An Automobile engine can be modelled as an idealized four-stroke Otto cycle, although it actually consists of 6 steps: 

Step 0: A fuel-air mixture is drawn into a cylinder at constant pressure (admission stroke). 

Step 1: Adiabatic and reversible (isentropic) compression of the air as the piston moves from the bottom (V₂) of the cylinder to the top (V₁)(compression stroke). 

Step 2: Isochoric (constant volume) heat transfer to the working gas from an external source while the piston is at V₁. (This represents an instantaneous combustion of the fuel-air mixture.) 

Step 3: Adiabatic and reversible (isentropic) expansion of the cylinder contents (power stroke).

Step 4: isochoric heat transfer in which heat is released by the gas to the atmosphere through the cylinder walls while the piston is at V₂. 

Step 5: isobaric release of exhaust gases to the atmosphere (exhaust stroke). Here steps 0 and 5 can be ignored as no appreciable work is obtained or consumed. Steps 1 to 4 produce the net shaft work to power the automobile. The heat source for the engine is the burning of the fuel; for simplicity this heat will be considered as coming from an external heat source.

a. Draw this cycle on a pressure-entropy diagram. 

b. Assuming that the working fluid behaves like an ideal gas of constant heat capacity, calculate the thermal efficiency of the cycle (net work divided by heat flow in) as a function of the specific heat ratio k = C_P/C_V and the compression ratio _γ = V₁/V₂, where V₁ and V₂ are the minimum an maximum volumes occupied by the gas, respectively.