Rotational Energy and Moment of Inertia of a nitrogen molecule

[Zacthor]

Problem 10-58a: The equilibrium separation between the nuclei of the nitrogen molecule (N2) is 0.190 nm and the mass of each nitrogen nucleus is 14.0 u, where u = 1.66E-27 kg. For rotational energies, the total energy is due to rotational kinetic energy. Approximate the nitrogen molecule as a rigid dumbbell of two equal point masses and calculate the moment of inertia about its center of mass.

Calculate the rotational energy levels for l = 1 and l = 2 using the relation El = l(l+1)hbar2/(2/I).

Anyone have any suggestions on how to tackle this problem?

 

[Shooting Star]

It seems to be an extremely simple problem Where are you getting stuck, actually?

The MI of a point mass about a given point at a distance r is mr^2. The CM of the two point masses here is obviously at the midpoint of the segment joining the the two masses.

 

[Moetasim]

I would approach this problem by first understanding the concepts of rotational energy and moment of inertia. Rotational energy is the energy associated with the rotation of an object, while moment of inertia is a measure of an object's resistance to rotational motion. In this case, the nitrogen molecule can be approximated as a rigid dumbbell with two equal point masses.

To calculate the moment of inertia of the nitrogen molecule, we can use the formula I = mr^2, where m is the mass of each nitrogen nucleus and r is the equilibrium separation between the nuclei. Plugging in the given values, we get I = 14.0 u * (0.190 nm)^2 = 4.99E-46 kg*m^2.

Next, we can use the given formula El = l(l+1)hbar^2/(2/I) to calculate the rotational energy levels for l = 1 and l = 2. Here, hbar is the reduced Planck's constant (h/2π). Plugging in the values, we get El = (1)(1+1)hbar^2/(2/4.99E-46) = 3.68E-24 J and El = (2)(2+1)hbar^2/(2/4.99E-46) = 1.47E-23 J for l = 1 and l = 2, respectively.

In summary, to tackle this problem, we first approximated the nitrogen molecule as a rigid dumbbell and calculated its moment of inertia. Then, we used the given formula to calculate the rotational energy levels for l = 1 and l = 2. This approach can be applied to other molecules or objects to calculate their rotational energy and moment of inertia.