxygen (ground level 3Σ) at elevated temperatures is attributed to a slight contribution of the first excited 1∆ level. We see that if there are only a few levels of significance, the heat capacity will ultimately fall to zero as all the factors (εj − εl )kBT become vanishingly small. Only when there is an effectively infinite set of levels of higher and higher energies do we get the classical limit of the equipartition principle. This is of course true for translation, rotation, and vibration. Because the translational heat capacity involves the failure of Maxwell–Boltzmann statistics, and hence of eqs 5 to 12, we consider rotation and translation first.

j=0

Rotational Heat Capacity The rotational energy of a diatomic molecule, regarded as a rigid rotor, is

0.2

ε rot ( j ) =

0.0 0.1

1

10

T / rot Figure 3. The probability distribution between the various rotational levels of a heteronuclear diatomic molecule as a function of T/θrot (all values of the rotational quantum number j allowed).

1.2

1.0

Crot /(NkB)

0.8

1

0

0.6

0.4

2

0 1

1

2 2

0.0 0.1

3

0 3

1

0.2

1

4

3 10

T / rot Figure 4. The rotational heat capacity CrotNkB as a function of Tθrot for a heteronuclear diatomic molecule showing the contribution of each jump j → j´ to the total heat capacity. (Except for 1 → 4, quantum jumps to levels higher than j = 3 are not shown.)

1074


•

j ( j + 1) h 2 8 π2 I

= j ( j + 1) kB θrot

(14)

where j is an integer quantum number ranging from zero to infinity, h is the Planck constant, I is the moment of inertia of the molecule, and θrot = h2(8π2IkB) is the “characteristic temperature” for rotation. (A real molecule is not completely rigid; there are higher terms in the energy owing to centrifugal stretching and rotation–vibration interaction. For precise calculation of heat capacities, these need to be included, but we ignore them here.) For heteronuclear diatomic molecules (for example, HD, HCl, BrCl), all j values are allowed, so the energy levels for successive j’s are proportional to 0, 2, 6, 12, 20, and so forth, while the degeneracies of each level (2j + 1) go up 1, 3, 5, 7, 9, and so forth. When these numbers are used to calculate the probabilities of the various levels, Figure 3 results; here for a particular “reduced temperature” Tθrot, a vertical line (of length unity) is divided into regions equal to ᏼ0, ᏼ1, ᏼ2, and so forth. Figure 4 shows the contributions of each jump (0 → 1, 0 → 2, 1 → 2, 0 → 3, 1 → 3, ands so forth.) to the heat capacity CrotNkB of a heteronuclear diatomic molecule (all values of the rotation quantum number j allowed) as a function of Tθrot, calculated from each term in eq 12, and the total heat capacity, which is the sum of these. We see that at low temperatures almost the entire contribution is from the 0 → 1 jump (compare this with the curve for the single jump for g1g0 = 3 in Figure 2) and that this produces a slight maximum (CrotNkB = 1.10) at Tθrot = 0.80. What is striking from Figures 3 and 4 is the feature that even when the heat capacity is approaching its classical limit, most of the molecules are still either in the ground level or the first excited level. It is not at all necessary that the molecules be widely distributed among a large number of lev-

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dency to excite molecules to still higher levels [ g4g2 = 95, while (ε4 − ε2)ε2 = 73] and the heat capacity declines. Of course, it does not decline to zero, but only to the classical limit because there is a spectrum of still higher levels to be filled. Again, only a few levels are occupied before the heat

1.0

j=4 0.8

j=2

ᏼ

0.6

j=0 0.4

0.2

0.0 0.1

10

1

T / rot Figure 5. The probability distribution between the various rotational levels of a homonuclear diatomic molecule as a function of Tθrot when only even values of the rotational quantum number j are allowed.

