R. Viswanathan, R.W. Schmude, Jr., and K.A. Gingerich (J. Phys. Chem. 100, 10784 (1996))

Discuss the limitations of the expressions qR = kT/hcB, qV = kT/hc#, and qE = gE.

Explain the origin of the symmetry number.

Explain the origin of residual entropy.

Use concepts of statistical thermodynamics to describe the molecular features that determine the magnitudes of the constant-volume molar heat capacity of a molecular substance.

Use concepts of statistical thermodynamics to describe the molecular features that lead to the equations of state of perfect and real gases.

Describe how liquids are investigated by using concepts of statistical thermodynamics.

Use concepts of statistical thermodynamics to describe the molecular features that determine the magnitudes of equilibrium constants and their variation with temperature.

Use the equipartition theorem to estimate the constant-volume molar heat capacity of (a) I2, (b) CH4, (c) C6H6 in the gas phase at 25C.

Use the equipartition theorem to estimate the constant-volume molar heat capacity of (a) O3, (b) C2H6, (c) CO2 in the gas phase at 25C.

Estimate the values of = Cp /CV for gaseous ammonia and methane. Do this calculation with and without the vibrational contribution to the energy. Which is closer to the expected experimental value at 25C?

Estimate the value of = Cp /CV for carbon dioxide. Do this calculation with and without the vibrational contribution to the energy. Which is closer to the expected experimental value at 25C?

Estimate the rotational partition function of HCl at (a) 25C and (b) 250C.

Estimate the rotational partition function of O2 at (a) 25C and (b) 250C.

Give the symmetry number for each of the following molecules: (a) CO, (b) O2, (c) H2S, and (d) SiH4, (e) CHCl3.

Give the symmetry number for each of the following molecules: (a) CO2, (b) O3, (c) SO3, (d) SF6, and (e) Al2Cl6.

Calculate the rotational partition function of H2O at 298 K from its rotational constants 27.878 cm1, 14.509 cm1, and 9.287 cm1. Above what temperature is the high-temperature approximation valid to within 10 per cent of the true value?

Calculate the rotational partition function of SO2 at 298 K from its rotational constants 2.027 36 cm1, 0.344 17 cm1, and 0.293 535 cm1. Above what temperature is the high-temperature approximation valid to within 10 per cent of the true value?

From the results of Exercise 17.5a, calculate the rotational contribution to the molar entropy of gaseous water at 25C.

From the results of Exercise 17.5b, calculate the rotational contribution to the molar entropy of sulfur dioxide at 25C.

Calculate the rotational partition function of CH4 (a) by direct summation of the energy levels at 298 K and 500 K, and (b) by the hightemperature approximation. Take B = 5.2412 cm1.

Calculate the rotational partition function of \(\mathrm{CH}_3 \mathrm{CN}\) (a) by direct summation of the energy levels at \(298 \mathrm{~K}\) and \(500 \mathrm{~K}\), and (b) by the high temperature approximation. Take \(A=5.28 \mathrm{~cm}^{-1}\) and \(B=0.307 \mathrm{~cm}^{-1}\).

The bond length of O2 is 120.75 pm. Use the high-temperature approximation to calculate the rotational partition function of the molecule at 300 K.

The NOF molecule is an asymmetric rotor with rotational constants 3.1752 cm1, 0.3951 cm1, and 0.3505 cm1. Calculate the rotational partition function of the molecule at (a) 25C, (b) 100C.

Plot the molar heat capacity of a collection of harmonic oscillators as a function of T/ V, and predict the vibrational heat capacity of ethyne at (a) 298 K, (b) 500 K. The normal modes (and their degeneracies in parentheses) occur at wavenumbers 612(2), 729(2), 1974, 3287, and 3374 cm1.

Plot the molar entropy of a collection of harmonic oscillators as a function of T/V, and predict the standard molar entropy of ethyne at (a) 298 K, (b) 500 K. For data, see the preceding exercise.

A CO2 molecule is linear, and its vibrational wavenumbers are 1388.2 cm1, 667.4 cm1, and 2349.2 cm1, the last being doubly degenerate and the others non-degenerate. The rotational constant of the molecule is 0.3902 cm1. Calculate the rotational and vibrational contributions to the molar Gibbs energy at 298 K.

