Carnot engine is

Explanation

When the working substance is an ideal gas and it is subjected to a cyclic Process as follows in order

(i)Isothermal expansion (AB)
(ii)Adiabatic Expansion (BC)
(iii)Isothermal Compression (CD)
(iv)Adiabatic compression (DA)
Process AB
`Q_1 rArr nRT_1 ln(V_B)/(V_A)`
Process CD
`Q_2 = nRT_2 lnV_D/V_C " , "Q_2 = -nRT_2 ln""V_C/V_D`
Efficiency `eta =("Netv ork done")/("H ea tabsorbed")=(Q_1 + Q_2)/Q_1= 1 + Q_2/Q_1`
` rArr" "eta =1 - (nRT_2 ln V_C //V_D)/(nRT_1 ln V_B //V_A)`
`eta = 1 0- (T_2)/(T_1) = (ln V_C//V_D)/(ln V_B //V_A)" ....(1)"`
Since points B and C lie on same adiabatic curve
`T_2 V_B^(gamma -1) = T_2 V_(BC)^(gamma-1)`
`T_1/T_2 = ((V_D)/(V_A))^(gamma-1)" ...(iii)"`
By Equation (ii) and (iii) `V_C/V_B = V_D/V_A`or`V_C /V_D = V_B /V_A`
So By Equation (i) `eta = 1 - T_2/ T_1`