1.6

1.4

1.2

1.0

Crot / (N kB)

els before essentially classical behavior is found. For example, at Tθrot = 1.92 where ε1 − ε0 = 1.04kBT the heat capacity CrotNkB is 1.01, within 1% of its limiting value; yet Figure 3 shows that only levels j = 0, 1, and 2 have any significant populations. (In spite of this, Figure 4 shows that jumps to j = 3 contribute as much as 19% to the heat capacity.) At a slightly higher temperature where the jump ε1 − ε0 = kBT, CrotNkB is 1.009, almost the classical limit. As noted earlier, because of the logarithmic abscissa, the integral from T = 0 to a selected temperature T * yields an area under the curve that is equal to the entropy (in units of NkB) at T *. Moreover, the corresponding area under the curve for an individual jump is the contribution of that jump to the entropy at T *. For homonuclear diatomic molecules (for example, HH, DD, 35Cl35Cl), the situation is more complicated. Quantum mechanics (as exemplified in the Pauli exclusion principle) requires that the wave function Ψ must be symmetric or antisymmetric with respect to the exchange of identical particles. In particular, the wave function must be symmetric with respect to exchange of nuclei of even mass number and antisymmetric with respect to exchange of nuclei of odd mass number. All vibrational wave functions are symmetric with respect to exchange, as are the electronic functions for most stable molecules, so one is left with the rotational functions and the nuclear-spin functions. The rotational functions are symmetric for even j’s and antisymmetric for odd j’s. If the nuclear spin is sn, the total number of spin functions is (2sn + 1)2, of which sn(2sn + 1) are antisymmetric and (sn + 1)(2sn + 1) are symmetric. For HH (sn = 12), there is only one antisymmetric spin function (“para-hydrogen”) that, because the mass number is odd (1), must couple with even-j rotational functions; conversely, there are three symmetric spin functions (“ortho-hydrogen”) that couple with odd-j rotational functions. For DD, with an even mass number (2) and sn = 1, the situation is quite different; the six symmetric spin functions (“ortho-deuterium”) couple with even j’s, and the three antisymmetric spin functions (“para-deuterium”) couple with odd j’s. Except for their effect on the allowed rotational levels, the spin functions contribute nothing to the heat capacity. However, the situation requires that we distinguish between the rotational heat capacities when only even j’s or odd j’s are allowed. Figures 5 and 7 show the probabilities as a function of reduced temperature for the two cases. Figures 6 and 8 show the contributions of the various allowed jumps to the total rotational heat capacity as a function of temperature. (Note that, although the energy spacings between levels 0 and 1, 0 and 2, and 1 and 3 are different, the definition of reduced temperature is unchanged.) Species restricted to even j’s (para-H2, ortho-D2) show a CrotNkB that rises steeply to a maximum of 1.47 at T = 2.1θrot before declining to the classical limit of 1.00. The reason for this unusual behavior is fairly obvious from Figures 5 and 6. The j = 0 level is nondegenerate ( g0 = 1), while the first excited level ( j = 2) has a degeneracy g2 = 5. This leads to a relatively sudden increase in the heat capacity at a temperature at which large numbers of molecules start to be excited to the j = 2 level. Note that this is not very different from the maximum in the electronic heat capacity (Figure 2) for g1g0 = 5. After this sudden rush, there is no great ten-

0

2

0.8

0.6

0.4

4

0 0.2

4

2 2

0.0 0.1

6 0 6

1

10

T / rot Figure 6. The rotational heat capacity CrotNkB as a function of Tθrot for a homonuclear diatomic molecule restricted to even values of j showing the contribution of each allowed jump to the total heat capacity.

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capacity is virtually at its classical limit; CNkB is 1.01 at Tθrot = 4.81 where ε2 − ε0 = 1.25kBT. There Figure 5 shows that only three levels ( j = 0, 2, and 4) have significant populations, and Figure 6 shows that jumps to higher levels contribute only 4% to the heat capacity. Moreover, when the spacing ε2 − ε0 = kBT, CrotNkB is 1.0016, even closer to 1.0000 than the corresponding all-j case. Species restricted to odd j’s (ortho-H2, para-D2) show no such peculiar behavior. Note that in this case g3g1 = 73

1.0

j5 j=3

0.8

ᏼ

0.6

j=1 0.4

0.2

0.0 0.1

10

1

T/ rot Figure 7. The probability distribution between the various rotational levels of a homonuclear diatomic molecule as a function of Tθrot when only odd values of the rotational quantum number j are allowed.

