Ideal Gas Cycle Analysis

00:00 12 marks

2. A sample of an ideal monatomic gas undergoes a cyclic process ABCA as shown in the pressure-volume (*P-V*) diagram. Process AB is isothermal, process BC is isobaric, and process CA is isovolumetric. The temperature at A is *T*_A = 300 K.

(a) Calculate the temperature of the gas at point B.
[1]

(b) Calculate the work done by the gas during the isothermal expansion AB. State your answer in terms of *P*₀ and *V*₀.
[2]

(d) Determine the thermal energy transferred to the gas during process CA.
[2]

(e) Calculate the efficiency of the heat engine represented by this cycle.
[3]

(f) State, with a reason, whether the entropy of the gas increases, decreases or stays the same during the isothermal expansion AB.
[2]

Part a

1 minutes 1 marks

Question

Calculate the temperature of the gas at point B.

Hint Hint

Hint

Identify the type of process connecting A and B.

Answer Answer
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Explanation Explanation

Answer Answer

Complete Answer

\\(T_B = 300\\) K

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Part b

3 minutes 2 marks

Question

Calculate the work done by the gas during the isothermal expansion AB. State your answer in terms of P₀ and V₀.

Hint Hint

Hint

Use W = nRT ln(V_f/V_i) and relate nRT to P_A V_A.

Answer Answer
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Explanation Explanation

Answer Answer

Complete Answer

\\(W_{AB} = 3 P_0 V_0 \\ln(3)\\)

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Part c

3 minutes 2 marks

Question

Calculate the change in internal energy of the gas during process BC.

Hint Hint

Hint

Use ΔU = (3/2)nRΔT and find T_C via ideal gas law.

Answer Answer
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Explanation Explanation

Answer Answer

Complete Answer

\\(ΔU_{BC} = -3 P_0 V_0\\)

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Part d

3 minutes 2 marks

Question

Determine the thermal energy transferred to the gas during process CA.

Hint Hint

Hint

Use Q=ΔU+W. W=0 for isovolumetric.

Answer Answer
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Explanation Explanation

Answer Answer

Complete Answer

\\(Q_{CA} = 3 P_0 V_0\\)

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Part e

5 minutes 3 marks

Question

Calculate the efficiency of the heat engine represented by this cycle.

Hint Hint

Hint

η=W_net/Q_in, sum works and positive heats.

Answer Answer
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Explanation Explanation

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Complete Answer

\\(η=\\frac{3\\ln(3)-2}{3(1+\\ln(3))}≈0.20\\)

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Part f

3 minutes 2 marks

Question

State, with a reason, whether the entropy of the gas increases, decreases or stays the same during the isothermal expansion AB.

Hint Hint

Hint

ΔS=Q/T for reversible isothermal, check sign of Q.

Answer Answer
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Explanation Explanation

Answer Answer

Complete Answer

Entropy increases because heat is added during the isothermal expansion.

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