Spring stiffness and Heat Capacity (Equipartition of energy)

[godiswatching_]

Homework Statement

At about 2000◦ K the heat capacity at constant volume increases to CV = 7/2kB per molecule from 3/2kB due to contributions from vibrational energy states. Use these observations to estimate the stiffness of the spring that approximately represents the inter-atomic force binding the molecule.

Relevant Equations

h-bar omega = 2KT

Here's a picture of what I tried. I was wondering if this is correct?

 

[Simon Bridge]

How did you use the observations they told you to?
You got to make it explicit.

Show your reasoning.

Usually, if you have laid out your reasoning to go with the formulae and equations, you'll be able to see if you got it right.

 

[godiswatching_]

How did you use the observations they told you to?
You got to make it explicit.

Show your reasoning.

Usually, if you have laid out your reasoning to go with the formulae and equations, you'll be able to see if you got it right.

my reasoning would be that the extra 2kBT energy comes from the vibrational energy. And since vibrational energy can be quantized using h-bar*omega we can equate those two. However, I am not sure if the 2kBT comes only from vibrational or vibrational AND rotational energies. And also if the mass we are using is the reduced mass of the system or just 2*mass of each hydrogen.

 

[mjc123]

Does the question say it's hydrogen? Hydrogen does not have Cv = 3k/2 below 2000K. In fact, I think the numbers in the question must be wrong. Cv = 3k/2 for a monatomic gas or a molecule at low temperatures where rotations are not activated. A diatomic molecule with 2 active rotational degrees of freedom has Cv = 5k/2. When the (one) vibrational mode becomes active, the Cv rises to 7k/2. Perhaps it is this last step that the question means to refer to.

 

[Simon Bridge]

my reasoning would be that the extra 2kBT energy comes from the vibrational energy.

... energy stored as vibration. OK. Since you are told that at some threshold temperature, an extra mode of heat store-age becomes available... good for an extra 2k_BT

And since vibrational energy can be quantized using h-bar*omega we can equate those two.

So far what you did matches your reasoning well.

However, I am not sure if the 2kBT comes only from vibrational or vibrational AND rotational energies.

Can you check? For example, how many degrees of freedom would you normally expect to get added for vibration as compared with vibration and rotation together? How much energy would you expect to be stored per degree of freedom?

And also if the mass we are using is the reduced mass of the system or just 2*mass of each hydrogen.

ie. is the quoted mass the reduced mass of the particle mass ... you'll have to decide that by context of the course, since it is not given. Which would normally be the case from your notes?

Notes:
... annotating your maths can save you marks if you make a mistake, and help the person marking not to make a mistake in assigning marks.
... the text of the question provided does not say what the molecule is or even if it is diatomic... is "hydrogen" or even "diatomic" a safe assumption?
... the idea here is to help you assess your own work so you can tell when you get good answers. (You are training to be able to solve problems nobody knows the right answer to, who will you ask then?)[/quote]