# Homework 15

- Page ID
- 204099

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

## Q1

If four distinguishable particles can each occupy energy levels 0, \(\epsilon\), \(2\epsilon\), and \(4\epsilon\). Calculate the entropy of the system if the total energy is

- \(3\epsilon\)
- \(4\epsilon\)
- \(8\epsilon\)
- \(12\epsilon\)
- \(16\epsilon\)

*One possible configuration for \(4\epsilon\);*

## Q2

Consider a system of \(N\) distinguishable,independent particles, each of which has only two accessible states; a ground state of energy 0 and an excited state of energy \(ε\). If the system is in equilibrium with a heat bath of temperature \(T\), calculate \(A\) ,\(U\), \(S\), and \(C_v\). Sketch \(C_v\) versus \(T\). How your results would change if \(ε_o\) were added to both energy values (i.e., a change in the zero of energy like a zero point energy) ?

## Q3

Calculate the value of \(n_x\), \(n_y\), \(n_z\) for the case \(n_x = n_y = n_z\) for a hydrogen **atom **in a box of dimension \(1\; cm^3\) if the particle has kinetic energy \(3k_BT/2\) for T =27 ºC. What significant fact does this calculation illustrate? How would this change if we considered a hydrogen molecule instead?

## Q4

Molecular nitrogen is heated in an electric arc and it is found spectroscopically that the (non-relative) populations of excited vibrational levels are

v=0 | v= 1 | v= 2 | v= 3 | v= 4 |
---|---|---|---|---|

1.00 | 0.2 | 0.04 | 0.008 | 0.005 |

Is the nitrogen in thermodynamic equilibrium with respect to vibrational energy? What is the vibrational temperature of the gas? Is this necessarily the same as the translational temperature?

## Q5

What is the entropy of 1 mole of helium at 298 K and 1 atm.

## Q6

Consider the gas phase \(N_2\) molecule at a temperature of 300 K.

- What is the most probable value of the rotational quantum number \(J\) if one ignores the fact that \(N_2\) is homonuclear?
- Determine the fraction of molecules in each \(J\) state from 0 to 9 at T=300 K ( just use the high temperature limit for \(q_{rot}\) in these calculations). Compare your results with what you determined in part a.
- What is the most probable vibrational quantum number \(v\) for this same temperature?