1.2

1.0

Crot /(NkB)

0.8

1

3

1

5

0.6

0.4

0.2

1 7 3 7 3 5

0.0 0.1

10

1

T / rot Figure 8. The rotational heat capacity CrotNkB as a function of Tθrot for a homonuclear diatomic molecule restricted to odd values of j showing the contribution of each allowed jump to the total heat capacity.

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•

and g5g3 = 117, while (ε5 − ε5)(ε3 − ε1) = 95; so no special consideration attaches to the first excited level; the probabilities are shown in Figure 7. As seen in Figure 8 the heat capacity CrotNkB rises gradually to the classical value of NkB; the 1 → 3 jump can be compared with the electronic heat capacity in Figure 1 for the degeneracy ratio 73. (Actually calculations show that CrotNkB has a trivial maximum of about 1.0005 at about T = 7θrot and then slowly decreases to 1.0000; this is probably virtually unmeasurable.) The heat capacity reaches 0.99 at Tθrot = 4.69 where ε3 − ε1 = 2.13kBT. Here only two levels ( j = 1 and j = 3) are significantly populated (Figure 7), although jumps to j = 5 contribute 22% to CrotNkB (Figure 8). Finally, when ε3 − ε1 = kBT, CrotNkB is 1.0002. Fortunately, for comparison to theory, ortho- and parahydrogen can be separated (“normal” hydrogen at room temperature is 34 ortho- and 14 para- because of the different spin degeneracies), and the two species do not interconvert in the absence of a catalyst. Because hydrogen has such a low boiling point, the various species of hydrogen (H2, θrot = 85.3 K; HD, θrot = 64.3 K; D2, θrot = 43.0 K) and their various modifications have been studied experimentally (1), and their total heat capacities are in complete agreement, within experimental error, with the theoretical calculations shown in Figures 4, 6, and 8. The heat capacities of “normal” hydrogen and “normal” deuterium (that is, in the absence of catalysts) are simply the appropriate high-temperature averages (contributions of 14 and 34, 23 and 13, respectively) of the heat capacities for even and odd j’s (that is, of Figures 6 and 8). In the presence of a catalyst, however, the ortho–para equilibrium shifts with temperature, so the calculation is more complex; the results (2) that might be observed for equilibrium hydrogen are shown in Figures 9 and 10. Equilibrium hydrogen shows a curious result, not only a maximum but also a minimum. At the lowest temperatures, the gas is entirely para-hydrogen, but the first jump is from j = 0 to j = 1, with a ratio g1g0 = 9, when one allows for the threefold spin degeneracy of ortho-hydrogen. Thus, the peak is very high (2.07 NkB at Tθrot = 0.575). Because the gas is becoming increasingly ortho-, the contributions of jumps from 0 to 2 and from 1 to 2 are relatively unimportant (as shown in Figure 10). The next important contribution is the jump from 1 to 3 with a maximum at a much higher temperature (Tθrot = 3.650). Compare this to the corresponding contributions in Figures 4 and 8, where, in spite of the differences in probabilities, the maximum in the 1 → 3 jump occurs at almost exactly the same temperature. The large temperature separation of these two jumps leads to the heat-capacity minimum of 0.684NkB at Tθrot = 1.67. The heat-capacity curve for equilibrium deuterium is entirely different. The ground rotational state ( j = 0) is that of ortho-deuterium, and the high-temperature distribution is given by a different ratio of spin degeneracies (6 for orthoto 3 for para-). The similarity noted above between the curves for the 1 → 3 jump in Figures 4, 8, and 10 and a similarity (not shown) for the 0 → 4 and higher jumps in Figures 4, 6, and 10 may seem surprising, but should not be. At these temperatures the heat capacity has nearly reached its classical value of NkB and the rotational partition function Qrot (eq 5) is approxi-

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Research: Science and Education 1.0

4

5

3 0.8

2

j=1

ᏼ

0.6

0.4

j=0 0.2

0.0 0.1

10

1

T / rot Figure 9. The probability distribution between the various rotational levels of “equilibrium“ hydrogen H2 as a function of Tθrot (jumps between para-H2 and ortho-H2 allowed).

