### P-V Diagram for the Carnot engine

1st law of thermodynamics (energy conservation): dE = dQ + dWon system
dE = dQ - dWby system

Step 1: Isothermal expansion at Th:
W1 = P dV = NkTh ln(VB/VA) > 0
E1 = 0
Qh = NkTh ln(VB/VA)

Step 2: Adiabatic expansion to Tc:
W2 = P dV = a V dV = a/(1-γ) [VC1-γ - VB1-γ] > 0
= Nk/(1-γ) [TC - TB] (use PVγ = a = NkT Vγ-1)
Q = 0
E2 = - W2 = Nk/(1-γ) [TB - TC]

Step 3: Isothermal compression Tc:
W3 = P dV = NkTc ln(VD/VC) < 0
E3 = 0
Qc = | NkTc ln(VD/VC) |

Step 4: Adiabatic compression to Th:
W4 = P dV = V dV = b/(1-γ) [VA1-γ - VD1-γ] < 0
= Nk/(1-γ) [TA - TD] = Nk/(1-γ) [TB - TC] = - W2
Q = 0
E4 = - W4 = Nk/(1-γ) [TB - TC] = - E2

After completing one cycle:

Wtot = W1 + W3 = NkTh ln(VB/VA) + NkTc ln(VD/VC) = Qh - Qc

The efficiency is:   η = 1 -
 Qc Qh
= 1 -
 NkTc ln(VC/VD) NkTh ln(VB/VA)

Now use rule for adiabats:   TVγ-1 = constant
Th VBγ-1= Tc VCγ-1
Th VAγ-1= Tc VDγ-1

Dividing gives:   VB/VA = VC/VD

Final result for the efficiency:   η = 1 -
 Tc Th

Important notes:

• The Carnot cycle applies for an impractical quasi-static process
• For a car, Tc ≈ 300K, while Th < 600K, the temperature where engine block softens.
• A Carnot car engine has an efficiency η < 0.5.
• Real engines have efficiencies η ≈ 0.2.