Question

The P-V diagram of an engine is shown in the figure below. The temperatures at points 1, 2, 3 and 4 are T₁, T₂, T₃ and T₄, respectively. 1→2 and 3→4 are adiabatic processes, and 2→3 and 4→1 are isochoric processes

Identify the correct statement(s). 
[γ is the ratio of specific heats C_p (at constant P) and C_v (at constant V)]

• A. 𝑇₁𝑇₃ = 𝑇₂𝑇₄
• B. The efficiency of the engine is \((\frac{p_1}{p_2})^{\frac{y-1}{y}}\)
• C. The change in entropy for the entire cycle is zero
• D. 𝑇₁𝑇₂ = 𝑇₃𝑇₄

Hint:

Apply the adiabatic and isochoric process relations.

Answer 1

The Correct Option is A, B, C

1. Adiabatic Process Relationships: 
- For \(1 \to 2\) (adiabatic process): \(T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1}.\)
- For \(3 \to 4\) (adiabatic process): \(T_3 V_3^{\gamma - 1} = T_4 V_4^{\gamma - 1}.\) 
- For adiabatic processes, pressure and temperature are related as: \(T_1 T_3 = T_2 T_4.\)
Hence, statement (A) is correct. 

2. Efficiency of the Engine: 
- The efficiency of a heat engine operating on a cycle is given by:
 \(\eta = 1 - \frac{Q_{{out}}}{Q_{{in}}}\), where \(Q_{{out}}\) and \(Q_{{in}}\) are the heat rejected and absorbed, respectively. 
- For this cycle, the efficiency can also be expressed using the pressure ratio for adiabatic processes as: \(\eta = 1 - \left(\frac{P_1}{P_2}\right)^{\frac{\gamma - 1}{\gamma}}.\)
Hence, statement (B) is correct. 

3. Entropy Change: 
- For the entire cyclic process, the system returns to its initial state. As entropy is a state function, the total entropy change over a complete cycle is zero: \(\Delta S_{{cycle}} = 0\).
 Hence, statement (C) is correct. 

4. Temperature Relationship: 
- The relationship \(T_1 T_2 = T_3 T_4\) is incorrect as it does not align with the conditions for adiabatic and isochoric processes. 
Hence, statement (D) is incorrect.