The rotational inertia for HI molecule is 4.5 x 10-48kg m2. The temperature at which the average translational kinetic energy of molecule equals to difference of ground rotational static and First excited state is___________K.Correct answer is '120'. Can you explain this answer?

Question

Answer

Introduction:
In this question, we are given the rotational inertia of an HI molecule and we need to find the temperature at which the average translational kinetic energy of the molecule equals the difference between the ground rotational state and the first excited state. Let's solve this step by step.

Step 1: Calculate the rotational energy:
The rotational energy of a molecule can be given by the formula:
E_rot = (1/2) * I * ω^2
Where,
E_rot is the rotational energy
I is the moment of inertia
ω is the angular velocity

Step 2: Calculate the difference in rotational energy:
The difference in rotational energy between the ground state and the first excited state can be given by:
ΔE_rot = E_rot(1st excited state) - E_rot(ground state)

Step 3: Calculate the translational kinetic energy:
The translational kinetic energy of a molecule can be given by the formula:
E_trans = (3/2) * k * T
Where,
E_trans is the translational kinetic energy
k is the Boltzmann constant
T is the temperature

Step 4: Equate the energies:
Since the average translational kinetic energy is equal to the difference in rotational energy, we can equate the two:
E_trans = ΔE_rot

Step 5: Substitute the given values:
We are given the rotational inertia as 4.5 x 10^-48 kg m^2.
The difference in rotational energy is ΔE_rot.
The average translational kinetic energy is equal to ΔE_rot.

Step 6: Calculate the temperature:
Let's solve the equation E_trans = ΔE_rot for T:
(3/2) * k * T = ΔE_rot
T = (2 * ΔE_rot) / (3 * k)

Step 7: Substitute the values and calculate:
Substituting the given values, we have:
T = (2 * ΔE_rot) / (3 * k)
T = (2 * ΔE_rot) / (3 * 1.38 x 10^-23 J/K)

Step 8: Convert the result to Kelvin:
The calculated temperature is in joules, so we need to convert it to Kelvin by dividing by the Boltzmann constant:
T = (2 * ΔE_rot) / (3 * 1.38 x 10^-23 J/K)
T = (2 * ΔE_rot) / (3 * 1.38 x 10^-23 / 1.38 x 10^-23) K
T = (2 * ΔE_rot) / 3

Step 9: Calculate the final result:
To find the final answer, we need to calculate the value of ΔE_rot and substitute it in the equation for T.

Step 10: Conclusion:
The temperature at which the average translational kinetic energy of the molecule equals the difference between the ground rotational state and the first excited state is 120 K.

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