From quantum mechanics, the single particle energy levels are given by

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1 The Heat Capacity of a Diatomic Gas 5.1 INTRODUCTION We have seen how statistical thermodynamics provides deep insight into the classical description of a monatomic ideal gas. We might have reason to hope, therefore, that the statistical model can resolve a thorny problem we encountered in the application of classical thermodynamics to a diatomic gas. The principle of equipartition of energy fails to give the observed value of the specific heat capacity. The explanation of this discrepancy was considered by Maxwell to be the most important challenge facing the statistical theory. In this chapter we shall see how the problem is solved. 5. THE QUANTIZED LINEAR OSCILLATOR Until now we have confined our attention to systems of particles that have translational degrees of freedom only. To deal with particles such as diatomic molecules, we need to investigate so-called internal degrees of freedom such as vibrations, rotations, and electronic excitations. We consider an assembly of N one-dimensional harmonic oscillators. We assume that the oscillators are loosely coupled in that the energy exchange among them is small. This means that each particle can oscillate nearly independently of the others. We further assume that each oscillator is free to vibrate in one dimension only with some natural frequency ". From quantum mechanics, the single particle energy levels are given by " j = ( j + 1 )h#, j = 0,1,,... (5.1) Note that the energies are equally spaced and that the ground state has zero-point energy equal to h". The states are nondegenerate in that g j =1 for all j. For the internal degrees of freedom, Boltzmann statistics applies. The assumption may seem questionable since Boltzmann statistics characterizes distinguishable particles (localized in a crystal lattice, for example) and would therefore appear to be inappropriate for the treatment of an assembly of diatomic molecules. However, in the dilute gas approximation, the number 1

2 of translational quantum states is so much larger than the number of particles ( g j >> N j ) that the great majority of states are unoccupied, a few are occupied by a single particle, and virtually none have a population greater than one. Thus, in our treatment of the internal degrees of freedom of diatomic molecules, we can regard the particles as differentiated by the translational quantum states that they occupy. We begin by evaluating the partition function Z = g j e "# j kt % = e " ( j + 1 )h& kt 5.) j= 0 j= 0 The temperature at which kt = h" is called the characteristic temperature " : " = h# k Using this in Equation (5.), we have Z = e " ( j + 1 )# T % = e "# T % e " j# T = e "# T 1+ e "# T + e "# T +... j= 0 j= 0 ( ) The sum in this expression is just an infinite geometric series of the form Then 1+ y + y +... = 1 1" y. Z = e"# T (5.3) "# T 1" e We shall also be concerned with the occupation numbers, or with N j N, the fraction of the total number of particles with energy " j. The Boltzmann distribution for g j =1 is kt N j N = e"# j Z = e "# j kt e T ( 1" e " T ) = ( 1" e " T )e "# j kt + T. (5.4) The exponent of the term outside the parentheses can be written

3 " # j kt + T = " ( j + 1 ) h% kt + h% kt = " j h% kt = " j T (5.5) Thus Equation (5.4) becomes N j N = e" j# T 1" e "# T ( ) (5.6) A sketch of Equation (5.6) for two temperatures shows that the lower the temperature, the more rapidly the occupation numbers decrease with j (Figure 5.1). At higher temperatures, more particles populate the higher energy levels. Figure 5.1 Fractional occupation numbers for quantized linear oscillators with (a) T = ", and (b) T = ". Next we compute the internal energy of the assembly of oscillators. The expression we obtained for U in terms of the derivative of lnw (Equation (3.7)) applies here because the Boltzmann distribution is the same as the Maxwell-Boltzmann distribution. We have #"ln Z & U = NkT "T ' (5.7) From eqn (5.3) lnz = " # T T " ln( 1" e"# ) 3

4 so ( ) 1 e # T ( ) " "T lnz = # T e # T % # T = # & 1 T + e# T ' ( 1 e # T U = Nk" ' & ) (5.8) % e " T #1( ) * + LIMIT: T " 0, U = Nk# = Nh, the zero-point energy. LIMIT: As T " # such that T " >>1 or " T <<1, we can expand the denominator of the second term in a Taylor series: e " T #1=1+ " ' & ) + 1 " ' & ) % T (! % T ( +...#1 * " T U " Nk# 1 + T ' & ) " NkT, % # ( T # >>1 This is exactly what we would expect for a diatomic molecule with two vibrational degrees of freedom, one associated with the kinetic energy and the other with the potential energy. Note that in our model we have suppressed any kinetic energy of translation. The variation of U with T is sketched in Figure 5.. Figure 5. ariation with temperature of the internal energy of an assembly of linear oscillators. 5.3 IBRATIONAL MODES OF DIATOMIC MOLECULES The most important application of these results is to the molecules of a diatomic gas. From classical thermodynamics, # = "U & "T ' for a reversible process. Using Equation (5.8), we have 4

