Ratio of Population of Two Rotational States

To calculate the approximate ratio of the population of two rotational states with J = 10 and J = 20, we need to consider the Boltzmann distribution and the formula for the population ratio.

1. Boltzmann Distribution:
The Boltzmann distribution describes the distribution of particles in different energy states at a given temperature. It is given by the formula:

P(E) = (1/Z) * exp(-E/kT)

Where:
P(E) is the probability of finding a particle in the energy state E,
Z is the partition function,
E is the energy of the state,
k is the Boltzmann constant (1.38 × 10^-23 J/K),
T is the temperature in Kelvin.

2. Population Ratio Formula:
The ratio of the population of two states, A and B, can be calculated using the formula:

Ratio = (NA/NB) = (PA/PB) = (exp(-EA/kT)/exp(-EB/kT)) = exp((EB - EA)/kT)

Where:
NA and NB are the populations of states A and B, respectively,
PA and PB are the probabilities of finding a particle in states A and B, respectively,
EA and EB are the energies of states A and B, respectively.

3. Calculation:
Given data:
Rotational constant, B = 0.2 cm^-1
kT = 209 cm^-1

The rotational energy, EJ, can be calculated using the formula:
EJ = BJ(J + 1)

For J = 10:
E10 = B * 10(10 + 1) = 0.2 * 10(11) = 2 cm^-1

For J = 20:
E20 = B * 20(20 + 1) = 0.2 * 20(21) = 8.4 cm^-1

Using the population ratio formula, we can calculate the ratio of populations:

Ratio = exp((EB - EA)/kT) = exp((8.4 - 2)/209) = exp(6.4/209)

Calculating this value gives us the approximate ratio of the population of the two rotational states.

In conclusion, to calculate the approximate ratio of the population of two rotational states with J = 10 and J = 20, we used the Boltzmann distribution and the population ratio formula. By plugging in the values for the rotational energy and the given temperature, we can calculate the ratio.