1+e θvib/t +e 2θvib/T +

Transcription

1 7Mar218 Chemistry 21b Spectroscopy & Statistical Thermodynamics Lecture # 26 Vibrational Partition Functions of Diatomic Polyatomic Molecules Our starting point is again the approximation that we can treat the molecular degrees of freedom independently, or H = H trans + H rot + H vib + H elec, 26.1) to yield an overall wavefunction that is the product of the solutions to the individual terms, or Ψ = ψ trans ψ rot ψ nucl )ψ vib ψ elec. Here we concentrate on the vibrational degrees of freedom, first for diatomic molecules and then for polyatomic systems and finally solids, treated as a very large molecule). For simplicity, we will use a harmonic oscillator model throughout and ignore the effects of anharmonicity. The Diatomic Harmonic Oscillator As we have seen before, the vibrational energy levels of a diatomic molecule are simply ǫ vib = v )hν vib, v=,1,2,... for a harmonic potential; where the vibrational frequency is ν vib = k/µ/2π), k being the force constant or second derivative of the interatomic potential) and µ the reduced mass. Each vibrational state is non-degenerate, and the zero-point energy is hν/2. Again, the reduced temperature for the system can be defined, and here is T/θ vib, where θ vib = hν vib /k. For the rotational degrees of freedom, even very light molecules possessed a θ B of at most a few 1 s of Kelvin. Here, the values of θ vib run from a high of 6 K for H 2 out to closer to 3 K for I 2. Thus, even for heavy diatomic species with weak bonds, T/θ vib is close to unity while T/θ vib 1 for typical diatomic composed of first row atoms. Furthermore, the zero-point energy, while real, cannot be accessed by the molecules in question when responding to changes in physical conditions. It must, however, be included when considering state functions such as the vibrational energy and would become the U) we have discussed previously). Including the zero-point energy, q vib becomes q vib T) = v e θ vibv+1/2)/t = e θ vib/2t ) 1+e θvib/t +e 2θvib/T ) For translation and rotation we typically invoke the classical limit approximation and replace the summation with an integral. Here, the terms in the exponent are often <1, and so the series converges rather rapidly and it is better to simply to evaluate the sum. SincewewillultimatelybeconcernedwithfunctionssuchaslnQ vib ), thezero-pointenergy multiplier in front becomes a negligible additive term for many species since ln1) =. The geometric series in the parenthesis of eq. 26.2) is a fairly simple one to examine as v see Apostol, Vol. I, pp ), and the closed form expression for q vib of a diatomic molecule under these conditions and ignoring zero-point energy is q vib T) = 1 1 e θ vib/t. 26.3) 252

2 As T/θ vib, q vib 1; and as T/θ vib, q vib T/θ vib. As noted above, for most molecules containing first row atoms or hydrogen, T/θ vib 1 near room temperature, and the vibrational partition function is close to unity. This is some two orders of magnitude smaller than q rot, and as we shall see helps explain the typically small vibrational contribution to state functions such as the entropy of a perfect gas. Polyatomic Systems and Normal Modes From Lecture #1, we know that for an N atom molecule there are 3N 6 vibrational degrees offreedom if it isnon-linear and 3N 5 vibrationaldegrees of freedom ifit is linear. In either case, the vibrational modes can be treated as independent harmonic oscillators, and so the full microcanonical vibrational partition function is simply the product of the partition functions associated with each normal mode, or for a non-linear molecule q vib T) = 3N 6 i=1 q NM i T), 26.4) where the index i runs over the normal modes. Recall that certain modes can have degeneracies larger than one, and these must be included in the appropriate qi NM. Unlike diatomic molecules, the vibrational frequencies in polyatomic systems can vary over a wide range. In CO 2, for example, the three distinct vibrational frequencies span the range from 1-34 K for θ vib ). Vibrations that are in some sense local, or characteristic of various functional groups, often have frequencies of 3-7 cm 1, and so yield individual terms in eq. 26.4) that are close to unity. As molecules become larger, however, the product of many terms of, say, 1.1, can lead to a sizeable q vib think about a protein, for example). Furthermore, in large systems there are often very soft vibrational degrees of freedom that are attached to what are called torsional modes. Their vibrational frequencies)/k can become quite close to or even less than 3 K, and when present they tend to dominate the vibrational partition function. If the barriers to these torsions or oscillations are high, the vibrational approach outlined here can be used. If they are extremely low, then a rotational approach can be used. If, however, the barriers are intermediate, as is almost always the case, the effects on the partition function must be numerically calculated see Maczek for additional details). Molecular/Vibrational Thermodynamic State Functions The localization of vibrations within molecules means that, like rotations, these degrees of freedom are distinguishable. Thus, the canonical partition function Q vib is Q vib = q N vib = ) N ) 1 e θ vib/t for a diatomic perfect gas. For a non-linear polyatomic system it is Q vib = q N vib = q 1 q 2 q 3N 6 ) N. 253

