Key examples — thermodynamics

Where the chapter’s machinery shows up across the bookshelf.

Example 1: Laplace’s adiabatic correction to the speed of sound

Newton’s 1687 calculation of the speed of sound assumed isothermal compression: $c_\text{N} = \sqrt{p/\rho} \approx 280\,\text{m/s}$ — about 15% below the measured 343 m/s. The discrepancy lasted 130 years. Laplace’s 1816 fix was to recognise that acoustic compressions are too fast for heat to conduct away, so they are adiabatic — and the slope $(\partial p/\partial \rho)_s$ is larger than $(\partial p/\partial \rho)_T$ by the factor $\gamma = 1.4$. The corrected speed $c = \sqrt{\gamma p/\rho} \approx 343\,\text{m/s}$ agrees with experiment. See Sound Ch 4.4.

Example 2: polytropic gas inside an oscillating bubble

The gas inside a Rayleigh–Plesset bubble oscillates between compression and rarefaction at frequencies up to MHz. Whether the compression is isothermal ($\kappa = 1$) or adiabatic ($\kappa = \gamma$) depends on the dimensionless ratio $\omega R^2/\alpha_g$, where $\alpha_g$ is the gas thermal diffusivity. For a 1 μm bubble in air at $\omega = 10^7\,\text{rad/s}$, this ratio is order unity, and $\kappa$ sits at an intermediate frequency-dependent value $\kappa(\omega)$ that the Cavitation book’s bubble-contents chapter develops. The polytropic envelope from the chapter is what makes this analysis tractable.

Example 3: enthalpy across a shock front

The Rankine–Hugoniot energy condition across a shock — derived from open-system conservation of mass, momentum, and energy with the shock as a control volume — is

where $h$ is the specific enthalpy and $u$ the flow velocity. The two states $1$ and $2$ are upstream and downstream of the shock. Enthalpy enters precisely because the shock is an open control volume — the flow-work term that closed-system $U$ would miss is exactly the $pV$ Legendre-transform that turns $U$ into $h$. See Sound Ch 10.5.

Example 4: γ from molecular structure of air

Air at room temperature has $\gamma = 7/5 = 1.4$, not $5/3$ (monatomic) or $9/7$ (full diatomic with vibrations). This is the equipartition counting from the chapter: N₂ and O₂ molecules have three translational + two rotational degrees of freedom active, but vibrational modes are frozen out by quantum statistics (the vibrational quantum exceeds $k_B T$ at room temperature). $\gamma = (f + 2)/f = 7/5$ for $f = 5$. The Sound book’s molecular-relaxation chapter develops the consequences for sound absorption when vibrational modes start to thaw at high frequencies.

Example 5: heat capacity of water

Liquid water has $c_p \approx 4.18\,\text{J/(g·K)}$ — roughly four times higher than most other liquids. This unusually large heat capacity comes from the vibrational degrees of freedom of hydrogen bonds (which are at energies comparable to $k_B T$ at room temperature, unlike covalent bonds), plus librations of the molecular rotation, plus the usual translational and rotational classical degrees of freedom. The combined active-DOF count is much higher than for a simpler liquid. This anomalously high $c_p$ is what makes water such an effective thermal buffer in climate and physiology. The intermolecular-forces chapter develops the hydrogen-bond picture.

Cross-book backlinks

• Sound Ch 4.4 — equation of state: isothermal vs adiabatic and the Newton-Laplace correction.
• Sound Ch 4.9 — speed of sound: $c^2 = (\partial p/\partial \rho)_s$ across all four derivations.
• Sound Ch 10.5 — shock formation: Rankine-Hugoniot enthalpy condition.
• Sound Ch 10.2 — molecular relaxation: when vibrational modes start to thaw.
• Cavitation Ch 3.2 — bubble contents: frequency-dependent polytropic exponent.