Thermodynamics: Laws, Processes & Engines

Thermodynamic System and State Variables

A thermodynamic system is a definite quantity of matter bounded by a surface. State variables (P, V, T, U) describe the system's condition. For an ideal gas: $PV = nRT$.

Internal energy ($U$): total kinetic energy of molecules. For an ideal gas, $U$ depends only on temperature: $U = nC_vT = \frac{f}{2}nRT$ where $f$ is degrees of freedom (3 for monatomic, 5 for diatomic at moderate T, 6 for polyatomic).

Zeroth Law of Thermodynamics

If systems A and B are each in thermal equilibrium with system C, then A and B are in thermal equilibrium with each other. This establishes the concept of temperature.

First Law of Thermodynamics

Energy conservation for thermodynamic systems: $\Delta Q = \Delta U + \Delta W$

• $\Delta Q$: heat added to the system (positive if absorbed)
• $\Delta U$: change in internal energy (positive if increased)
• $\Delta W$: work done by the system (positive if gas expands)

Work Done by a Gas

$W = \int_{V_1}^{V_2} P \, dV$

Work equals the area under the P-V curve. Work is path-dependent (not a state function).

Thermodynamic Processes

1. Isothermal Process ($T$ = constant, $\Delta U = 0$): $PV = \text{constant} \quad \Rightarrow \quad W = nRT \ln\frac{V_2}{V_1} = nRT \ln\frac{P_1}{P_2}$ $\Delta Q = \Delta W = nRT \ln\frac{V_2}{V_1}$

2. Adiabatic Process ($\Delta Q = 0$): $PV^\gamma = \text{constant}, \quad TV^{\gamma-1} = \text{constant}, \quad TP^{-\gamma/(\gamma-1)} = \text{constant}$ $W = \frac{P_1V_1 - P_2V_2}{\gamma - 1} = \frac{nR(T_1 - T_2)}{\gamma - 1}$ $\Delta U = -W = nC_v(T_2 - T_1)$

3. Isobaric Process ($P$ = constant): $W = P\Delta V = P(V_2 - V_1) = nR\Delta T$ $\Delta Q = nC_p\Delta T, \quad \Delta U = nC_v\Delta T$

4. Isochoric Process ($V$ = constant, $W = 0$): $\Delta Q = \Delta U = nC_v\Delta T$

5. Polytropic Process ($PV^n = \text{constant}$): $W = \frac{nR(T_1 - T_2)}{n - 1}, \quad C_{\text{poly}} = C_v\frac{\gamma - n}{1 - n}$

Specific Heats of Ideal Gas

$C_v = \frac{f}{2}R, \quad C_p = C_v + R = \frac{f+2}{2}R$ $\gamma = \frac{C_p}{C_v} = \frac{f+2}{f}$

Second Law of Thermodynamics

Kelvin-Planck statement: No engine can convert heat entirely into work in a cyclic process. Some heat must be rejected to a cold reservoir.

Clausius statement: Heat cannot spontaneously flow from a colder body to a hotter body without external work.

Heat Engines

A heat engine absorbs heat $Q_H$ from a hot reservoir, converts part to work $W$, and rejects $Q_C$ to a cold reservoir.

Efficiency: $\eta = \frac{W}{Q_H} = 1 - \frac{Q_C}{Q_H}$

Carnot Engine (Maximum Efficiency)

Operating between temperatures $T_H$ and $T_C$ (Kelvin): $\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}$

Carnot cycle: isothermal expansion → adiabatic expansion → isothermal compression → adiabatic compression.

Refrigerator (Reverse Heat Engine)

Coefficient of Performance: $\text{COP} = \frac{Q_C}{W} = \frac{Q_C}{Q_H - Q_C} = \frac{T_C}{T_H - T_C}$

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