1. The problem statement, all variables and given/known data
A monatomic ideal gas has pressure
p_1 and temperature T_1. It is contained in a cylinder of volume V_1 with a movable piston, so that it can do work on the outside world.
Consider the following three-step transformation of the gas:
1. The gas is heated at constant volume until the pressure reaches Ap_1 (where A >1).
2. The gas is then expanded at constant temperature until the pressure returns to p_1.
3. The gas is then cooled at constant pressure until the volume has returned to V_1.
It may be helpful to sketch this process on the pV plane.
How much heat DeltaQ_1 is added to the gas during step 1 of the process?
Express the heat added in terms of p_1, V_1, and A.
How much work W_2 is done by the gas during step 2?
Express the work done
in terms of p_1, V_1, and A.
How much work W_3 is done by the gas during step 3?
If you've drawn a graph of the process, you won't need to calculate an integral to answer this question.
Express the work done in terms of p_1, V_1, and A.
2. Relevant equations
C_V = 12.47
R = 8.31
3. The attempt at a solution
I tried Q = p_1*V_1*(C_V/R) = 1.5*Ap_1*V_1 but I was told this is the final internal energy, not the change in internal energy. so I worked out that
Q = [1.5*p_1*V_1*(AT_1-T_1)] / T_1 but the answer does not depend on AT_1 or T_1
all I've got so far is
W = nRT*ln(V_f/V_i) = pV*ln(V_f/V_i)
but thats about as far as I get.
I got Ap_1*V_1 but this is what the value would be if it were coming from V = 0. So I re-arranged pV=nRT to eventually get
W = p_1[(p_1V_1)/(Ap_1) - V_1]
but this is also wrong how do I take into account the initial state, wouldn't I just be able to write W = (Ap_1V_1) - V_1 ?
Could someone please set me on the right path, I have been up late each night this week trying to work this out.