2.5

2.0

Crot /(N kB)

1.5

1.0

1

0 0.5

3

1 2

1 0

2

0.0 0.1

5

1 0

3 10

1

T / rot Figure 10. The rotational heat capacity CrotNkB as a function of Tθrot for “equilibrium” hydrogen showing the contribution of each jump j → j´ to the total heat capacity. Only the major contributions in this temperature range are shown; thus 2 → 3 and 0 → 4 are omitted. At higher temperatures, the principal contributions are between two odd-j levels.

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mately either Tθrot (all j’s allowed) or (12)(Tθrot) (even j’s or odd j’s). Thus, in this temperature region, the probabilities (see eq 6) for levels restricted to even j’s or odd j’s are twice those for the corresponding level when all j’s are allowed; consequently the contribution of a particular jump is approximately four times that for all j’s, and the maxima occur at approximately the same reduced temperature. The situation for equilibrium hydrogen at temperatures where the heat capacity has reached its classical value is more complicated. Because of the 3:1 ratio, the probability of finding a molecule in an even-j level is 12 that for the simple all-j case, while that for an odd-j level is 32 that for the allj case. Thus, the probabilities for an even–even jump are 14 that for the all-j case, that for an odd–odd jump are 94 times that for the all-j case, and that for even–odd or odd–even jump is 34 that for the all-j case. Other homonuclear diatomic molecules should in principle show the same effects as those shown by gaseous hydrogen and its isotopes, but for them the boiling point Tb is always far above the rotational characteristic temperature θrot. Regardless of the symmetry or lack thereof, the hightemperature heat capacity of a diatomic molecule is NkB, and the enthalpy H is similarly independent. However, this is not so for the entropy S or the free energy (A or G ). Because the number of states allowed is reduced by a factor of two, the entropy is reduced by NkBln 2, and the free energy is increased by NkBT ln2. For polyatomic molecules, one must distinguish between linear molecules with two degrees of rotational freedom and all other molecules with three degrees of rotational freedom. For all such molecules, the boiling points are sufficiently high that only the classical limits for CrotNkB (1 and 32, respectively) would be observed. The low-temperature behavior of spherical top molecules like methane has been calculated by MacDougall (3). For methane, there are (2sn + 1) spin functions, so for hydrogen with its half-integral spin, there are sixteen possible functions, which combine into a quintet, three triplets, and two singlets, which produce three different heat capacity curves at low temperatures. However, it is unlikely that quantum mechanical deviations could be observed. CH4 boils at 112 K and melts at 91 K where the vapor pressure is 101 atm. At 60 K, where the vapor pressure is 2 × 104 atm, the calculated CrotNkB is less than 0.1 higher than the classical value of 1.5; the maximum in the rotational heat capacity occurs at 30 K, where the vapor pressure is less than 1011 atm. Moreover, the three spin isomers apparently interconvert (4). Again, the high-temperature heat capacity and the enthalpy are unaffected by the symmetry, but the entropy is reduced by NkBlnσ and the free energy is increased by NkBT lnσ. Here the symmetry number σ is the factor by which the number of states available to a particular spin function is reduced. Quite generally, this is the number of distinct rotations (including the identity operation) that bring the molecule to a new position that would be different if the atoms were all different, but is not because some atoms are identical. Obviously, for linear molecules like hydrogen, as we have seen, the symmetry number σ can only be 1 or 2. However, for other molecules, higher values are possible. Thus, for methane σ = 12. (A more detailed discussion of symmetry numbers has been given by Winn in ref 4.)