5 or = Nk" # ( #T e" T 1) 1, # = Nk " & T ' e " T ( e " T )1) (5.9) LIMIT: As T " #, at very high temperatures T " >>1 or " T <<1, and " Nk # ' & ) % T ( 1 # ' = Nk. & ) % T ( That is, the heat capacity approaches the constant Nk. LIMIT: As T " 0, T " <<1 or " T >>1, we have e " T >>1 and e " T ( e " T #1) 1 e " T = e#" T so that " Nk # ' & ) % T ( e *# T The rate at which the exponential factor approaches zero as T " 0 is greater than the rate of growth of " T ( ) that " 0 as T " 0, consistent with the third law. The variation of with T is shown schematically in Figure 5.3. At an extremely low temperature T << " or kt << h" ; the oscillators (particles) are frozen into the ground state, in that virtually none are thermally excited at any one time. We have U = Nh" and = 0. This is the extreme quantum limit. The discrete quantum nature of the states totally dominates the physical properties. 5

6 At the opposite extreme of temperature for which T >> " or kt >> h", we reach a classical limit. Here the expressions for U and do not involve h". Planck s constant, the scale of the energy level separation, is irrelevant. Oscillators of any frequency have the same average energy kt. The dependence of U on kt is associated with the classical law of energy equipartition: each degree of freedom Figure 5.3 ariation with temperature of the heat capacity of an assembly of linear oscillators. contributes NkT to the internal energy. But the question left unanswered by the classical law is: when is a degree of freedom excited and when is it frozen out? The total internal energy of a diatomic molecule is made up of four contributions that can be treated separately: (1) the kinetic energy associated with the translational motion of the center of the mass of the molecule (this is 3 kt, the same as that for a monatomic molecule); () the rotational energy due to the rotation of the two atoms about the center of mass of the molecule; (3) the vibrational motion of the two atoms along the axis joining them; and (4) the energy of excitation of the atomic electrons. The last three are internal modes of possessing energy. Because the four contributions can vary independently, it follows that the partition function is a product of the corresponding factors. This can be seen by noting that the internal energy is a function of the logarithm of Z. Having considered the vibrational motion in Section 5.3, we turn our attention to the rotational contribution. 5.4 ROTATIONAL MODES OF DIATOMIC MOLECULES 6

7 The rotation of a diatomic molecule is modeled as the motion of a quantum mechanical rigid rotator. The rotation takes place about an axis through the center of mass of the molecule and perpendicular to the line joining the two atoms. We take I as the moment of inertia of the molecule about this axis; I = µr 0 where µ = m 1 m (m 1 + m ) is the reduced mass of the two atoms and r 0 is the equilibrium value of the distance between the nuclei. Quantum mechanics states that the allowed values of the square of the angular momentum are l(l +1)h, where h is Planck s constant divided by " and l = 0,1,,3,... Recall from classical mechanics that the rotational energy is 1 I", where " is the angular velocity. The angular momentum L is I" so the energy is L I. The quantized energy levels are therefore " l = l(l +1) h I (5.10) We define a characteristic temperature for rotation: " rot = h Ik (5.11) so that " l = l(l +1)k# rot (5.1) Experimental values of " rot are given in Table 5.1 for several gases; they are found from infrared spectroscopy, in which the energies required to excite the molecules to higher rotational states are measured. The energy levels of equation (5.1) are degenerate; quantum mechanics gives g l = (l +1) states for level l corresponding to different possible directions of the angular momentum vector. Given these results, we can write down the partition function for rotation: Z = g l e "# l kt = (l +1)e "l(l +1)% rot T (5.13) l l 7

