Key examples — kinetic theory

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

Example 1: the kinetic-theory route to the speed of sound

The Sound book derives $c^2 = \gamma R T / M$ four different ways. The kinetic-theory route (Sound Ch 4.7) starts from $p = \tfrac13 n m\langle v^2\rangle$ and $\langle\varepsilon\rangle = \tfrac32 k_B T$ to give

with $\gamma = (f + 2)/f$ for $f$ active degrees of freedom — the equipartition counting from the chapter’s table. For diatomic air with $f = 5$ this gives $c \approx 0.68\, v_\text{rms}$. The collective wave is slower than the random thermal motion that drives it; this is the kinetic statement of why “sound speed” is an emergent property of the equilibrium gas.

Example 2: Brownian-motion floor on hearing thresholds

A pressure receiver of area $A$ exposed to a gas in thermal equilibrium sees fluctuating force $\delta F$ of variance $\langle\delta F^2\rangle = 4\rho c k_B T A / \pi$ over a bandwidth (Sivian–White, 1933). At audible-band bandwidths, the equivalent acoustic pressure is on the order of a few μPa — roughly the actual auditory threshold near 1–4 kHz. Hearing is, to within a factor of order unity, as sensitive as the molecular noise of the air will allow. The Brownian-motion picture from the chapter is what sets this floor. See Sound Ch 1.3.

Example 3: nucleation rate from the Boltzmann factor

The classical-nucleation-theory rate of bubble birth in a metastable liquid is

where $\Delta G^* = 16\pi\sigma^3/3(\Delta p)^2$ is the height of the free-energy barrier built in the free-energy chapter. The Boltzmann factor is the same machinery from the kinetic-theory chapter — the probability that a thermal fluctuation has enough energy to clear the barrier. For pure water at room temperature, $\Delta G^*/k_BT$ is $\sim 1000$, so $J$ is effectively zero — homogeneous nucleation is forbidden, and the Cavitation book develops the heterogeneous-nucleation story.

Example 4: hair-cell channel gating

A mechanotransduction channel in an inner-ear hair cell switches between open (O) and closed (C) states under a mechanical bias $x$ (tip-link deflection). The open probability follows the Fermi function

a direct Boltzmann-factor weighting of the two states. The width of the sigmoid in $x$ is $k_B T/(K_\text{TL} d) \approx 100\,\text{nm}$, set entirely by $k_BT$ and the spring + gating-swing geometry. See Hearing Ch 4.6.

Example 5: vibrational modes are frozen out

Why is $\gamma = 7/5$ for air at room temperature, not $9/7$? N₂ has three translational, two rotational, and one vibrational degree of freedom — in principle. But the vibrational quantum is $\hbar\omega_\text{vib} \approx 0.29\,\text{eV}$, far larger than $k_B T \approx 0.025\,\text{eV}$ at 290 K. The vibrational state has population $\sim e^{-12}$ in the excited level — effectively zero. By the Boltzmann-factor argument, the vibrational mode contributes nothing to the heat capacity until the gas is heated to several thousand kelvin. This is the first place classical equipartition meets quantum mechanics.

Cross-book backlinks

• Sound Ch 1.1 — air at rest: pressure from molecular momentum delivery.
• Sound Ch 1.2 — kinetic theory: the molecular picture.
• Sound Ch 1.3 — Brownian motion: the noise floor on hearing.
• Sound Ch 4.7 — kinetic-theory route to speed of sound: derivation of $c$ from $\langle v^2\rangle$.
• Sound Ch 10.2 — molecular relaxation: vibrational unfreezing as a sound-absorbing mechanism.
• Hearing Ch 4.6 — hair-cell transduction: two-state Boltzmann gating.
• Cavitation Ch 2.1 — homogeneous nucleation: nucleation rate from the Boltzmann factor.