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Vibrational Heat Capacity According to quantum mechanics, the energy of a harmonic oscillator is given by ε vib (v ) = v +

1 hν = 2

1 kB θ vib 2

v +

(15)

Q vib = e

hν − 2 kBT

= e

1 − e

θ − vib 2T

1− e

1.0

v8

3

hν − kBT

−1

θ − vib T

−1

=

1 2 sinh z

v=0

0.2

0.0 0.1

θ vib T

θvib T +

1.2

1

1.0

θvib T

2

e

θvib T 2

=

z sinh z

2

0.8

0.6

0.4

0

−1

which can be expanded in a power series in (θvibT )

0

2 0.0

θ vib T 12

2

+

θ vib T 240

0.1

4


•

1

2 0 3 0 4 0 5 1

10

T / vib …

(16)

(Similar attempts to develop power series for the rotational heat capacities lead to misleading results because of the pe1078

10

= z coth z

0.2

C vib = 1 − N kB

1

T / vib

θ − vib T

Further differentiation yields an explicit (closed-form) expression for Cvib,

e

7

v=1

0.4

Cvib /(N kB)

e

1 + 2 e

C vib = NkB

5

Figure 11. The probability distribution between the various vibrational levels (states) v of a harmonic oscillator as a function of Tθvib.

d lnQvib U vib = NkBT dlnT =

4

2

0.6

in which the characteristic temperature for vibration θvib = hνkB and z = θvib(2T ). Differentiation leads to an expression for the energy Uvib.

θ vib T

6

0.8

ᏼ

where v is the vibrational quantum number (0, 1, 2, 3, etc.) and ν is the fundamental frequency of vibration; the levels are nondegenerate. In quantum mechanics the energy of the ground state (v = 0) must be (12)hν because even in this state the molecule is not at rest at the minimum of the potential energy curve. However, in statistical mechanics the energies can all be expressed relative to the ground-state energy ε0, so the (12)hν is irrelevant to the heat capacity and could be discarded; however we shall carry out the derivations retaining it. The simple progression of harmonic-oscillator states permits the explicit evaluation of the sum in the vibrational partition function Qvib:

culiar way in which Euler–Maclaurin summations are carried out; see ref 6.) We return to eqs 6 and 12 to calculate the probabilities of the various levels (states in this case) shown in Figure 11 and the contributions of the various jumps to the heat capacity in Figure 12. Because the vibrational energy levels, unlike those for rotation, are nondegenerate and evenly spaced, the vibrational heat capacity Cvib rises monotonically with temperature from zero to the classical value of NkB.

Figure 12. The vibrational heat capacity CvibNkB as a function of Tθvib showing the contribution of each jump (v → v′) to the total heat capacity. Higher quantum jumps (0 → 6, 1 → 2, 1 → 3, 2 → 3, etc.) are not shown; all have much lower maxima than those shown.

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Moreover, this even spacing allows higher vibrational levels to be more easily excited. Figure 11 shows that, by the time CvibNkB reaches 0.99 at Tθvib = 2.88, many states are excited; even at Tθvib = 1.27 where CvibNkB is 0.95, 1.7% of the molecules are in states above v = 4. When the vibrational spacing ∆ε = k BT, C vibNk B is only 0.921, and, when CvibNkB = 0.99, ∆ε is 0.35kBT. This is very different from the rotational case, but still one would hardly say that ∆ε is very small by comparison to kBT. However, unlike the situation with rotation, there is no molecule that has all its vibrational degrees of freedom classically excited at room temperature, although iodine (θvib = 306.9 K) is very nearly so (CvibNkB is 0.94 at 298 K as compared to the limiting value of 1.00). In contrast, hydrogen with its extremely stiff bond (θvib = 5986 K) would not reach its classical harmonic-oscillator limit below 5000 K. (Actually, as will be seen in Figure 14, the vibrational heat capacity rises somewhat faster and higher because of anharmonicity.) Polyatomic molecules have several fundamental vibration frequencies (3n − 5 for linear molecules and 3n − 6 for others, where n is the number of atoms in the molecule). Polyatomic molecules always have one or more bond-bending vibrations; these are generally of lower frequency than bondstretching vibrations and are therefore more nearly classically excited. The vibrational heat capacity is then the sum of terms, one for each fundamental frequency. No vibrational degree of freedom is truly a harmonic oscillator. There are always a finite number of quantum states below the continuum corresponding to dissociation; moreover, because the spacing of levels gets closer together, this can ultimately lead to a vibrational heat capacity in excess of the classical limit. For most molecules (but not hydrogen), this is not a serious problem; the heat capacity reaches its classical limit long before anharmonicities contribute significantly. Translational Heat Capacity Even at temperatures well below those at which ordinary gases can be observed, the number of translational energy states with energies below kBT far exceeds the number of molecules N. Therefore, the probability that any particular state is occupied is very low. Thus, unlike rotational and vibrational degrees of freedom, the translational degrees of freedom are essentially classical down to these low temperatures. The translational partition function can then be written as Q trans =

