# Homework 15

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

1. $$3\epsilon$$
2. $$4\epsilon$$
3. $$8\epsilon$$
4. $$12\epsilon$$
5. $$16\epsilon$$ 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.

1. What is the most probable value of the rotational quantum number $$J$$ if one ignores the fact that $$N_2$$ is homonuclear?
2. 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.
3. What is the most probable vibrational quantum number $$v$$ for this same temperature?