An O3 molecule is angular, and its vibrational wavenumbers are 1110 cm1, 705 cm1, and 1042 cm1. The rotational constants of the molecule are 3.553 cm1, 0.4452 cm1, and 0.3948 cm1. Calculate the rotational and vibrational contributions to the molar Gibbs energy at 298 K.

The ground level of Cl is 2P3/2 and a 2P1/2 level lies 881 cm1 above it. Calculate the electronic contribution to the heat capacity of Cl atoms at (a) 500 K and (b) 900 K.

The first electronically excited state of O2 is 1g and lies 7918.1 cm1 above the ground state, which is 3g . Calculate the electronic contribution to the molar Gibbs energy of O2 at 400 K.

The ground state of the Co2+ ion in CoSO47H2O may be regarded as 4T9/2. The entropy of the solid at temperatures below 1 K is derived almost entirely from the electron spin. Estimate the molar entropy of the solid at these temperatures.

Estimate the contribution of the spin to the molar entropy of a solid sample of a d-metal complex with S = 52 .

Calculate the residual molar entropy of a solid in which the molecules can adopt (a) three, (b) five, (c) six orientations of equal energy at T = 0.

Suppose that the hexagonal molecule C6HnF6 n has a residual entropy on account of the similarity of the H and F atoms. Calculate the residual for each value of n.

Calculate the equilibrium constant of the reaction I2(g)52 I(g) at 1000 K from the following data for I2: # = 214.36 cm1, B = 0.0373 cm1, De = 1.5422 eV. The ground state of the I atoms is 2P3/2, implying fourfold degeneracy.

Calculate the value of K at 298 K for the gas-phase isotopic exchange reaction 2 79Br81Br 79Br795Br + 81Br81Br. The Br2 molecule has a nondegenerate ground state, with no other electronic states nearby. Base the calculation on the wavenumber of the vibration of 79Br81Br, which is 323.33 cm1.

In a spectroscopic study of buckminsterfullerene C60, F. Negri, G. Orlandi, and F. Zerbetto (J. Phys. Chem. 100, 10849 (1996)) reviewed the wavenumbers of all the vibrational modes of the molecule. The wavenumber for the single Au mode is 976 cm1; wavenumbers for the four threefold degenerate T1u modes are 525, 578, 1180, and 1430 cm1; wavenumbers for the five threefold degenerate T2u modes are 354, 715, 1037, 1190, and 1540 cm1; wavenumbers for the six fourfold degenerate Gu modes are 345, 757, 776, 963, 1315, and 1410 cm1; and wavenumbers for the seven fivefold degenerate Hu modes are 403, 525, 667, 738, 1215, 1342, and 1566 cm1. How many modes have a vibrational temperature V below 1000 K? Estimate the molar constant-volume heat capacity of C60 at 1000 K, counting as active all modes with V below this temperature.

Treat carbon monoxide as a perfect gas and apply equilibrium statistical thermodynamics to the study of its properties, as specified below, in the temperature range \(100-1000 \mathrm{~K}\) at 1 bar. \(\tilde{v}=2169.8 \mathrm{~cm}^{-1}, B=1.931 \mathrm{~cm}^{-1}\), and \(D_0=11.09 \mathrm{eV}\); neglect anharmonicity and centrifugal distortion. (a) Examine the probability distribution of molecules over available rotational and vibrational states. (b) Explore numerically the differences, if any, between the rotational molecular partition function as calculated with the discrete energy distribution and that calculated with the classical, continuous energy distribution. (c) Calculate the individual contributions to \(U_{\mathrm{m}}(T)-U_{\mathrm{m}}\) (100 K), \(C_{V, \mathrm{~m}}(T)\), and \(S_{\mathrm{m}}(T)-S_{\mathrm{m}}(100 \mathrm{~K})\) made by the translational, rotational, and vibrational degrees of freedom.

Calculate and plot as a function of temperature, in the range 300 K to 1000 K, the equilibrium constant for the reaction CD4(g) + HCl(g)5 CHD3(g) + DCl(g) using the following data (numbers in parentheses are degeneracies): #(CHD3)/cm1 = 2993(1), 2142(1), 1003(3), 1291(2), 1036(2); #(CD4)/cm1 = 2109(1), 1092(2), 2259(3), 996(3); #(HCl)/cm1 = 2991; #(DCl)/cm1 = 2145; B(HCl)/cm1 = 10.59; B(DCl)/cm1 = 5.445; B(CHD3)/cm1 = 3.28; A(CHD3)/cm1 = 2.63, B(CD4)/cm1 = 2.63.

