59917.3MEANENERGIES(e)Theoverall partitionfunctionThepartition functions for eachmode of motion of a molecule are collected inTable17.3at the end of the chapter.Theoverall partition function istheproductofeach contribution. For a diatomic molecule with no low-lying electronically excitedstates and T>SR=8E(17.23)1-e-710TAExample17.4CalculatingathermodynamicfunctionfromspectroscopicdataCalculate the value of G-Ge(0) for H,O(g) at 1500 K given thatA=27.8778 cm-l,B=14.5092cm-1,andC=9.2869cm-landtheinformationinExample17.3.Method The starting point is eqn 17.9.For the standard value, we evaluate thetranslational partition function at p (that is, at 1o' Pa exactly).The vibrationalpartition function was calculated in Example 17.3. Use the expressions in Table17.3for theother contributions.Answer Because m= 18.015 u, it follows that qTe/N = 1.706× 10°. For the vibrational contribution we have alreadyfound thatqV=1.352.From Table17.2we seethat=2,so therotational contribution is q=486.7.Therefore,G-G(0)=(8.3145JK-mol-)×(1500K)×ln((1.706×10%)×486.7×1.352)=317.3 kj mol-lComment17.1Self-test 17.4 Repeat the calculation for CO,. The vibrational data are given inThetext's web site contains links to[366.6k] mol-"]Self-test17.3;B=0.3902cm-on-linedatabases ofatomicandmolecularspectra.Overall partitionfunctions obtained from eqn17.23areapproximatebecause theyassume that the rotational levels are very close together and that the vibrational levelsareharmonic.These approximations areavoided byusing theenergylevelsidentifiedspectroscopically and evaluating the sums explicitly.UsingstatisticalthermodynamicsWe can now calculate any thermodynamic quantityfrom a knowledge of the energylevels of molecules: we have merged thermodynamics and spectroscopy.In this section, we indicate how todo the calculationsforfour important properties.17.3MeanenergiesItis often usefultoknowthemean energy,(e),ofvarious modesofmotion.When themolecular partition function can befactorized into contributions from eachmode,themeanenergyofeachmodeM (fromeqn16.29)is1(agM(eM):M=T,R,V,orE(17.24)qM）
17.3 MEAN ENERGIES 599 Comment 17.1 The text’s web site contains links to on-line databases of atomic and molecular spectra. (e) The overall partition function The partition functions for each mode of motion of a molecule are collected in Table 17.3 at the end of the chapter. The overall partition function is the product of each contribution. For a diatomic molecule with no low-lying electronically excited states and T >> θR, q = gE (17.23) Example 17.4 Calculating a thermodynamic function from spectroscopic data Calculate the value of G7 m − G7 m(0) for H2O(g) at 1500 K given that A = 27.8778 cm−1 , B = 14.5092 cm−1 , and C = 9.2869 cm−1 and the information in Example 17.3. Method The starting point is eqn 17.9. For the standard value, we evaluate the translational partition function at p7 (that is, at 105 Pa exactly). The vibrational partition function was calculated in Example 17.3. Use the expressions in Table 17.3 for the other contributions. Answer Because m = 18.015 u, it follows that qm T7 /NA = 1.706 × 108 . For the vibrational contribution we have already found that qV = 1.352. From Table 17.2 we see that σ = 2, so the rotational contribution is qR = 486.7. Therefore, G7 m − G7 m(0) = −(8.3145 J K−1 mol−1 ) × (1500 K) × ln{(1.706 × 108 ) × 486.7 × 1.352} = −317.3 kJ mol−1 Self-test 17.4 Repeat the calculation for CO2. The vibrational data are given in Self-test 17.3; B = 0.3902 cm−1 . [−366.6 kJ mol−1 ] Overall partition functions obtained from eqn 17.23 are approximate because they assume that the rotational levels are very close together and that the vibrational levels are harmonic. These approximations are avoided by using the energy levels identified spectroscopically and evaluating the sums explicitly. Using statistical thermodynamics We can now calculate any thermodynamic quantity from a knowledge of the energy levels of molecules: we have merged thermodynamics and spectroscopy. In this section, we indicate how to do the calculations for four important properties. 17.3 Mean energies It is often useful to know the mean energy, ε, of various modes of motion. When the molecular partition function can be factorized into contributions from each mode, the mean energy of each mode M (from eqn 16.29) is εM = − V M = T, R, V, or E (17.24) D F ∂qM ∂β A C 1 qM D F 1 1 − e−T/θ v A C D F T σθ R A C D F V Λ3 A C