2π m kBT h2

3 2

V

where m is the mass of the molecule and V is the volume. From this one can show that Utrans is 3NkBT2, and thus the translation of such an ideal gas contributes 3NkBT2 to CV, the heat capacity at constant volume, and 5NkBT2 to Cp, the heat capacity at constant pressure, in agreement with the equipartition principle. However, as we have noted earlier, the entropy is the integral of CdlnT and, if it is to remain finite, even the translational heat capacity must go to zero at sufficiently low temperatures. www.JCE.DivCHED.org

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Even if one could observe gases at still lower temperatures, the deviations from classical behavior would be due to the failure of Maxwell–Boltzmann statistics—that is, to a need to regard the molecules as indistinguishable—rather than to the discrete character of the energy levels. As noted earlier in the discussion of rotational levels, quantum mechanics requires that the wave function Ψ be antisymmetric with respect to the exchange of a pair of identical elementary particles (electrons, protons, neutrons). For systems of independent molecules, this means that two identical molecules, each containing an odd number of elementary particles, cannot occupy the same quantum state; such molecules (for example, HD, 14N16O, H12C14N) obey Fermi–Dirac statistics and are called “fermions”. Molecules containing even numbers of elementary particles (for example, 4He, 12CH4, and all homonuclear diatomic molecules such as H2, D2, 16O2, etc.) have no restrictions on occupancy of quantum states; they obey Bose–Einstein statistics and are called “bosons”. For a particle in a box (any shaped box), the number of states G(ε) with energy less than ε is given by the equation π 8m ε G (ε) = 6 h2

3 2

gV

where g is not a translational degeneracy, but one that multiplies the number of states at each level, such as the spin degeneracy of an electron, or that of the molecule resulting from nuclear spins. At the normal boiling point of a liquid, the gas volume V = NkBT(1 atm). At the normal boiling point of helium, T = 4.2 K, g(He4) = 1, so G(3kBT2)N = 10.3. Figure 13 shows the translational heat capacities CVNkB of a Bose–Einstein gas and a Fermi–Dirac gas. The characteristic temperature θtrans is given by the equation θ trans

h2 N = 2 πm kB gV

2 3

(17)

where m is the mass of the molecule and NV is the number of molecules per unit volume, that is, the molecular density. Note that, unlike all other characteristic temperatures, θtrans is not a measure of the spacing of the levels, that is, proportional to (ε1 − ε0); rather it is a measure of the difference between the Nth level and the ground state and turns out to be directly proportional to (εN − ε0). Note further that θtrans is a function of volume, so, unlike the other components of the heat capacity Ctrans we calculate (for fixed θtrans) is strictly a constant-volume heat capacity. The initial departure from classical behavior can be expressed as a power series in (θtransT )32, and, as the temperature falls, the Bose–Einstein heat capacity increases above the classical value, while the Fermi–Dirac heat capacity falls. However, below a temperature T = 0.527θtrans, the series fails to converge for the Bose–Einstein gas. Examination of the mathematical details (7) shows that below this temperature a significant fraction of all the molecules is in the ground state (momentum quantum numbers kx = ky = kz = 1); one thus speaks of a “condensation in momentum space”. Because there is a discontinuity in the temperature derivative of Ctrans, that is, a discontinuity in a third derivative of the (Helmholtz) free energy, one is tempted to call this a “third-order

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1079


2.0

1.5

Ctrans /(N kB)

Bose−Einstein Fermi−Dirac 1.0

0.5

0.0 0.01

0.1

1

10

T / trans Figure 13. The translational heat capacity CtransNkB for fixed volume V as a function of Tθtrans for Bose–Einstein and Fermi–Dirac statistics.