3 For a diatomic molecule, the expression for the vibrational energy of a mole of oscillators, neglecting the zero-point contribution, becomes ) ln U vib,m = E vib = kt 2 Qvib Rθ = vib T e 1 ). 26.6) If zero-point is included, we need to add Avogadro s number times the individual molecular zero-point energy, or for a diatomic molecule V U vib,m = E vib = Rθ vib e 1 ) + N Ahν vib 2. At high temperatures, and ignoring the zero-point energy, the diatomic vibrational energy for one mole is U vib,m = RT, as expected from the equipartition principle. For a polyatomic molecule, we would predict RT per each normal mode. This, of course, can become quite large in bio)polymers. Near room temperature, however, many of the vibrational modes will have characteristic vibrational frequencies that are difficult to excite, and so the energy content will fall far short of that predicted by equipartition. The heat capacity at constant volume is given by the temperature derivative of U vib, and so the zero-point energy or reference energy, if you will) is unimportant, so that C V, vib,m = R θvib T ) 2 e θ vib/t e 1 ) ) This very famous result is known as the Einstein equation, and we ll come back to it in greater detail in the next section. For now, note that at as T this expression goes to C V, vib,m R as predicted by equipartition, and that as T, C V, vib,m. This latter result is perfectly understandable from the quantum chemical perspective of a two level system as the temperature becomes very low it is impossible to access the excited vibrational state, and so the system cannot respond to a change in temperature), but was quite perplexing to physical chemists at the turn of the last century. For the entropy we use, as before, S vib = U vib T + k ln Q vib 26.8) or S vib,m R = e 1 ) ln 1 e θ vib/t ) 26.9) for diatomic molecules. Again, if 1, as is typical near room temperature, S vib,m /R is close to zero, and the entropy of a perfect gas is dominated by the translational and rotational degrees of freedom as we saw numerically for F 2 in Lecture #25). Vibrations and Crystalline Solids As we saw in Lecture #13, in a macroscopic solid the translational and rotational degrees of freedom are unimportant, and vibrational effects come to the fore. To remind 254

4 ourselves of how to proceed in this case, consider the N atom variant of the 1D coupled oscillator we examined in Lecture #9 where N N A ) and generalized in Lecture #13 that is depicted in Figure Clearly, the interatomic interactions cannot be neglected, indeed they are central to the problem at hand. It is, nevertheless, possible to use the weakly interacting machinery we have set up to consider the problem. #1 k #2 #N m m m Figure 26.1 A schematic of a one-dimensional array of coupled oscillators as a simple model for solids, where N N A. As with our treatment of normal modes in a polyatomic molecule, we begin by expanding the potential energy surface about its equilibrium configuration. The first derivatives are zero since we are at a potential minimum, and we truncate the expansion at the quadratic term. Defining the potential energy at the minimum as U) where the bold face stresses that this is a multi-dimensional system), the second derivatives of the potential with the force constants k ij, and the displacement coordinates as ξ i, we have Uξ 1,ξ 2,...,ξ N ) U) + 1 k ij ξ i ξ j. 26.1) 2 From such an analysis we know that there would be 3N - 6 3N vibrational frequencies since N N A, the approximation here is an extremely good one!), and that the analog of eq. 26.2) for the crystalline vibrational partition function becomes i,j 3N Q vib T) = e U)/kT q vib,j, where q vib,j is the harmonic oscillator partition function for each normal mode j. In a reasonable sized molecule it is possible to enumerate the vibrational frequencies and assign them to distinct modes. Here, since N is so large, the vibrational frequencies blend into a continuum. For single atom crystals, these become the acoustic branch derived in eq. 13.7). Here, we introduce a spectral function gν)dν which denotes the number of normal mode frequencies between ν and ν+dν. The function gν) is normalized such that j=1 gν)dν = 3N ) Inserting this into the partition function, including the zero point energy, and the expressions for the vibrational energy E vib and C V for N = N A yields [ ] hνe hν/kt E vib = U) + ) + hν gν)dν 26.12) 1 e hν/kt 2 255

