A heat engine with 0.300 mol of a monatomic ideal gas initially fills a 3000 cm3

Question

A heat engine with 0.300 mol of a monatomic ideal gas initially fills a 3000 cm3 cylinder at 800 K. The gas goes through the following closed cycle - Isothermal expansion to 4000 cm - Isochoric cooling to 100 K - Isothermal compression to 3000 cm - Isochoric heating to 800 K 0.300 mol of a monatomi How much work does this engine do per cycle? Express your answer with the appropriate units. alueUnits Submit My Answers Give Up Part B What is its thermal efficiency? Express your answer with the appropriate units alueUnits Submit My Answers Give Up

Explanation / Answer

Here,
no of mole, n = 0.300
initial volume of cycliner, v1 = 3000 cm^3 = 0.003 m^3
temperature, T = 800 k

Work done BY the gas is given by the integral
W = p*dV from initial to final volume

Part 1:
In the isochoric steps of the process no work is done, cause volume does no change, i.e
W2 = W3 = 0

From Ideal gas, under isothermal expansion
P1*v1 = p2*v2
p2 = p1*v1/v2

Work done in that portion will be :
W = p1*V1* ln(V2/V1)

or by using ideal gas law,
w = n*R*T1 * ln(v2/v1) ------------------------(1)

For the expansion process.
W1 = 0.3 * 8.3145 * 800 * ln(0.004/0.003)
w1 = 574.064 J

Work done by the gas in compression process is
W3 = 0.3 * 8.3145 * 100 * ln(0.003/0.003)
W3 = -249.435 J   

So the net work done by the engine per cycle is:
W = W1 + W2 + W3 + W4
W = 574.064 + 0 + 0 -249.435
W = 324.63 J

Part 2 :
Change of internal energy of an ideal gas is given by:
delU = n*Cv*delT

For a monatomic gas
Cv = (3/2)*R

also,
delU = Q - W

For the isothermal steps
delU = Q - W = 0
Q = W

For the isochoric steps
W = 0
Q = delU = (3/2)*n*R*delT -----------------(2)

Hence:
Q1 = W1 = 574.064 J

Using eqn 2,
Q2 = (3/2)*0.3*8.3145 * (100 - 800)
Q2 = -2619.068 J

Q3 = W3 = -249.435 J

Q4 = (3/2)*0.3*8.3145 * (800 - 100)
Q4 = 2619.068 J

So the total heat transferred to the engine is
Qin = Q1 + Q4
Qin = 574.064 + 2619.068
Qin = 3193.132 J

Therefore Thermal Efficiency will be equal to,
n = W / Qin
n = 324.63 / (3193.132)
n = 0.102
n = 10.2 %

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