60017STATISTICALTHERMODYNAMICS2:APPLICATIONS(a) The mean translational energyTo see a pattern emerging, we consider first a one-dimensional system oflength X, for1.5which q-X/A, withA=h(β/2元m)1/2.Then, if we note thatAis a constanttimes β/2,A(ax)-B1/2_ d ((eT=IKT(17.25a)dpB1/2xaA2B8o,2)For a molecule free to move in three dimensions, the analogous calculation leads to(eT)=kT(17.25b)Both conclusions are in agreement with the classical equipartition theorem (see0.5Molecular interpretation 2.2) that the mean energy of each quadratic contributionto the energy is kT. Furthermore, the fact that the mean energy is independent ofthe size ofthe container is consistent with the thermodynamicresultthatthe internalenergy of a perfect gas is independent of its volume (Molecular interpretation 2.2).(b)Themean rotational energyTIOaThe mean rotational energy of a linear molecule is obtained from the partition func-Fig.17.s The mean rotational energy ofation given in eqn 17.13.When thetemperature islow(T<,),the seriesmustbenonsymmetrical linearrotorasafunctionsummed term by term, which givesoftemperature.AthightemperaturesqR = 1 + 3e-2 phcB + 5e-6phcB + ...(T ), the energy is linearlyproportional to the temperature, inHenceaccord with the equipartition theoremhcB(6e-2PhcB + 30e6βhcB +-.)Exploration Plot the temperature国(eR)=(17.26a)dependence of the mean rotational1 +3e-2βhcB+ 5e-6βhcB+***energyforseveral valuesoftherotationalconstant (for reasonable values of theThis function is plotted in Fig. 17.8. At high temperatures (T> g), qR is given byrotationalconstant,see theData section)eqn 17.15, andFrom your plots,estimatethe temperature1 dqRd1at which the mean rotational energy begins(eR)=--ohcβB=kT(17.26b)to increase sharply.qRdβdβhcβBβ(gR is independent of V, so the partial derivatives have been replaced by completederivatives.)The high-temperature result is also in agreement with the equipartitiontheorem,forthe classical expression for theenergy ofa linear rotor is Ek=I+I,g.(Thereis no rotation around thelineofatoms.)Itfollowsfromtheequiparti-tion theorem that the mean rotational energy is 2×kT=kT.(c)The mean vibrational energyThe vibrational partition function in the harmonic approximation is given in eqn17.19. Because q is independent of the volume, it follows thathcve-Bhcepdqd(17.27)dβdβ(i-e-c(1 eβhc)2and hencefromhcVe-BhephcVe-Bhep1 dgv(eV)=(1-e-βho1-e-Bhep(1 e-βhe)2qapthathcv(ev):(17.28)Bhcv-1
600 17 STATISTICAL THERMODYNAMICS 2: APPLICATIONS (a) The mean translational energy To see a pattern emerging, we consider first a one-dimensional system of length X, for which qT = X/Λ, with Λ = h(β/2πm) 1/2. Then, if we note that Λ is a constant times β1/2, εT = − V = −β1/2 = = 1 –2 kT (17.25a) For a molecule free to move in three dimensions, the analogous calculation leads to ε T = 3 –2kT (17.25b) Both conclusions are in agreement with the classical equipartition theorem (see Molecular interpretation 2.2) that the mean energy of each quadratic contribution to the energy is 1 –2 kT. Furthermore, the fact that the mean energy is independent of the size of the container is consistent with the thermodynamic result that the internal energy of a perfect gas is independent of its volume (Molecular interpretation 2.2). (b) The mean rotational energy The mean rotational energy of a linear molecule is obtained from the partition function given in eqn 17.13. When the temperature is low (T < θR), the series must be summed term by term, which gives qR = 1 + 3e−2βhcB + 5e−6βhcB + · · · Hence εR = (17.26a) This function is plotted in Fig. 17.8. At high temperatures (T >> θR), qR is given by eqn 17.15, and εR =− =−σhcβB = = kT (17.26b) (qR is independent of V, so the partial derivatives have been replaced by complete derivatives.) The high-temperature result is also in agreement with the equipartition theorem, for the classical expression for the energy of a linear rotor is EK = 1 –2 I⊥ωa 2 + 1 –2 I⊥ωb 2 . (There is no rotation around the line of atoms.) It follows from the equipartition theorem that the mean rotational energy is 2 × 1 –2 kT = kT. (c) The mean vibrational energy The vibrational partition function in the harmonic approximation is given in eqn 17.19. Because qV is independent of the volume, it follows that = =− (17.27) and hence from εV =− =−(1 − e−βhc# ) − = that εV = (17.28) hc# eβhc# − 1 hc#e−βhc# 1 − e−βhc# 5 6 7 hc#e−βhc# (1 − e−βhc# ) 2 1 2 3 dqV dβ 1 qV hc#e−βhc# (1 − e−βhc# ) 2 D F 1 1 − e−βhc# A C d dβ dqV dβ 1 β 1 σhcβB d dβ dqR dβ 1 qR hcB(6e−2βhcB + 30e−6βhcB + · · · ) 1 + 3e−2βhcB + 5e−6βhcB + · · · 1 2β D F 1 β1/2 A C d dβ D F X Λ ∂ ∂β A C Λ X 0 0 1 2 T/ R 0.5 1 1.5" # R /hcB Fig. 17.8 The mean rotational energy of a nonsymmetrical linear rotor as a function of temperature. At high temperatures (T >> θR), the energy is linearly proportional to the temperature, in accord with the equipartition theorem. Exploration Plot the temperature dependence of the mean rotational energy for several values of the rotational constant (for reasonable values of the rotational constant, see the Data section). From your plots, estimate the temperature at which the mean rotational energy begins to increase sharply.