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Q.
A fixed mass of oxygen gas performs a cyclic process ABCA as shown in the figure. Efficiency
of the cycle will be
NTA AbhyasNTA Abhyas 2020Thermodynamics
Solution:
$A \rightarrow B$ isochoric as $P$ is directly proportional to $T$ $W _{ AB }=0$
$Q _{ AB }= n C _{ v }(3 T - T )$
$=\left(\frac{5}{2}\right) nR 2 T =5 nRT$
$B \rightarrow C$ isothermal and thus $Q = W$ as there won't
be any change in internal energy. $W _{ BC }= nR 3 T \ln \left(\frac{ P _{ B }}{ P _{ C }}\right)=3 nRTln 3$
$=3 n R T \ln 3=Q_{B C}$
$C \rightarrow$ Aisobaric process $W _{ CA }= P \left( V _{ A }- V _{ C }\right)= nR \left( T _{ A }-3 T \right)=-2 nRT$
Efficiency of a cyclic process is given by $\eta=\frac{\text { Work done in cyclic process }}{\text { Heat supplied to the system }}=\frac{ W }{ Q _{\text {in }}}$
Efficiency $\eta=\frac{3 nRT _{ A } \ln 3-2 nRT _{ A }}{5 nRT _{ A }+3 nRT _{ A } \ln 3}$
$=\frac{3 \ln 3-2}{5+3 \ln 3}$
