9.5 Problems for This Chapter
507
Isotopologue M/kg mol
−1
rot /K vib /K
H 2
2.018
85.30
6215
T 2
6.032
28.51
3593
HT
4.024
56.90
5076
From the relation between the depth (−D e ) of the potential well for a diatomic
molecule, the zero-point vibrational energy (
1
2 hν osc ), and the experimentally
accessible dissociation energy D 0 (H 2 ) = 431.79 kJ mol
−1 determine D 0 values
for the T 2 and HT hydrogen isotopologues. Obtain an expression for evaluating
the equilibrium constant, K P (T ), for the tritium–hydrogen exchange reaction,
and employ it to evaluate K P (T ) at temperatures T = 300 K, 500 K, and 1000 K.
3. The equilibrium constant K eq characterizing the isotopic exchange of a deuterium atom between a deuterated acid and water may be evaluated using
spectroscopic data for the molecules involved in the reaction. Calculate values
of the equilibrium constant at temperatures of 300 and 1000 K for the gas-phase
isotopic exchange reaction
DCl(g) + H 2 O(g) HDO(g) + HCl(g) ,
using the following approximate values for relevant spectroscopic data (in
cm −1 ):
• H 2 O: ν 1 (a 1 ) = 3657, ν 2 (a 1 ) = 1595, ν 3 (b 2 ) = 3756, A 0 = 27.9, B 0 = 14.5,
C 0 = 9.28.
• HDO: ν 1 (a ) = 2727, ν 2 (a ) = 1402, ν 3 (a ) = 3707, A 0 = 23.4, B 0 = 9.10,
C 0 = 6.41.
• HCl: ν(a) = 2886, B 0 = 10.4; DCl: ν(a) = 2091, B 0 = 5.39.
All vibrational modes are nondegenerate.
4. Obtain an expression for the equilibrium constant K eq (T ) for the isotopic
exchange reaction
CD 4 (g) + HCl(g) CHD 3 (g) + DCl(g) .
Make a plot of K eq (T ) vs. T to illustrate the temperature dependence of K eq (T )
in the temperature range 700 K ≤ T ≤ 1000 K using the relevant approximate
values for the vibrational frequencies and rotational constants (in cm −1 ):
• CD 4 : ν 1 (a 1 ) = 2109, ν 2 (e) = 1092, ν 3 (f 2 ) = 2259, ν 4 (f 2 ) = 996, B 0 =
2.63.
• CHD 3 : ν 1 (a 1 ) = 2293, ν 2 (a 1 ) = 2142, ν 3 (a 1 ) = 1003, ν 4 (e) = 2263,
ν 5 (e) = 1291, ν 6 (e) = 1036, A 0 = 2.63, B 0 = 3.28.
• HCl: ν(a) = 2886, B 0 = 10.4; DCl: ν(a) = 2091, B 0 = 5.39.