7

6

dissociation to 2H at p = 1 atm

anharmonicity, rotational stretching

Cp /(N kB)

5

rotational maximum (even j ’s)

4

harmonic vibration

GAS rigid rotation

gas nonideality

3

LIQUID

2

translation 1

SOLID

intermolecular vibrations

0

100

101

10 2

10 3

10 4

T/ K Figure 14. The total heat capacity CpNkB of para-H2 at a constant pressure of 1 atm as a function of temperature T showing the contributions of translation, rotation, and vibration (including anharmonicity) to the gas and including the heat capacities of solid and liquid and the contribution of dissociation at high temperatures.

1080


•

phase transition”. However, on an isothermal plot of pressure versus volume, the pressure remains constant for any volume less than that corresponding to θtrans (eq 17); thus, the “condensation” is actually first-order. This behavior is not observed in real gases, which condense (in position space) to liquids first, but the unusual properties of the low-temperature “superfluid” phase of 4He (helium-II) below the λ transition at 1.1 K are closely related to its being a Bose–Einstein fluid. Since 1995, research has produced at very low temperatures (nanokelvins, 109 K) a Bose–Einstein condensation of magnetically trapped alkali-metal atoms. Measurements (8) of the fraction of molecules in the ground state of 89Rb as a function of temperature show good agreement with the predictions for a finite number of noninteracting bosons in the three-dimensional harmonic potential produced by the magnets; indeed the observed fraction of molecules in the ground state differs only slightly from that predicted for a three-dimensional box. However, the variation of the energy as a function of temperature differs from that predicted for a three-dimensional box; the harmonic potential predicts (and experiment confirms) a discontinuity in the heat capacity itself, rather than in its derivative. Conversely, Fermi–Dirac statistics apply to the “free” electrons in metals. At sufficiently low temperatures, the heat capacity of the “free-electron gas” in metals should be proportional to the first power of the temperature (not apparent in Figure 13 because of the logarithmic scale), an effect that is observed experimentally in metals at low temperatures where the T 3-dependence of the vibrational heat capacity of crystals (the Debye effect) is negligible. Note that for both the Bose–Einstein gas and the Fermi– Dirac gas, Ctrans goes to zero as the temperature goes to zero. Note further that the areas under the two curves are the same, so at normal temperatures the translational entropies of a Fermi–Dirac gas and a Bose–Einstein gas of the same molecular mass are the same. (These translational entropies do not include any spin degeneracy of the substance, e.g., that of the electron or of ortho-hydrogen.) The Heat Capacity of Para-Hydrogen We can combine all these separate heat capacities to show the total constant-pressure heat capacity CpNkB of para-H2 at a pressure of 1 atm (Figure 14). For this molecule, the characteristic temperatures are θrot = 85.35 K, θvib = 5986 K, and θel = 131,923 K, and the rotational levels are restricted to even j’s. H2 is a Bose–Einstein gas, but, since it boils at 20.3 K, the translational heat capacity is classical; instead, we show the heat capacities of the solid (9) and the liquid (10). It should be noted that the heat capacity of liquid hydrogen is unusual; most liquids show heat capacities higher than that of the gas and are only slightly temperature dependent. Moreover, at 1 atm and low temperatures, there is a contribution from nonideality of the gas (essentially from the second virial coefficient B ) that must be added to the ideal translational term (11). The vibrational anharmonicity would ultimately produce a vibrational heat capacity well in excess of the classical value (12); however, at 1 atm hydrogen will start to dissociate into atoms at about 2000 K.

