Quantized angular momentum of diatomic gas molecule: Bohr Model

[frankR]

Here is the problem:

A diatomic gas molecule consists of two atoms of mass m separated by a fixed distance d rotating about an axis as shown. Assuming that its angular momentum is quantized just as in the Bohr atom, determine a) the quantized angular speed, b) the quantized rotational energy.

Note: The diagram consists of two point masses of mass m rotating about an axis with angular speed &omega separated by a distance d.


Here is my solution:

The assumption made by Bohr under his model of the hydrogen atom: angular momentum is quantized according to L = nh/(2&pi)

The following model of quantized &omega and E of the diatomic molecule will use the same assumption.

L = 2mvr = nh/(2&pi)

Substitute: v = r&omega

2m(r&omega)r = nh/(2&pi)

Substituing: r = 1/2d, and solving for &omega we find:

&omega = nh/(&pi md²)

For rotational E:

E =1/2I&omega²

I = 2mr²

Substituting: r = 1/2d into I

I = 1/2md²

Substituting I and &omega² in E:

We find:

E = n²h²/(4m&pi²d²)


Is my solution correct?

Thanks

 

[quantumdude]

Originally posted by frankR
Is my solution correct?

Yes.

 

[M98Ranger]

for sharing your solution. Your solution looks correct. You have correctly applied the Bohr's assumption of quantized angular momentum to the diatomic gas molecule and have derived the quantized angular speed and rotational energy. Your substitution of the moment of inertia and angular speed into the formula for rotational energy is also correct. Great job!