2P  

Consider a small system that is interacting with a reservoir at temperature 400 K, pressure 108 Pa, and chemical potential ‒0.3 eV.

(a) To go from state 1 to state 2, the small system must take an additional 0.03 eV of energy and 10‒29 m3 of volume from the reservoir. How many times more probable is it for the small system to be in state 1 than state 2?

(b) To go from state 1 to state 3 the small system must take 0.4 eV of energy and one particle (but no extra volume) from the reservoir. How many times more probable is it that the small system is in state 1 than state 3?

(c) To go from state 1 to state 4, the small system must take no energy but one particle and 10‒27 m3 of volume from the reservoir. How many times more probable is it that the small system is in state 1 than state 4?

Question 4: The Ensemble of Pressures: Imagine that you bring a system into contact with a
reservoir of heat and work (or Thermal and volume reservoirs). The wall connecting the system
and the reservoir is diathermal and can also move.
a) Using the same formalism used to find the Boltzmann factor (Chapter 3) and the Gibbs
factor (Chapter 5), derive in detail an expression for the equivalent factor related to the
reservoir in question. We can baptize this factor as “Pressure factor”.
b) Find the partition function for the problem.
c) Find expressions for U (average internal energy) and <V> (average volume).

In answering this set of nine questions, you are encouraged to draw a PV diagram, with P (pressure) on the y axis and V (volume) on the x axis. Plot the three points A, B, and C on this diagram.

Consider 3.00 liters of an ideal (monatomic) gas at a pressure of 31.0 atm and a temperature of 356 K. Call this state of the system A. Using the ideal gas law, calculate the number of moles of gas present in the system.

Number of moles, n = 3.19

The temperature of the system is reduced, keeping the volume constant at 3.00 liters, until the pressure in the system equals 16.0. Call this state of the system B. Calculate the temperature at this new state B in degrees K.

Temperature at state B = 184

Now the gas is allowed to expand at constant pressure (16.0 atm) until the temperature is again equal to 356 K. Call this state of the system C. Calculate the volume of the gas at state C.

Volume at state C = 5.81

Now calculate the work done on or by the system when the system moves back from state C to state A along the path CBA. Enter your answer in joules with the correct sign.

Work along the path CBA, w = 4.56E+3

Calculate the heat absorbed or liberated by the system when the system moves from state C to state B. Note that the molar heat capacity of an ideal gas under constant pressure is (5/2)R. Enter your answer in joules with the correct sign.

Heat along the path C to B, q = -1.14E+4

Using your results from the previous two questions, calculate the heat absorbed or liberated by the system as it moves from state B to state A. Enter your answer in joules with the correct sign.

Heat along the path B to A, q = 6.84E+3

What is the change in the internal energy of the system as the system travels down the isotherm from state A to state C?

Change in internal energy down the isotherm = 0.000

How much work is done by the system as the system travels down the isotherm from state A to state C? Enter your answer in joules with the correct sign.

Work down the isotherm, w = -6.23E+3

How much heat must be absorbed or liberated by the system as it moves down the isotherm from A to C? Enter your answer in joules with the correct sign.

Heat down the isotherm, q = 6.23E+3