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The total heat capacity in the dissociation region can be calculated from the equation Cp NkB

=

α (2C1 − C 2 ) C2 + NkB N kB +

α 2

∆H ° RT

(1 − α ) 2

2

(18)

where Cp is the total heat capacity of N molecules of all the H2 whether dissociated or not, while C2 is the heat capacity of N molecules of undissociated H2, and C1 is the heat capacity of N atoms of completely dissociated H. ∆H is the molar enthalpy of dissociation of H2, while α is the degree of dissociation calculated from the equilibrium constant K using the equation 4α2

p p °

(

= K 1 − α2

)

(19)

where p = 1 atm and, in our case, p = p. All the necessary data (up to 5000 K) are available (12, 13). Were the heat capacity to be measured at a constant pressure of 1 atm at temperatures above 2000 K, the necessary input of energy to produce the dissociation would lead to a very large value as shown (14) in Figure 14. Indeed there would be a maximum value of about 45 for CpNkB near T = 3800 K. Summary We have seen how the heat capacity of an ideal gas can be expressed as the sum of individual jumps from one quantum state to a higher one. In addition, we have seen that the rotational heat capacity has nearly reached its classical limit of NkB before any energy spacing becomes less than kBT and before more than two or three states are significantly populated. Because of the uniform spacing of the states of a harmonic oscillator, the vibrational heat capacity shows a less extreme effect, but, when it has nearly reached the classical limit, the spacing is still a significant fraction of kBT.

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Literature Cited 1. Heat capacity measurements on ortho- and para-hydrogen are summarized in Farkas, A. Orthohydrogen, Parahydrogen and Heavy Hydrogen; Cambridge University Press: Cambridge, 1935; pp 33– 37. Since the experimental results agreed with theory within experimental error, few, if any, later experiments have been reported. 2. Mains, G. J.; Larson, J. W.; Hepler, L. G. J. Phys. Chem. 1984, 88, 1257–1261. 3. MacDougall, D. P. Phys. Rev. 1931, 38, 2296–2298. 4. Winn, J. S. Physical Chemistry; Harper Collins: New York, 1995; pp 859–864. 5. Curl, R. F., Jr.; Kasper, J. V. V.; Pitzer, K. S.; Sathianandan, K. J. Chem. Phys. 1966, 44, 4636. 6. Mayer, J. E.; Mayer, M. G. Statistical Mechanics, 1st ed.; Wiley: New York, 1940; p 153. (The comments are about Qrot, but apply equally well to Crot.) 7. See, for example, McQuarrie, D. A. Statistical Thermodynamics; Harper & Row: New York, 1973; pp 171–177. 8. Ensher, J. R.; Jin, D. S.; Mathews, M. R.; Wieman, C. E.; Cornell, E. A. Phys. Rev. Lett. 1996, 77, 4984–4987. 9. Krause, J. K.; Swenson, C. A. Phys. Rev. 1980, 21, 2533–2548, Table IX, p 2542. 10. Roder, H. M.; Weber, L. A.; Goodwin, R. D. Thermodynamic and Related Properties of Parahydrogen from the Triple Point to 100 K at Pressures to 340 Atmospheres; NBS Monograph 94; National Bureau of Standards: Washington DC, 1965; p 36. 11. Heat capacity data for the gas at 1 atm from 20 to 100 K taken from Krause, J. K.; Swenson, C. A. Phys. Rev. 1980, 21, 2533–2548. 12. Wagman, D. D.; Kilpatrick, J. E.; Taylor, W. J.; Pitzer, K. S.; Rossini, F. D. J. Res. Nat. Bur. St. 1984, 34, 143–161. 13. Rossini, F. D.; Pitzer, K. S.; Arnett, R. L.; Braun, R. M.; Pimentel, G. C. Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds; American Petroleum Institute: Washington DC, 1953. 14. Calculated to 5,000 K using heat capacity, heat of dissociation and equilibrium constants from Wagman, D. D.; Kilpatrick, J. E.; Taylor, W. J.; Pitzer, K. S.; Rossini, F. D. J. Res. Nat. Bur. St. 1984, 34, 143–161 and Rossini, F. D.; Pitzer, K. S.; Arnett, R. L.; Braun, R. M.; Pimentel, G. C. Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds; American Petroleum Institute: Washington DC, 1953. Extrapolations to 10,000 K are straightforward because the enthalpy of dissociation is very nearly constant over a wide range of temperatures.

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1081

The Heat Capacity of Ideal Gases

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