- #1

nicky04

- 3

- 0

## Homework Statement

The potential energy V between the two hydrogen atoms ( mH = 1.672*10^(-24) g) in a

hydrogen molecule is given by the empirical expression V = V0 [ exp(-2a(r-r0)) - 2exp(-a(r-r0))] where

V0 = 7*10^(-12) erg, a = 2*10^8 cm-1, r0 = 8*10^-9 cm.

(a) Estimate the temperatures at which rotation and vibration begin to contribute to the specific

heat of hydrogen gas.

(b) Give the approximate values of Cv (in terms of the gas constant R for the following

temperatures: 25 K, 250 K, 2500 K, 10000 K.

## Homework Equations

I am not sure since I am not sure about my starting point.

## The Attempt at a Solution

Basically I can think of two ways to do this. Either I could calculate the rotational temperature given by

T_rot = h_bar^2 / 2 k I , with I = M*r0^2 and the reduced mass M = mh / 2.

The vibrational temperature is given by

T_vib = h_bar omega / k.

What is confusing me is the fact that the given energy potential is not being used. I thought, maybe I need it to calculate omega? I started solving the equation of motion for a harmonic oscillator with and additional potential, but with this potential it is not that easy to solve and I have a feeling that it's wrong anyway. Is there another way to calculate omega from V? I also tried it by saying

omega = sqrt( k / m) and

F=-kx.

With the given potential I could calculate F but then it depends on r and I am not sure my finaly solution should depend on r?

Another way would be to look at the partition functions for the rotation and the vibration. The first question there is, can I assume the atoms are in the ground state? Because the functions depend on l and n. Further I am not sure how to include the potential again, should I just add it in the exponent? Once I had the partition functions I could derive the specific heats and set them equal to the known specific heats depending on R.

Does any of this make sense or should I rethink it all over again?