The working substance of an engine is 1.00 mol of diatomic gas. The engine operates in a cycle consisting of three steps: (1) an adiabatic expansion from an initial volume of 9.00 L to a pressure of 1.00 atm and a volume of 30.6 L, (2) a compression at constant pressure to its original volume of 9.00 L, and (3) heating at constant volume to its original pressure. Find the efficiency of this cycle.
The thermal efficiency of a cycle process is ratio the net work
done by the engine to the heat input.
? = W/Q_in
To solve this problem it is sufficient to consider the only the
heat transfer in each step of the process.
For a cycle process the overall change in internal energy is
zero:
?U = Q - W = 0
So net heat flow to and work done by the engine have same
magnitude:
W = Q = Q_in - Q_out
Hence,
? = (Q_in - Q_out)/Q_in = 1 - (Q_out/Q_in)
The process consists of three steps:
- Step (1)
This step is adiabatic, so there is no heat
transfer:
Q? = 0
For the following calculations you need to find the initial
pressure of this process.
For a reversible, adiabatic process
P?V^? = constant
=>
P_initial?V_initial^? = P_final?V_final^?
=>
P_initial = P_final ? (V_final/V_initial)^?
? is the heat capacity ratio. For a diatomic gas.
? = Cp / Cv = 7/5 = 1.4
Hence,
P_initial = 1 atm ? (39.6/9)^1.4 = 7.96 atm = 806,400
Pa
- Step (2)
The heat transferred to a system in a constant pressure process
equals its change in enthalpy. Change in enthalpy for an ideal gas
is given by:
?H = n?Cp??T
Using ideal gas law you can express change in temperature in terms
of change in volume:
T = P?V/(n?R)
=>
?T = (P/(n?R))??V
=>
?H = (Cp/R)?P??V
Molar heat capacity for a diatomic ideal gas is (7/2)?R.
Therefore
Q? = ?H? = (7/2)?P??V
= (7/2) ? 1?101325 Pa ? (9
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