Problem Set 2

Problem Set 2

Due Monday, February 19, uploaded to Blackboard by 11pm

Problem 1 [10 points total]
Prove the following relationship between the specific heat at constant pressure, C_p, and the coefficient of thermal expansion α.
(∂C_p/∂p)_T = −TV[α² + (∂α/∂T)_p]
Problem 2 [15 points total]
In a particular engine a gas is compressed in the initial stroke of the piston. You may assume the gas behaves like an ideal gas. Measurements of the instantaneous temperature, carried out during the compression, reveal that the temperature increases according to the relation:
T = (V/V_o)^ηT_o
where T_o and V_o are the initial temperature and volume and η is a constant. The gas is compressed to the volume V₁ (where V₁ < V_o). Assume that the gas is a monatomic ideal gas of N atoms, and assume the process is quasi-static (i.e. the system is always instantaneously in equilibrium).
a) Calculate the mechanical work done on the gas. [5 points]
b) Calculate the change in the total energy of the gas. [5 points]
c) Calculate the heat transfer Q to the gas. For what value of η is Q = 0? Show that this corresponds to the case of adiabatic compression (see Problem Set 1, problem 4). [5 points]
[Hint: you may use the facts you know about an ideal gas, i.e. pV = Nk_BT, and E = (3/2)Nk_BT.]
Problem 3 [20 points total]
A thermodynamic engine can be thought of as a cyclic process by which heat (TdS) pumped from a "hot" reservoir into a system (the engine) is converted into mechanical work (pdV) done by the engine. The most famous such example is the Carnot cycle. Another such cycle is the Otto cycle, schematically shown in the diagram below as a directed path in the volume-entropy plane. The Otto cycle is a rough approximation to the operation of a gasoline engine.

The gas is compessed adiabatically (constant entropy) in the step A to B. Then it is heated isochorically (constant volume) in the step B to C (this step corresponds to the combustion of the gas in the gas engine). The gas is then expanded adiabatically in step C to D (this is the power stroke). Finally the gas is cooled isochorically to bring it back to its initial state; this is step D to A.
Step B to C extracts heat Q_BC from the "hot" reservoir to heat the gas, while step D to A returns heat Q_DA = − Q_AD to the "cold" reservoir (usually the ambient atmosphere). The difference is the work done by the engine.
If the engine efficiency is defined as ratio of the net heat converted to work, divided by the heat withdrawn from the hot reservoir
ε = (Q_BC − Q_AD)/Q_BC
then assuming the gas is an ideal gas, show that the efficiency of the Otto cycle is given by,
ε = 1 − (V_B/V_A)^{(C_p − C_V)/C_V}
Problem 4 [10 points total]
Consider a box containing N extremely relativistic particles, moving so fast that their speed can be taken as essentially the speed of light c, but the relativisitc factor γ = 1/(1-v²/c²)^1/2 is finite. For such particles the energy-momentum relation is well approximated by ε = pc. Equivalently, you can think of these as massless particles (think photon!) which travel with speed c, carry momentum p, and have energy ε = pc.
Using a kinetic theory for such particles, as in Notes 2-1, find the relation between the pressure P, volume V, number of particles N, and the average energy per particle <ε>. You should assume that a particle is equally likely to be traveling in any spatial direction. If we assume that the ideal gas law continues to hold for such a gas (it does), what is the relation between the average energy per particle and the temperature?