Final answer:
To find the ratio of the average rotational kinetic energies per O2 molecule to per N2 molecule, we can use the Boltzmann distribution and the formula for rotational kinetic energy. This involves calculating integrals and using the moments of inertia for each molecule.
Step-by-step explanation:
To find the ratio of the average rotational kinetic energies per O2 molecule to per N2 molecule, we need to use the formula for rotational kinetic energy and the Boltzmann distribution. The rotational kinetic energy of a molecule is given by E = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity. The average rotational kinetic energy per molecule can be calculated by taking the integral of the rotational kinetic energy over the velocity distribution. Applying the Boltzmann distribution, which relates the probability of a molecule having a certain kinetic energy to the temperature, to both O2 and N2 molecules, we can find the ratio of their average rotational kinetic energies.
Let's denote the ratio of the average rotational kinetic energy per O2 molecule to per N2 molecule as R. According to the Boltzmann distribution, the probability of a molecule having a certain kinetic energy is proportional to e^(-E/kT), where E is the energy, k is the Boltzmann constant, and T is the temperature. The ratio of the average rotational kinetic energies can be written as:
R = (∫(1/2)Iω^2 * e^(-Eo₂/kT) dω)/(∫(1/2)Iω^2 * e^(-En₂/kT) dω)
Since the ratio of temperatures is given as T₂/T₁ = 1/2, we can substitute T₂ = (1/2)T₁ into the Boltzmann distribution. Using this, we can simplify the ratio R to:
R = (∫(1/2)Iω^2 * e^(-Eo₂/(k(T/2))) dω)/(∫(1/2)Iω^2 * e^(-En₂/(k(T/2))) dω)
Now, we need to find the moments of inertia for O2 and N2 molecules and then solve the integrals to calculate the ratio R.