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Work Integral in the pV Plane
The diagram shows the pressure and volume of an ideal gas during one cycle of an engine. As the gas proceeds from state 1 to state 2, it is heated at constant pressure. It is then cooled at constant volume, until it reaches state 3. The gas is then cooled at constant pressure to state 4. Finally, the gas is heated at constant volume until it returns to state 1.
Part A  
Find W_12, the work done by the gas as it expands from state 1 to state 2.
Part A.1  

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Part A.2  

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Express the work done in terms of p_0 and V_0.
ANSWER:
  W_12 =  9*p_0*V_0 
Part B  
Find W_23, the work done by the gas as it cools from state 2 to state 3.
Hint B.1 Volume of the gas

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Express your answer in terms of p_0 and V_0.
ANSWER:
  W_23 =  0 
Part C  
Find W_34, the work done by the gas as it is compressed from state 3 to state 4.
Express your answer in terms of p_0 and V_0.
ANSWER:
  W_34 =  -3*p_0*V_0 
Part D  
Find W_41, the work done by the gas as it is heated from state 4 to state 1.
Express your answer in terms of p_0 and V_0.
ANSWER:
  W_41 =  0 
Part E  
What is W_net, the total work done by the gas during one cycle?
Express your answer in terms of p_0 and V_0.
ANSWER:
  W_net =  6*p_0*V_0 
Notice that the net work done by the gas during this cycle is equal to the area of the rectangle that appears in the pV (pressure-volume) diagram. (The width of this rectangle is 3V_0, and its height is 2p_0.) In other words, the net work done by this gas is equal to the net area under its pV curve. For a cycle, this is equivalent to the (signed) area enclosed by the cycle.
Part F  
When the gas is in state 1, its temperature is T_1. Find the temperature T_3 of the gas when it is in state 3. (Remember, this is an ideal gas.)
Part F.1 Equation of state in terms of p_0 and V_0

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Express T_3 in terms of T_1.
ANSWER:
  T_3 =  (4/3)*T_1 
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