350
6 Molecular Systems
for the NO molecule can be approximated rather well simply by taking the
electronic partition function to have the form
z el (T ) = ω e1 + ω e2 exp{− e2 /(k B T )} ,
in which e2 = 119.82 cm −1 .
Obtain an expression for the contribution, U el (T ), of the excited electronic
level to the internal energy U(T ) as a function of temperature. From your result
for U el (T ) obtain an expression for and discuss the behaviour of the electronic
contribution to C V as a function of temperature. Plot C V as a function of T
for NO, and obtain an expression for the electronic contribution to the entropy,
S el (T ). Check that S el → Nk B ln 2 for T → 0 and that S el → Nk B ln 4 for
T → ∞.
23. Determine the contribution of the electronic degrees of freedom of the doubly
degenerate ground electronic term of the NO molecule to the standard molar
entropy of NO. Evaluate S el (T ) for temperatures 50 and 298.15 K. Compare
your 298.15 K result with R ln 4: why should you expect this to be a meaningful
comparison?
24. Starting from Eq. (6.2.59) for the rotational partition function for a heteronuclear diatomic molecule, show that the rotational contribution to the molar heat
capacity at constant volume may be written in the form
C V ,rot (T ) = R[[ε
2
rot (T ) − −ε rot
2 (T )] ,
in which the average f of f (j) is defined as f ≡
j p j f (j), with p j
given by Eq. (6.2.62b), while ε rot is defined as ε rot (j ) ≡ j rot /T , with j
the rotational energy Bj (j + 1) and B ≡ ¯
h 2 /(2I ) is the rotational constant.
Compare your computed results with the experimental values 0.2013 at 20 K,
0.9698 at 40 K, 1.0403 at 77.5 K, 1.0201 at 90.2 K, and 1.0000 at 293 K on a
plot of C V ,rot (T )/R vs. temperature.
25. Compute the rotational partition function for methane (CH 4 ) at temperatures
T = 50 K, 100 K, 300 K, and 500 K by direct summation over rotational states
using Eq. (E.2.3) and compare the values obtained with those determined from
the classical formula of Eq. (6.3.4). As CH 4 is spherical top molecule, the
three principal moments-of-inertia are all equal. The rotational constant for
CH 4 is B = 5.2412 cm −1 , and the symmetry number for CH 4 is 12. Compare
your summations (converged to two decimal places) with the classical (or high
temperature) values for z rot (T ).
26. Nuclear spin statistical effects are important for the computation of the intensities of spectroscopic transitions, which depend upon the individual fractional
populations of the spectroscopic states. For bulk thermodynamic properties that
depend less directly upon the fractional populations and more directly upon the
partition function, we should expect the effects of nuclear spin statistics to be
relatively small (with the exception of the hydrogen isotopologues).
6 Molecular Systems
for the NO molecule can be approximated rather well simply by taking the
electronic partition function to have the form
z el (T ) = ω e1 + ω e2 exp{− e2 /(k B T )} ,
in which e2 = 119.82 cm −1 .
Obtain an expression for the contribution, U el (T ), of the excited electronic
level to the internal energy U(T ) as a function of temperature. From your result
for U el (T ) obtain an expression for and discuss the behaviour of the electronic
contribution to C V as a function of temperature. Plot C V as a function of T
for NO, and obtain an expression for the electronic contribution to the entropy,
S el (T ). Check that S el → Nk B ln 2 for T → 0 and that S el → Nk B ln 4 for
T → ∞.
23. Determine the contribution of the electronic degrees of freedom of the doubly
degenerate ground electronic term of the NO molecule to the standard molar
entropy of NO. Evaluate S el (T ) for temperatures 50 and 298.15 K. Compare
your 298.15 K result with R ln 4: why should you expect this to be a meaningful
comparison?
24. Starting from Eq. (6.2.59) for the rotational partition function for a heteronuclear diatomic molecule, show that the rotational contribution to the molar heat
capacity at constant volume may be written in the form
C V ,rot (T ) = R[[ε
2
rot (T ) − −ε rot
2 (T )] ,
in which the average f of f (j) is defined as f ≡
j p j f (j), with p j
given by Eq. (6.2.62b), while ε rot is defined as ε rot (j ) ≡ j rot /T , with j
the rotational energy Bj (j + 1) and B ≡ ¯
h 2 /(2I ) is the rotational constant.
Compare your computed results with the experimental values 0.2013 at 20 K,
0.9698 at 40 K, 1.0403 at 77.5 K, 1.0201 at 90.2 K, and 1.0000 at 293 K on a
plot of C V ,rot (T )/R vs. temperature.
25. Compute the rotational partition function for methane (CH 4 ) at temperatures
T = 50 K, 100 K, 300 K, and 500 K by direct summation over rotational states
using Eq. (E.2.3) and compare the values obtained with those determined from
the classical formula of Eq. (6.3.4). As CH 4 is spherical top molecule, the
three principal moments-of-inertia are all equal. The rotational constant for
CH 4 is B = 5.2412 cm −1 , and the symmetry number for CH 4 is 12. Compare
your summations (converged to two decimal places) with the classical (or high
temperature) values for z rot (T ).
26. Nuclear spin statistical effects are important for the computation of the intensities of spectroscopic transitions, which depend upon the individual fractional
populations of the spectroscopic states. For bulk thermodynamic properties that
depend less directly upon the fractional populations and more directly upon the
partition function, we should expect the effects of nuclear spin statistics to be
relatively small (with the exception of the hydrogen isotopologues).