The exchange of deuterium between acid and water is an important type of equilibrium, and we can examine it using spectroscopic data on the molecules. Calculate the equilibrium constant at (a) 298 K and (b) 800 K for the gas-phase exchange reaction H2O + DCl5HDO + HCl from the following data: #(H2O)/cm1 = 3656.7, 1594.8, 3755.8; #(HDO)/cm1 = 2726.7, 1402.2, 3707.5; A(H2O)/cm1 = 27.88; B(H2O)/cm1 = 14.51; C(H2O)/cm1 = 9.29; A(HDO)/cm1 = 23.38; B(HDO)/cm1 = 9.102; C(HDO)/cm1 = 6.417; B(HCl)/cm1 = 10.59; B(DCl)/cm1 = 5.449; #(HCl)/cm1 = 2991; #(DCl)/cm1 = 2145.

Derive the SackurTetrode equation for a monatomic gas confined to a two-dimensional surface, and hence derive an expression for the standard molar entropy of condensation to form a mobile surface film.

For H2 at very low temperatures, only translational motion contributes to the heat capacity. At temperatures above R = hcB/k, the rotational contribution to the heat capacity becomes significant. At still higher temperatures, above V = h/k, the vibrations contribute. But at this latter temperature, dissociation of the molecule into the atoms must be considered. (a) Explain the origin of the expressions for R and V, and calculate their values for hydrogen. (b) Obtain an expression for the molar constant-pressure heat capacity of hydrogen at all temperatures taking into account the dissociation of hydrogen. (c) Make a plot of the molar constant-pressure heat capacity as a function of temperature in the high-temperature region where dissociation of the molecule is significant.

Derive expressions for the internal energy, heat capacity, entropy, Helmholtz energy, and Gibbs energy of a harmonic oscillator. Express the results in terms of the vibrational temperature, V and plot graphs of each property against T/ V.

Suppose that an intermolecular potential has a hard-sphere core of radius r1 and a shallow attractive well of uniform depth out to a distance r2. Show, by using eqn 17.42 and the condition << kT, that such a model is approximately consistent with a van der Waals equation of state when b << Vm, and relate the van der Waals parameters and the JouleThomson coefficient to the parameters in this model.

(a) Show that the number of molecules in any given rotational state of a linear molecule is given by NJ = C(2J + 1)ehcBJ( J+1)/kT, where C is independent of J. (b) Use this result to derive eqn 13.39 for the value of J of the most highly populated rotational level. (c) Estimate the temperature at which the spectrum of HCl shown in Fig. 13.44 was obtained. (d) What is the most highly populated level of a spherical rotor at a temperature T?

A more formal way of arriving at the value of the symmetry number is to note that is the order (the number of elements) of the rotational subgroup of the molecule, the point group of the molecule with all but the identity and the rotations removed. The rotational subgroup of H2O is {E, C2}, so = 2. The rotational subgroup of NH3 is {E, 2C3}, so = 3. This recipe makes it easy to find the symmetry numbers for more complicated molecules. The rotational subgroup of CH4 is obtained from the T character table as {E, 8C3, 3C2}, so = 12. For benzene, the rotational subgroup of D6h is {E, 2C6, 2C3, C2, 3C2 , 3C2}, so = 12. (a) Estimate the rotational partition function of ethene at 25C given that A = 4.828 cm1, B = 1.0012 cm1, and C = 0.8282 cm1. (b) Evaluate the rotational partition function of pyridine, C5H5N, at room temperature (A = 0.2014 cm1, B = 0.1936 cm1, C = 0.0987 cm1).

Although expressions like _ _ = d ln q/d are useful for formal manipulations in statistical thermodynamics, and for expressing thermodynamic functions in neat formulas, they are sometimes more trouble than they are worth in practical applications. When presented with a table of energy levels, it is often much more convenient to evaluate the following sums directly: q =j ej =j jej =j (j)2ej (a) Derive expressions for the internal energy, heat capacity, and entropy in terms of these three functions. (b) Apply the technique to the calculation of the electronic contribution to the constant-volume molar heat capacity of magnesium vapour at 5000 K using the following data: Term 1S 3P0 3P1 3P2 1P1 3S1 Degeneracy 1 1 3 5 3 3 #/cm1 0 21 850 21 870 21 911 35 051 41 197

Show how the heat capacity of a linear rotor is related to the following sum: () = J,J { ( J) ( J )}2g( J)g( J )e{ ( J)+ ( J )} by C = 12 Nk 2( ) where the (J ) are the rotational energy levels and g( J ) their degeneracies. Then go on to show graphically that the total contribution to the heat capacity of a linear rotor can be regarded as a sum of contributions due to transitions 01, 02, 12, 13, etc. In this way, construct Fig. 17.11 for the rotational heat capacities of a linear molecule.

