Show that efficiency of Caront's ideal heat engine is `eta=(1-(T_(2))/(T_(1)))`.

Explanation

Carnot.s ideal heat engine completes a sequence of isothermal and adiabatic expansion and isothermal and adiabatic compression. The working substance returns to intial state. Heat absorbed by the system from the sourece at `T_(1)K` is
`W_(1)=Q_(1)=muRT_(1)log_(e)((V_(2))/(V_(1)))." ".......(1)`
Total work done `W=W_(1)+W_(2)+W_(3)+W_(4)`
Where `W_(1)&W_(3)` refres to isothermal process and `W_(2)&W_(4)` refers to adiabatic process.
`W_(2)=(muRT)/((gamma-1))(T_(1)-T_(2))" "` [adiabatic expansion]
and `W_(4)=(-muRT)/((gamma-1))(T_(1)-T_(2))" "` [adiabatic compression]
So that `W_(2)+W_(4)=0`
For an adiabatic expansion,
`((T_(1))/(T_(2)))=((V_(3))/(V_(2)))^(gamma-1)" ".......(2)`
and adiabatic compression,
`((T_(1))/(T_(2)))=((V_(4))/(V_(1)))^(gamma-1)" ".......(3)`
From (2) & (3) `(V_(3))/(V_(2))=(V_(4))/(V_(1))or(V_(3))/(V_(4))=(V_(2))/(V_(1))" "............(4)`
For isothermal expasion and compression, total work done `=W_(1)+W_(3)=muRT_(1)log_(e)((V_(2))/(V_(1)))-muRT_(2)log((V_(2))/(V_(4)))`.
Applying (2) & (3) we get `W=muR(T_(1)-T_(2))log_(e)((V_(2))/(V_(1)))" "............(5)`
Hence efficiency `eta=(W)/(Q_(1))=((5))/((1))=(T_(1)-T_(2))/(T_(1))`
i.e., `eta=1-(T_(2))/(T_(1))" also "eta=1-(Q_(2))/(Q_(1))" as "T_(2)propQ_(2)&T_(1)propQ_(1)`.