Interpretation:
The ratio of the population of the levels with J=2 and J=1 of HCl at 25οC has to be calculated.
Concept introduction:
Statistical
q≡∑igie−β∈i
It is also known as canonical ensemble partition function.
Answer to Problem 13A.1ST
The ratio of the population of the levels with J=2 and J=1 of HCl at 25οC is 1.359_.
Explanation of Solution
The expression for the energy of the state with quantum number J is shown below.
εJ=hc˜BJ(J+1) (1)
Where,
- h is Plank’s constant.
- c is the velocity of light.
- ˜B is rotational constant.
- J is the rotational quantum number.
The energy of the state with quantum number J=1 is calculated by substituting J=1 in equation (1) as shown below.
ε1=hc˜B×1(1+1)=2hc˜B
The energy of the state with quantum number J=2 is calculated by substituting J=2 in equation (1) as shown below.
ε2=hc˜B×2(2+1)=6hc˜B
The expression for the Boltzmann population ratio is shown below.
NiNj=e−β(εi−εj) (2)
Where,
- NiNj is the ratio of the population of the upper state and lower state.
- εi is the energy of the state with a quantum number i.
- εj the energy of the state with a quantum number j.
- β=1/kT, here k is Boltzmann’s constant and T is the temperature.
The energy separation of the states with J=2 and J=1 is shown below.
ε2−ε1=6hc˜B−2hc˜B=4hc˜B
Substitute the values of i=2 and j=1 in equation (2) as shown below.
N2N1=e−β(ε2−ε1)
Substitute the value of ε2−ε1 in the above equation.
N2N1=e−4hc˜Bβ
The upper level with rotational quantum number has five-fold degeneracy and the lower level has three-fold degeneracy. So the relative population of the levels is calculated as shown below.
N2N1=53e−4hc˜Bβ
Substitute β=1/kT in the above equation as shown below.
N2N1=53e−4hc˜BkT (3)
The value of kT/hc is 207.22 cm−1 at 25οC and the value of ˜B is 10.591 cm−1. Substitute the value of kT/hc and ˜B in equation (3) as shown below.
N2N1=53e−4hc˜BkT=53e−4×10.591 cm−1207.22 cm−1=53e−42.364207.22=53e−0.204
The above equation is further solved as shown below.
N2N1=53e−0.204=53×0.8154=1.359_
Therefore, the ratio of the population of the levels with J=2 and J=1 of HCl at 25οC is 1.359_.
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Chapter 13 Solutions
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