# Three step gas cycle

## Homework Statement

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.

Part 1-
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.

Part 2-
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.

Part 3-
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.

R = 8.31

## The Attempt at a Solution

Part 1-
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

Part 2-
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.

Part 3-
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 ?

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Consider the following three-step transformation of the gas:

1. The gas is then expanded at constant temperature until the pressure returns to p_1.
2. The gas is then cooled at constant pressure until the volume has returned to V_1.
I think the first step isn't mentioned.

I think the first step isn't mentioned.
thanks for point that out!! i fixed it!!! now its a three step cycle.

But i still dont know how to figure this problem out!