5 and C V = hν/kt) 2 e hν/kt gν)dν 1 e hν/kt ) ) These expressions contain the relevant physics, the difficult part is to evaluate gν). We have encountered one of the most famous approximations that leads to the relationship in eq. 26.7), the Einstein equation. This approximation says that all normal frequencies are the same, that is gν) = 3Nδν ν E ), where ν E is called the Einstein frequency for a given crystal. Defining the Einstein temperature as Θ E = hν E /k and inserting the approximation for gν) into eq ) yields, for a mole of oscillators, ) 2 ΘE e Θ E/T C V,m crystal) = 3R T e θ E /T 1 ) 2, 26.14) where the multiplicative factor of three compared to the diatomic arises because the atoms in a realistic crystal are free to vibrate in three dimensions. The Einstein relationship successfully predicts that the heat capacity should go to zero at low temperature, and this was a major step forward in the early days of quantum mechanics since the low temperature heat capacity behavior of metals was impossible to understand from a classical perspective. However, the low temperature behavior is predicted to behave as C V,m Einstein) 3R θe T ) 2 e Θ E/T, while the experimental observation is C V,m exp.) T 3 as T. The exponential term clearly dominates the low temperature character of the Einstein equation, and so the discrepancy between theory and experiment is, in fact, quite large at low temperature. The culprit is the extremely crude assumption made for the frequency dependence of gν) to derive the Einstein equation, namely that the vibrational behavior of crystalline normal modes can be approximated by a single frequency. The Debye model briefly outlined in eq. 13.9) is much more realistic model in that it examines the low frequency behavior of crystals when they are treated as a continuous elastic body. This approach weds the quantum nature of the vibrational partition function embodied in eq. 26.1) with a mechanically realistic implementation of the crystalline vibrational spectrum as modeled by the continuous function gν). Specifically, if the crystal is treated as a cube and the number of standing waves/unit frequency interval is determined this is the same calculation that is used to examine the Rayleigh-Jeans limit of the blackbody equation), we can recast the density of states expression derived in Lecture #13 to yield: 256

6 gν)dν = 9N νd 3 ν 2 dν ν ν D = ν > ν D, 26.15) where ν D is called the Debye frequency for the crystal, and leads in turn to the Debye temperature, or Θ D = hν D /k. While the actual form of the acoustic branch frequencies does not exactly follow this behavior, the lowest frequencies do and it is precisely these modes that matter most at low temperature. By inserting the Debye formula for gν), it is found that the molar heat capacity at constant volume is given by where x = hν/kt) ) 3 T ΘD /T x 4 e x C V,m Debye) = 9R Θ D e x 2 dx ) 1) Figure 26.2 Left) Data for the molar heat capacity of Al, Cu, and Pb as a function of reduced temperature are plotted as symbols. The solid curve presents a Debye heat capacity fit to the data. Right) Experimental acoustic mode spectrum of Fe. Note that below 3 THz the expected g ν 2 for the Debye model is met 3 THz=1 cm 1 14 K). Figure 26.2 presents a fit of the Debye equation to heat capacity data for metals. At high temperatures, both the Einstein and Debye molar heat capacities approaches 3R, the Dulong and Petit value motivated by the equipartition principle. At low temperatures, however, x, and the integral in eq ) becomes definite. Its value is x 4 e x /e x 1) 2 dx = 4π 2 /15. Thus, the limiting form of the Debye formula becomes ) 3 C V,m Debye,T ) 12π4 T R, 26.17) 5 Θ D which does correspond to the low temperature behavior of the heat capacity of monatomic metals. Clearly, by plotting C V /3R versus T/Θ D, C V s for all such systems can be shown on one plot! Experimentally, the Debye temperatures of metals are found to be a few hundred Kelvin Θ D,Pb = 88 K, Θ D,Na = 15 K, Θ D,Cu = 315 K, Θ D,Fe = 42 K, and Θ D,Be = 1 K, for example). The actual form of the acoustic spectrum for iron is shown at right in Fig At sufficiently low temperature, only the longest wavelength acoustic modes are excited, and the structure in the actual density of states becomes unimportant. 257