Set up a calculation like that in Problem 17.19 to analyse the vibrational contribution to the heat capacity in terms of excitations between levels and illustrate your results graphically in terms of a diagram like that in Fig. 17.11.

Determine whether a magnetic field can influence the value of an equilibrium constant. Consider the equilibrium I2(g)52 I(g) at 1000 K, and calculate the ratio of equilibrium constants K(B)/K, where K(B) is the equilibrium constant when a magnetic field B is present and removes the degeneracy of the four states of the 2P3/2 level. Data on the species are given in Exercise 17.14a. The electronic g value of the atoms is 43. Calculate the field required to change the equilibrium constant by 1 per cent. 1 q2

The heat capacity ratio of a gas determines the speed of sound in it through the formula cs = ( RT/M)1/2, where = Cp /CV andMis the molar mass of the gas. Deduce an expression for the speed of sound in a perfect gas of (a) diatomic, (b) linear triatomic, (c) nonlinear triatomic molecules at high temperatures (with translation and rotation active). Estimate the speed of sound in air at 25C.

An average human DNA molecule has 5 108 binucleotides (rungs on the DNA ladder) of four different kinds. If each rung were a random choice of one of these four possibilities, what would be the residual entropy associated with this typical DNA molecule?

It is possible to write an approximate expression for the partition function of a protein molecule by including contributions from only two states: the native and denatured forms of the polymer. Proceeding with this crude model gives us insight into the contribution of denaturation to the heat capacity of a protein. It follows from Illustration 16.4 that the total energy of a system of N protein molecules is E = where is the energy separation between the denatured and native forms. (a) Show that the constant-volume molar heat capacity is CV,m = Hint. For two functions f and g, the quotient rule of differentiation states that d( f /g)/dx = (1/g)df/dx ( f /g2)dg/dx. (b) Plot the variation of CV,m with temperature. (c) If the function CV,m(T) has a maximum or minimum, derive an expression for the temperature at which it occurs.

R. Viswanathan, R.W. Schmude, Jr., and K.A. Gingerich (J. Phys. Chem. 100, 10784 (1996)) studied thermodynamic properties of several boronsilicon gas-phase species experimentally and theoretically. These species can occur in the high-temperature chemical vapour deposition (CVD) of silicon-based semiconductors. Among the computations they reported was computation of the Gibbs energy of BSi(g) at several temperatures based on a 4 ground state with equilibrium internuclear distance of 190.5 pm and fundamental vibrational wavenumber of 772 cm1 and a 2P0 first excited level 8000 cm1 above the ground level. Compute the standard molar Gibbs energy G7m (2000 K) G7m (0).

The molecule Cl2O2, which is believed to participate in the seasonal depletion of ozone over Antarctica, has been studied by several means. M. Birk, R.R. Friedl, E.A. Cohen, H.M. Pickett, and S.P. Sander (J. Chem. Phys. 91, 6588 (1989)) report its rotational constants (actually cB) as 13 109.4, 2409.8, and 2139.7 MHz. They also report that its rotational spectrum indicates a molecule with a symmetry number of 2. J. Jacobs, M. Kronberg, H.S.P. Mller, and H. Willner (J. Amer. Chem. Soc. 116, 1106 (1994)) report its vibrational wavenumbers as 753, 542, 310, 127, 646, and 419 cm1. Compute G7m (200 K) G7m (0) of Cl2O2.

J. Hutter, H.P. Lthi, and F. Diederich (J. Amer. Chem. Soc. 116, 750 (1994)) examined the geometric and vibrational structure of several carbon molecules of formula Cn. Given that the ground state of C3, a molecule found in interstellar space and in flames, is an angular singlet with moments of inertia 39.340, 39.032, and 0.3082 u 2 (where 1 = 1010 m) and with vibrational wavenumbers of 63.4, 1224.5, and 2040 cm1, compute G7m (10.00 K) G7m (0) and G7m (1000 K) G7m (0) for C3.