Laws of Thermodynamics & Carnot Engine Efficiency
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1. The Four Laws in Brief
- Zeroth Law (Thermal Equilibrium): If system $A$ is in thermal equilibrium with $B$, and $B$ is with $C$, then $A$ is in thermal equilibrium with $C$. This establishes the definition of temperature.
- First Law (Conservation of Energy):
$$\Delta U = Q - W$$
Where $\Delta U$ is the change in internal energy, $Q$ is heat added to the system, and $W$ is work done by the system ($W = \int P \, dV$). - Second Law (Entropy & Spontaneity): The entropy of an isolated system never decreases over time:
$$\Delta S_{\text{universe}} \ge 0$$
No cyclic heat engine can convert 100% of input heat into mechanical work (Kelvin-Planck statement). - Third Law (Absolute Zero): As temperature approaches absolute zero ($T \to 0\text{ K}$), the entropy of a pure crystalline substance approaches a constant minimum value (zero).
2. Work Done in Thermodynamic Processes (Ideal Gas)
| Process Type | Defining Condition | Work Done ($W$) | Internal Energy Change ($\Delta U$) |
|---|---|---|---|
| Isochoric | Volume constant ($dV = 0$) | $W = 0$ | $\Delta U = Q = n C_v \Delta T$ |
| Isobaric | Pressure constant ($dP = 0$) | $W = P(V_2 - V_1) = n R \Delta T$ | $\Delta U = n C_v \Delta T$ |
| Isothermal | Temperature constant ($dT = 0$) | $W = nRT \ln\left(\frac{V_2}{V_1}\right)$ | $\Delta U = 0 \implies Q = W$ |
| Adiabatic | No heat exchange ($Q = 0$) | $W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1}$ | $\Delta U = -W$ |
Where $\gamma = \frac{C_p}{C_v}$ is the heat capacity ratio.
3. The Carnot Cycle & Efficiency
The Carnot cycle is the theoretical upper limit on the efficiency of any heat engine operating between two temperatures $T_H$ (hot reservoir) and $T_C$ (cold reservoir):
$$\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H} = \frac{T_H - T_C}{T_H}$$
Critical Rule: All temperatures in thermodynamic efficiency calculations must be converted to Kelvin ($T_{\text{Kelvin}} = T_{\text{Celsius}} + 273.15$).
The 4 Stages of the Carnot Cycle:
- Reversible Isothermal Expansion at $T_H$ (Heat absorbed $Q_H$)
- Reversible Adiabatic Expansion ($T_H \to T_C$)
- Reversible Isothermal Compression at $T_C$ (Heat rejected $Q_C$)
- Reversible Adiabatic Compression ($T_C \to T_H$)