Thermodynamics

Zeroth Law of Thermodynamics: if two bodies A and B are each in thermal equilibrium with a third body C, then A and B are in thermal equilibrium with each other. This seemingly obvious principle is the logical foundation of temperature measurement: it allows us to use a thermometer (C) as an objective reference. Without it, we could not assert that two objects have the same temperature without direct contact.

Thermometric scales:

• Kelvin (K) — absolute scale: $T(K) = T(°C) + 273.15$. Absolute zero (0 K = -273.15°C) is the theoretical temperature where all thermal motion ceases. Every other scale is linear with respect to this one.
• Celsius (°C) — $0°C$ = ice melting at 1 atm, $100°C$ = water boiling at 1 atm. A difference of 1°C = 1 K.
• Fahrenheit (°F) — $T(°F) = \tfrac{9}{5}T(°C) + 32$, used in the USA. $32°F$ = ice melting, $212°F$ = water boiling.

Thermal expansion: as temperature increases, materials expand because particle kinetic energy grows and average interatomic distances increase. For solids: $\Delta L = \alpha L_0 \Delta T$ (linear), $\Delta A = 2\alpha A_0 \Delta T$ (surface), $\Delta V = \beta V_0 \Delta T$ (volumetric, with $\beta \approx 3\alpha$ for solids).

Real-world examples: mercury or alcohol thermometers exploit liquid expansion in a capillary. Electronic thermistors use resistance variation with $T$. Gas thermometers at constant volume measure $P \propto T$ (Gay-Lussac's law) and are the most precise.
Connection: the Zeroth Law defines the concept of temperature, the fundamental variable for the ideal gas equation of state (Theory §2) and for all subsequent developments in thermodynamics.

Empirical gas laws (17th-19th century):

• Boyle (1662): $PV = \text{constant}$ at constant $T$. If you compress a gas to half its volume, the pressure doubles. Explanation: molecules hit the walls with the same kinetic energy, but in a smaller volume collisions are more frequent.
• Charles (1787): $V/T = \text{constant}$ at constant $P$. Heating a gas at constant pressure causes the volume to increase proportionally to $T$. Example: a hot-air balloon — hot air expands, becomes less dense than cold air, and rises.
• Gay-Lussac (1802): $P/T = \text{constant}$ at constant $V$. In a rigid container (pressure cooker), increasing temperature makes pressure rise linearly.

Synthesis — Equation of State: combining the three laws gives $PV = nRT$, where $n$ is the number of moles and $R = 8.314$ J/(mol·K) is the universal gas constant. Temperature must always be in Kelvin because 0 K is the true thermal zero.

Microscopic form: $PV = Nk_B T$, with $N$ the number of molecules and $k_B = 1.381\times10^{-23}$ J/K Boltzmann's constant. The two constants are related by $R = N_A k_B$, where $N_A = 6.022\times10^{23}$ mol⁻¹ is Avogadro's number.

Model limits: the ideal gas equation holds for rarefied gases at room temperature. At high pressures or temperatures near liquefaction, more accurate models such as Van der Waals (Theory §4) are required.

Microscopic model: kinetic theory explains macroscopic properties (P, T) in terms of molecular motion. Assumptions: point-like molecules in random motion, elastic collisions with walls, no mutual interactions.

Pressure arises from the momentum change of molecules hitting the walls. Each collision transfers $2mv_x$ to the wall; integrating over all molecules yields $P = \frac{1}{3}\rho\langle v^2\rangle = \frac{1}{3}(Nm/V)\langle v^2\rangle$.

Temperature is proportional to mean translational kinetic energy: $\langle KE\rangle = \frac{3}{2}k_B T$. Higher T means faster molecules. At 300 K, N₂ molecules have $v_{rms} \approx 517$ m/s, faster than the speed of sound (343 m/s)!

Equipartition of energy: each quadratic degree of freedom contributes $\frac{1}{2}k_B T$ to internal energy.

• Monatomic gas (He, Ar): 3 translational degrees of freedom → $U = \frac{3}{2}nRT$, $C_v = \frac{3}{2}R$.
• Diatomic gas (N₂, O₂, H₂): 3 translational + 2 rotational (at room T) → $U = \frac{5}{2}nRT$, $C_v = \frac{5}{2}R$. Vibrational degrees are excited only at high temperatures.
• What $C_v$ and $C_p$ are: the molar specific heats, i.e. the heat to raise 1 mole by 1 K. $C_v$ is at constant volume (the gas does not expand, does no work: all the heat goes into internal energy, $Q=nC_v\Delta T=\Delta U$); $C_p$ is at constant pressure (the gas expands and does work $p\Delta V$, so more heat is needed).
• Mayer's relation: $C_p - C_v = R$. It follows from the first law at constant $P$: $nC_p\Delta T = \underbrace{nC_v\Delta T}_{\Delta U} + \underbrace{nR\Delta T}_{W=p\Delta V}$, hence $C_p = C_v + R$. $C_p$ is always larger than $C_v$ (by exactly $R$) because at constant pressure you also pay the expansion work.

Root-mean-square speed: $v_{rms} = \sqrt{3RT/M}$. It is useful for estimating gas diffusion rates and comparing molecular speeds at different temperatures.

Why do we need a more accurate model? The ideal gas equation fails when:

• Pressure is high — molecules are close enough to feel intermolecular forces.
• Temperature is low — kinetic energy is insufficient to overcome mutual attractions.
• Volume is small compared to the total molecular volume.

Van der Waals equation (1873): $(P + a/V^2)(V - b) = RT$ (per mole). Two corrections:

• $a/V^2$ — internal pressure term: attractive forces between molecules reduce the measured pressure compared to the ideal gas. $a$ is specific to each gas (large for polar molecules).
• $b$ — covolume: the actually available volume is $V-b$ because molecules have a finite volume. $b$ is about 4× the Avogadro molecular volume.

Critical state and liquefaction: for $T < T_c$ the Van der Waals isotherm shows an unstable region (∂P/∂V > 0) corresponding to liquid-gas phase separation.

• $T_c = 8a/(27Rb)$ — critical temperature, above which the gas cannot be liquefied by compression alone.
• $P_c = a/(27b^2)$ — critical pressure.
• $V_c = 3b$ — critical molar volume.

Example: CO₂ has $T_c = 31°C$ (304 K), so at room temperature it can be liquefied by compression. Helium has $T_c = 5.2$ K, requiring cryogenic cooling before liquefaction.
Connection: below $T_c$, Van der Waals isotherms describe the phase transition, connecting to the study of phase changes (Theory Transformations §4 — latent heat).