A cycle followed by an engine (made of one mole of perfect gas in… - Vidyadip

Question

A cycle followed by an engine (made of one mole of perfect gas in a cylinder with a piston) is shown in Fig.

A to B : volume constant B to C : adiabatic C to D : volume constant D to A : adiabatic $V_C= V_D= 2V_A= 2V_B$

In which part of the cycle heat is supplied to the engine from outside?
In which part of the cycle heat is being given to the surrounding by the engine?
What is the work done by the engine in one cycle? Write your answer in term of $P_A, P_B, V_A$.
What is the efficiency of the engine?[$\gamma=\frac{5}{3}$ for the gas], $\big(\text{C}_\text{v}=\frac{3}{2}\text{R for one mole}\big)$

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Answer

(a) A to B
(b) C to D
(c) $\text{W}_\text{AB}=\int\limits^\text{B}_\text{A}\text{pdV}=0;\text{W}_\text{CD}=0.$
Simillarly. $\text{W}_\text{BC}=\Big[\int\limits^\text{C}_\text{B}\text{pdV}=\text{k}\int\limits^\text{C}_\text{B}\frac{\text{dV}}{\text{V}^\text{r}}=\text{k}\frac{\text{V}^\text{-r+1}}{-\text{R}+1}\Big]^{\text{V}_{\text{C}}}_{\text{V}_\text{B}}$
$= \frac{1}{1-\gamma}(\text{P}_\text{c}\text{V}_\text{c}-\text{P}_\text{B}\text{V}_\text{B})$
Simillarly, $\text{W}_\text{DA}=\frac{1}{1-\gamma}(\text{P}_\text{A}\text{V}_\text{A}-\text{P}_\text{D}\text{V}_\text{D})$
Now $\text{P}_\text{C}=\text{P}_\text{B}\Big(\frac{\text{V}_\text{B}}{\text{V}_\text{C}}\Big)^\gamma=2^{-\gamma}\text{P}_\text{B}$
Simillarly, $\text{P}_\text{D}=\text{P}_\text{A}2^{-\gamma}$
Total work done $=\text{W}_\text{BC}+\text{W}_\text{DA}$
$=\frac{1}{1-\gamma}\big[\text{P}_\text{B}\text{V}_\text{B}\big(2^{-\gamma+1}-1\big)-\text{P}_\text{A}\text{V}_\text{A}\big(2^{-\gamma+1}-1\big)\big]$
$=\frac{1}{1-\gamma}\big(2^{1-\gamma}-1\big)\big(\text{P}_\text{B}-\text{P}_\text{A}\big)\text{V}_\text{A}$
$=\frac{3}{2}\big(1-\Big(\frac{1}{2}\Big)^\frac{2}{3}\big)\big(\text{P}_\text{B}-\text{P}_\text{A}\big)\text{V}_\text{A}$

Heat supplied during process A, B

$\text{d}\text{Q}_\text{AB}=\text{d}\text{U}_\text{AB}$
$\text{Q}_\text{AB}=\frac{3}{2}\text{n}\text{R}\big(\text{T}_\text{B}-\text{T}_\text{A}\big)\text{V}_\text{A}$
$\text{Efficiency}=\frac{\text{Net Work done }}{\text{Heat Supplied}}=\Big[1-\Big(\frac{1}{2}\Big)^\frac{2}{3}\Big]$

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