8 The important quantity in this expression is the argument of the exponential. For T << " rot, virtually all the molecules are in the few lowest rotational states, and the series can be truncated with negligible error after the first two or three terms. It can be shown that in this case both the internal energy and the heat capacity at constant volume vanish as the temperature approaches zero (see Problem 5-4). For all diatomic gases except hydrogen, the rotational characteristic temperature is of the order of 10 K or less. Since these gases have liquefied (indeed, solidified) at such low temperatures, it is always true that T >> " rot at ordinary temperatures. Hence a great many closely spaced energy states are excited and the sum in Equation (5.13) may be replaced by an integral. We write x = l(l +1) and treat x as a continuous variable. Note that dx = (l +1)dl. In this approximation, Z = exp ("x# rot T) )dx = T, T >> # 0 rot (5.14) # rot This result is too large by a factor of for homonuclear molecules such as H, O, and N. For two identical nuclei, two opposite positions of the molecular axis correspond to the same state of the molecule. This slight modification has no effect on the thermodynamic properties of the system such as the internal energy and the heat capacity. Using Equation (5.14) in #"ln Z & U = NkT "T ' # " = NkT { "T lnt ) ln* & % rot} ( ' we find that fort >> " rot, U rot = NkT, and,rot = Nk, T >> " rot. (5.16) Thus, at significantly high temperatures, the rigid rotator exhibits the equipartition of energy between two degrees of freedom. At very low temperatures such that " rot >> T, the partition function, Equation (5.13), can be expanded as Z = (l +1)e "l(l +1)# rot T % =1+ 3e "# rot T + 5e "6# rot T +... l= 0 8

9 Retaining only the first two terms, we can write T ( ) lnz " ln 1+ 3e # rot Since the exponential term is small compared with unity, we can use the approximate relation ln(1+ ") # " for " <<1, and obtain the result lnz " 3e # rot T. Hence #"ln Z & U = NkT "T ' Figure 5.4 ariation with temperature of the heat capacity of an assembly of rigid rotators. = 6Nk) rot e *) rot T, ) rot >> T (5.18) and #,rot = "U & "T ' = 3Nk ) # & rot e *) rot T, ) T ' rot >> T. (5.19) Again, we see that approaches zero as T approaches zero, as it must. Figure 5.4 shows schematically the contribution of rotation to the heat capacity as a function of temperature. 5.5 ELECTRONIC EXCITATION The electronic partition function can be written Z " g 0 + g 1 e # e T (5.0) where g 0 and g 1 are, respectively, the degeneracies of the ground state and the first excited state, and " e is the energy separation of the two lowest states divided by Boltzmann s constant k. For most gases, the higher electronic states are not excited (" e ~ K for hydrogen) and Z " g 0. Thus the partition function at practical temperatures makes no contribution to the external energy or heat capacity. 9

10 5.6 THE TOTAL HEAT CAPACITY We are finally in a position to account fully for the behavior of as a function of T for a diatomic gas at temperatures in the usual range of interest. Assuming that electronic excitations are negligible, we have =,tr +,rot +,vib Thus, at ordinary temperatures, = 3 Nk + Nk + Nk # " & vib T ' e " vib T * e " vib ( T )1) = Nk, 5 + # " & vib, T ' + e " vib T - / e " vib ( T )1) /. (5.1) Table 5.1 gives values for " rot and " vib for several diatomic gases. At the lowest temperatures at which the system exists in the gaseous phase, the molecular motion is solely translational, contributing 3Nk = 3nR to the heat capacity. As the temperature increases and approaches " rot, rotational quantum states are excited. Eventually, for T >> " rot, the rotational contribution becomes equal to nr, corresponding to two rotational degrees of freedom. Therefore, the heat capacity steps up to 5 3 nr in the region" rot << T << " vib. Since room temperature lies within this range, this is the value usually given for the heat capacity of a diatomic gas. 10

11 TABLE 5.1 Characteristic temperatures of rotation and vibration of diatomic molecules. Substance " rot (K) " vib (K) H O.1 39 N HCl CO NO Cl At elevated temperatures, the higher vibrational states are excited and the heat capacity exhibits two additional degrees of freedom, rising asymptotically to a value of 7nR for T >> " vib. The characteristic temperature of vibration depends on the bond strength between the two atoms of the molecule and on their masses. In some cases, the diatomic molecules disassociate as the vibrational energy overcomes the bonding energy. Figure 5.5 alues of nr for hydrogen ( H ) as a function of temperature. The temperature scale is logarithmic. Experimental values of the heat capacity at constant volume are close to predicted values over a wide range of temperatures. alues of nr are shown for hydrogen on a logarithmic scale in Fig 5.5. An understanding of the temperature variation of the heat capacity of diatomic gases is surely an outstanding triumph of quantum statistical theory. The exact theory is apparently so firmly established that heat capacities of gases can be computed theoretically from optical measurements, more accurately than they can be measured experimentally (using calorimetry). 11