6.7 Problems for This Chapter
351
Obtain a discrete summation expression for z rot (T ) that includes the
centrifugal-distortion energy, with the rotational energy of a CH 4 molecule
given in cm −1 units as
j = j (j + 1)B − [j (j + 1)]
2 D .
Compute the values of z rot (T ) for temperatures 50 K, 100 K, 300 K, and
500 K using B 0 = 5.2412 cm −1 and D 0 = 1.23 × 10 −4 cm −1 . Compare the
results of this calculation with the rigid-rotor values obtained in Problem 25
via z rot−nuc (T ) = 16z DS
rot (T ) with the values z DS
rot−nuc (T ) 41.993, 116.421,
598.639, and 1287.523 computed for CH 4 at temperatures 50 K, 100 K, 300 K,
500 K, respectively, by Robiette and Dang-Nhu [49] and the closed-form values
z CF
rot−nuc (T ) given via Eq. (6.3.23).
27. Calculate the standard molar entropy of ClO 2 at 298.15 K (NATP). The ClO 2
molecule is a nonlinear triatomic molecule (point symmetry group C 2v ), and
has one unpaired electron (ground electronic term symbol 2 B 1 ), so that it is
paramagnetic. Its rotational constants have been determined from spectroscopy
to be B A = 1.737 cm −1 , B B = 0.332 cm −1 , and B C = 0.278 cm −1 , while
the fundamental vibrational oscillator frequencies are ν 1 = 945.5 cm −1 , ν 2 =
447.4 cm −1 , and ν 3 = 1110.5 cm −1 . Remember that standard conditions (NTP)
are P = 1 bar, T = 273.15 K, and V
◦ = 22.711 dm
3 mol
−1 .
28. Linear triatomic molecules have four independent vibrational degrees of freedom, each characterized by a natural vibrational frequency ν i (i = 1, 2, 3, 4)
in Hz [equivalently wave number ν i in the non-SI unit cm −1 ] and, in the SHO
approximation, a corresponding energy vib,i ≡ (v i +
1
2 )hν i [or i = (v i +
1
2 )hcν i ], with h the Planck constant and v i the vibrational quantum number for
vibrational degree of freedom i. The corresponding SHO vibrational partition
function for a triatomic linear molecule is hence given by
z vib (T ) =
4
i=1
e − vib,i /(2T )
1 − e vib,i /T ,
in which the characteristic vibrational temperature vib,i is defined as vib,i ≡
vib,i /k B . Express z vib (T ) in terms of hyperbolic functions, and obtain an
expression for the temperature derivative of z vib . [Hint: use the technique of
logarithmic differentiation.]
29. When the temperature is either very high or very low (by which we mean
that k B T is either much larger than or much smaller than the fundamental
oscillator energies or, equivalently, that the temperature T is either much larger
or smaller than the characteristic temperatures vib, ), then we can introduce
approximations for the functions appearing in the expression for z vib (T ) given
in Problem 28, and thereby obtain appropriate approximate expressions for the
vibrational partition function itself. Obtain such expressions for the individual
z vib,i (T ) valid for these two cases.
351
Obtain a discrete summation expression for z rot (T ) that includes the
centrifugal-distortion energy, with the rotational energy of a CH 4 molecule
given in cm −1 units as
j = j (j + 1)B − [j (j + 1)]
2 D .
Compute the values of z rot (T ) for temperatures 50 K, 100 K, 300 K, and
500 K using B 0 = 5.2412 cm −1 and D 0 = 1.23 × 10 −4 cm −1 . Compare the
results of this calculation with the rigid-rotor values obtained in Problem 25
via z rot−nuc (T ) = 16z DS
rot (T ) with the values z DS
rot−nuc (T ) 41.993, 116.421,
598.639, and 1287.523 computed for CH 4 at temperatures 50 K, 100 K, 300 K,
500 K, respectively, by Robiette and Dang-Nhu [49] and the closed-form values
z CF
rot−nuc (T ) given via Eq. (6.3.23).
27. Calculate the standard molar entropy of ClO 2 at 298.15 K (NATP). The ClO 2
molecule is a nonlinear triatomic molecule (point symmetry group C 2v ), and
has one unpaired electron (ground electronic term symbol 2 B 1 ), so that it is
paramagnetic. Its rotational constants have been determined from spectroscopy
to be B A = 1.737 cm −1 , B B = 0.332 cm −1 , and B C = 0.278 cm −1 , while
the fundamental vibrational oscillator frequencies are ν 1 = 945.5 cm −1 , ν 2 =
447.4 cm −1 , and ν 3 = 1110.5 cm −1 . Remember that standard conditions (NTP)
are P = 1 bar, T = 273.15 K, and V
◦ = 22.711 dm
3 mol
−1 .
28. Linear triatomic molecules have four independent vibrational degrees of freedom, each characterized by a natural vibrational frequency ν i (i = 1, 2, 3, 4)
in Hz [equivalently wave number ν i in the non-SI unit cm −1 ] and, in the SHO
approximation, a corresponding energy vib,i ≡ (v i +
1
2 )hν i [or i = (v i +
1
2 )hcν i ], with h the Planck constant and v i the vibrational quantum number for
vibrational degree of freedom i. The corresponding SHO vibrational partition
function for a triatomic linear molecule is hence given by
z vib (T ) =
4
i=1
e − vib,i /(2T )
1 − e vib,i /T ,
in which the characteristic vibrational temperature vib,i is defined as vib,i ≡
vib,i /k B . Express z vib (T ) in terms of hyperbolic functions, and obtain an
expression for the temperature derivative of z vib . [Hint: use the technique of
logarithmic differentiation.]
29. When the temperature is either very high or very low (by which we mean
that k B T is either much larger than or much smaller than the fundamental
oscillator energies or, equivalently, that the temperature T is either much larger
or smaller than the characteristic temperatures vib, ), then we can introduce
approximations for the functions appearing in the expression for z vib (T ) given
in Problem 28, and thereby obtain appropriate approximate expressions for the
vibrational partition function itself. Obtain such expressions for the individual
z vib,i (T ) valid for these two cases.