Name: Second Law of Thermodynamics
Uploaded: 2019-07-19
Duration: 8 min 37 s
Description: The statement of the second law of thermodynamics is defined in detail. The video also explains the quantity called entropy and helps to understand the law i...

Question

In a cycle $A B C A$ consisting of isothermal expansion $A B$ , isobaric compression $B C$ and adiabatic compression $C A$ , find the efficiency of cycle (Given: $T_{A} = T_{B} = 400 K, γ = 1.5$ )

Accepted Answer

For process $A \to C, P_{0} V_{0}^{γ} = \frac{P_{0}}{2} V_{C}^{γ} \Rightarrow V_{C} = 2^{\frac{1}{γ}} V_{0}$

Also, $T_{0} V_{0}^{γ - 1} = T_{C} {(2^{\frac{1}{γ}} V_{0})}^{γ - 1} \Rightarrow T_{C} = T_{0} 2^{\frac{1 - γ}{γ}}$

Total work done $= W = W_{A B} + W_{B C} + W_{C A}$

The process $A \to B$ is an isothermal process, so $W_{A B} = n R T_{0} \ln \frac{2 V_{0}}{V_{0}}$ = $n R T_{0} \ln 2$

The process $B \to C$ is an isobaric process, so

$W_{B C} = \frac{P_{0}}{2} (2^{\frac{1}{γ}} V_{0} - 2 V_{0})$ = $\frac{P_{0} V_{0}}{2} (2^{\frac{1}{γ}} - 2) = \frac{n R T_{0}}{2} (2^{\frac{1}{γ}} - 2)$

The process $C \to A$ is an adiabatic process, so

$W_{C A} = - \frac{n R}{γ - 1} (T_{0} - T_{C})$ = $\frac{- n R}{γ - 1} (T_{0} - T_{0} 2^{\frac{1 - γ}{γ}})$ = $\frac{- n R T_{0}}{γ - 1} (1 - 2^{\frac{1 - γ}{γ}})$

Hence, total work done in cyclic process,

$W = n R T_{0} \ln 2 + \frac{n R T_{0}}{2} (2^{\frac{1}{γ}} - 2) - \frac{n R T_{0}}{γ - 1} (1 - 2^{- \frac{(γ - 1)}{γ}})$

The heat is absorbed only in process $A \to B$ .

Hence, $Q_{A B} = W_{A B} [∵ Δ U_{A B} = 0] = n R T_{0} \ln 2$

So $η = \frac{W}{Q_{A B}} = \frac{n R T_{0} [\ln 2 + \frac{1}{2} (2^{\frac{1}{γ}} - 2) - \frac{1}{γ - 1} (1 - 2^{- \frac{γ - 1}{γ}})]}{n R T_{0} \ln 2}$

= 1 $+ \frac{2^{\frac{1}{γ} - 1} - 1 - 2 (1 - 2^{\frac{- 1}{3}})}{\ln 2}$

=1+ $\frac{2^{\frac{- 1}{3}} - 1 - 2 + 2 \times 2^{\frac{- 1}{3}}}{\ln 2}$

= 1 - $\frac{3 - 3 \times 2^{\frac{- 1}{3}}}{\ln 2}$

= $1 - \frac{3 (1 - \frac{1}{2^{\frac{1}{3}}})}{\ln 2}$

In a cycle $A B C A$ consisting of isothermal expansion $A B$ , isobaric compression $B C$ and adiabatic compression $C A$ , find the efficiency of cycle (Given: $T_{A} = T_{B} = 400 K, γ = 1.5$ )

Important Questions on Thermodynamics

Learn and Prepare for any exam you want !

In a cycle ABCA consisting of isothermal expansion AB, isobaric compression BC and adiabatic compression CA, find the efficiency of cycle (Given: TA=TB=400 K, γ=1.5)

Important Questions on Thermodynamics

Learn and Prepare for any exam you want !

In a cycle $A B C A$ consisting of isothermal expansion $A B$ , isobaric compression $B C$ and adiabatic compression $C A$ , find the efficiency of cycle (Given: $T_{A} = T_{B} = 400 K, γ = 1.5$ )