In a heat engine, 2.00 mol of a diatomic gas are taken through the cycle ABCA, a
ID: 1778335 • Letter: I
Question
In a heat engine, 2.00 mol of a diatomic gas are taken through the cycle ABCA, as shown in the figure. (The PV diagram is not drawn to scale.) At A the pressure and temperature are 3.50 atm and 600 K. The volume at B is twice the volume at A. The segment BC is an adiabatic expansion, and the segment CA is an isothermal compression. (a) What is the volume of the gas at A? (b) What are the volume and temperature of the gas at B? (c) What is the temperature of the gas at C? (d) What is the volume of the gas at C? (e) How much work is done by the gas in each of the three segments of the cycle? k) (A to B) kJ (B to C) (f) How much heat is absorbed or released by the gas in each segment of this cycle? k) (A to B) kJ (B to C) kJ (C to A)Explanation / Answer
(A) Volume of the gas at A is
We know the ideal gas equation is P*V1 = n*R*T1
V1 = n*R*T1 / P
= (2 moles * 0.0821 L atm/molK * 600K)/3.50 atm
= 28.148 L
(B) Volume of gas at B is
The volume at B is twice the volume at A
= 2* 28.148 L = 56.296 L
Temperature at B is
P*V2 = n*R*T2
T2 = P*V2 / n*R
= (3.50 atm * 56.296 L) / 2 moles * 0.0821Latm /molK
= 1199.975 K = 1200 K
(C) The segment BC is an adiabatic expansion,
T2 = 1200 K, V2 = 56.296 L, As the gas is expanding adiabatically,
the temperature becomes again the original temperature.
Hence the temperature at C is 600 K
(D) the volume of the gas at C is
T2*V2^gamma-1 = T1 *V3^gamma-1
T2 / T1 = (V3/V2)^gamma-1
1200/600 = (V3/56.296 L)^1.4-1 (since gamma = 1.4 for a diatomic gas)
2 = (V3/56.296 L)^0.4
V3 = 318.458 L
(E) Work done from A to B is
- P*(V2-V1)
= -3.50 atm(56.296 - 28.148 )lit
= - 98.518 lit atm
= 9982.828 J
work done by the gas from B to C is
w = [nR / GAMMA-1 ](T2 -T1)
= ( 2 moles * 8.314 J ) / 0.4 (1200 - 600)K
= 24942 J
work done by the gas from C to A
W = P *(V3-V1)
= 3.50 atm ( 318.458 - 28.148)L
= 102959.893 J
(F) heat is absorbed by the gas from A to B is
= 9982.828 J ( this is equal to the work done from A to B)
During adiabatic expansion there is no absorption or release of heat from B to C.
From C to A that is isothermal compression of the gas.
Q = W = work done by the gas from C to A = 102959.893 J
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