Thermodynamics

Zeroth Law: If two systems are each in thermal equilibrium with a third, they are in thermal equilibrium with each other (defines temperature).

First Law of Thermodynamics: Energy conservation including heat. $$\Delta Q = \Delta U + \Delta W$$ where $\Delta Q$ = heat added to system, $\Delta U$ = change in internal energy, $\Delta W$ = work done by the system. Sign convention: $Q > 0$ if heat added; $W > 0$ if system does work on surroundings.

For an ideal gas: $\Delta U = nC_V\Delta T$ (depends only on temperature).

Work done in expansion: $W = \int P\,dV$ (area under P-V curve).

Thermodynamic Processes (Quasi-static):

Adiabatic relation: $PV^\gamma = \text{const}$; also $TV^{\gamma-1} = \text{const}$ and $TP^{(1-\gamma)/\gamma} = \text{const}$, where $\gamma = C_P/C_V$.

Polytropic process: $PV^n = \text{const}$, generalizing all of the above.

Second Law of Thermodynamics:

• Kelvin-Planck statement: No process can have its sole effect that of converting heat from a reservoir into work.
• Clausius statement: Heat cannot flow spontaneously from a colder to a hotter body.

Heat Engine: Converts heat into work cyclically. Absorbs $Q_1$ from hot reservoir at $T_1$, rejects $Q_2$ to cold reservoir at $T_2$, does work $W = Q_1 - Q_2$.

Efficiency: $\eta = \dfrac{W}{Q_1} = 1 - \dfrac{Q_2}{Q_1}$.

Carnot Engine: Reversible engine operating between two reservoirs. Maximum efficiency: $$\eta_{\text{Carnot}} = 1 - \frac{T_2}{T_1}$$ ($T$ in Kelvin). Carnot cycle: isothermal expansion → adiabatic expansion → isothermal compression → adiabatic compression.

Refrigerator (heat pump in reverse): Extracts $Q_2$ from cold reservoir, supplies $W$, rejects $Q_1$ to hot reservoir.

Coefficient of Performance (COP): $$\text{COP}_{\text{refrigerator}} = \frac{Q_2}{W} = \frac{Q_2}{Q_1-Q_2} = \frac{T_2}{T_1-T_2}$$

Entropy $S$: state function whose change is $\Delta S = \int dQ_{\text{rev}}/T$. For reversible process: $\Delta S_{\text{universe}} = 0$. For irreversible: $\Delta S_{\text{universe}} > 0$.

Ideal Gas Equation: $PV = nRT = NkT$, where $n$ = moles, $N$ = molecules, $k = R/N_A = 1.38 \times 10^{-23}$ J/K (Boltzmann constant), $N_A = 6.022 \times 10^{23}$.

Pressure of Ideal Gas (kinetic interpretation): $$P = \frac{1}{3}\rho \overline{v^2} = \frac{1}{3}\frac{N}{V}m\overline{v^2} = \frac{2}{3}\frac{N}{V} \cdot \frac{1}{2}m\overline{v^2}$$

This gives mean translational KE per molecule: $$\overline{KE}_{\text{trans}} = \frac{3}{2}kT$$

Mean Speeds of gas molecules at temperature $T$:

Ratio: $v_p : \bar v : v_{\text{rms}} = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3} \approx 1 : 1.128 : 1.225$.

Maxwell-Boltzmann Distribution: Speeds of molecules follow: $$f(v) = 4\pi n\left(\frac{m}{2\pi kT}\right)^{3/2} v^2 e^{-mv^2/(2kT)}$$

Gas Laws (special cases of ideal gas equation):

• Boyle's law (T const): $PV = \text{const}$
• Charles's law (P const): $V/T = \text{const}$
• Gay-Lussac's law (V const): $P/T = \text{const}$
• Avogadro's law: equal volumes contain equal molecules at same $P, T$

Degrees of Freedom $f$: independent ways a molecule can store energy.

• Monatomic (e.g., He, Ar): $f = 3$ (translational only)
• Diatomic (e.g., $O_2, N_2$) at moderate T: $f = 5$ (3 trans + 2 rot)
• Diatomic at high T: $f = 7$ (+ 2 vibrational)
• Polyatomic (non-linear): $f = 6$

Equipartition theorem: Each quadratic degree of freedom contributes $\dfrac{1}{2}kT$ to mean energy per molecule.

Internal energy: $U = \dfrac{f}{2}nRT$.

Molar Specific Heats:

Mayer's Relation: $C_P - C_V = R$.

Mean Free Path: Average distance a molecule travels between collisions. $$\lambda = \frac{1}{\sqrt{2}\pi d^2 n}$$ where $d$ = molecular diameter, $n$ = molecules per unit volume. $\lambda \propto 1/P$ at constant $T$.

Mixture of gases: Effective $C_V$ for mixture of $n_1$ moles ($C_{V_1}$) and $n_2$ moles ($C_{V_2}$): $$C_V^{\text{mix}} = \frac{n_1 C_{V_1} + n_2 C_{V_2}}{n_1 + n_2}$$