3.4 Adiabatic processes and the speed of sound

An adiabatic process exchanges no heat with its surroundings. Compress a gas adiabatically and the work done on it has nowhere to go but its internal energy, so the gas heats; its pressure then rises faster with falling volume than it would at constant temperature. That extra steepness is captured by a single exponent, $\gamma$ , and it sets the speed of every pressure wave the gas can carry.

The adiabatic equation of state

▶ pV^γ = const for an adiabatic ideal gas Derivation

For an adiabatic process $\delta Q = 0$ , so the first law is $dU = -p\,dV$ . For an ideal gas the internal energy depends only on temperature, $dU = n c_v\, dT$ , and the equation of state gives $p = nRT/V$ . Substituting,

n c_v\, dT \;=\; -\frac{nRT}{V}\,dV \quad\Longrightarrow\quad \frac{dT}{T} \;=\; -\frac{R}{c_v}\,\frac{dV}{V}.

Integrating gives $T V^{R/c_v} = \text{const}$ . By Mayer’s relation $R/c_v = \gamma - 1$ , so $T V^{\gamma-1} = \text{const}$ , and replacing $T$ by $pV/(nR)$ ,

p V^\gamma \;=\; \text{const}. \;✓

V = 1.00 P = 1.00 γ = c_p/c_v = 1.40

The adiabatic curve is *steeper* than the isothermal one at the same state. Compressing a gas adiabatically heats it (raises P faster than V drops); compressing isothermally lets the heat flow away and is comparatively soft. For diatomic air γ = 7/5 = 1.4; for monatomic helium γ = 5/3 ≈ 1.67.

The interactive plots the isothermal curve $pV = \text{const}$ and the adiabatic curve $pV^\gamma = \text{const}$ through the same state point. The adiabatic curve is steeper — by exactly the factor $\gamma$ in logarithmic slope — because adiabatic compression also heats the gas. The two coincide only in the limit $\gamma\to1$ , a gas with so many internal degrees of freedom that compression barely raises its temperature.

The polytropic family

Isothermal and adiabatic are the two ends of a continuum. A polytropic process obeys $pV^\kappa = \text{const}$ for an exponent $\kappa$ between $1$ (isothermal, perfect thermal contact) and $\gamma$ (adiabatic, perfect insulation). The value of $\kappa$ a real process takes is set by the competition between its timescale and the time for heat to diffuse across the system: fast compared with thermal diffusion is adiabatic, slow is isothermal.

κ1.40

regimeadiabatic (γ)

Polytropic exponent κ = 1.40

The polytropic family interpolates between isothermal (κ = 1, perfect heat exchange with surroundings) and adiabatic (κ = γ, no heat exchange). Real processes sit somewhere in between, with κ chosen by the ratio of process timescale to thermal-diffusion timescale: slow processes are isothermal, fast processes are adiabatic. Oscillating bubble interiors hover at frequency-dependent κ that the Cavitation book exploits.

Slide $\kappa$ from $1$ to $\gamma$ and the curve sweeps between the two limits. A gas bubble pulsating in a liquid and a parcel of air rising in the atmosphere both sit somewhere on this envelope, their $\kappa$ fixed by how their oscillation or ascent time compares with their thermal-diffusion time.

The speed of sound

A small pressure disturbance travels through a fluid at a speed set by the fluid’s stiffness against compression, divided by its density:

c^2 \;=\; \left(\frac{\partial p}{\partial\rho}\right)_s,

where the subscript $s$ marks the derivative at constant entropy — adiabatic, because the compressions and rarefactions of a sound wave are too fast for heat to even out between them. Using the adiabatic relation $p \propto \rho^\gamma$ ,

c^2 \;=\; \gamma\,\frac{p}{\rho} \;=\; \frac{\gamma R T}{M}.

H₂ c = 1301 m/s v_rms = 1904 c/v_rms = 0.683

He c = 1007 m/s v_rms = 1351 c/v_rms = 0.745

air c = 343 m/s v_rms = 502 c/v_rms = 0.683

CO₂ c = 268 m/s v_rms = 407 c/v_rms = 0.658

Temperature T = 293 K (20 °C) Overlay v_rms (dashed)

Sound speed is always √(γ/3) ≈ 0.68 (diatomic) or √(5/9) ≈ 0.75 (monatomic) of the rms thermal speed — and this ratio is gas-independent for each γ. Lighter gases give faster sound; the curves are √T-shaped, just like thermal speeds. CO₂ has lower γ (more active rotational modes), shifting it slightly relative to air.

The sound speed rises as $\sqrt{T}$ and falls with molecular mass — light gases (H₂, He) carry sound faster. The ratio $c/v_\text{rms} = \sqrt{\gamma/3}$ is gas-independent for a given $\gamma$ (about $0.68$ for a diatomic gas): sound travels somewhat slower than the typical molecule, because a disturbance can spread no faster than the molecules that carry it. Treating the compression as isothermal instead would replace $\gamma$ by $1$ and underestimate the speed by $\sqrt\gamma \approx 18\%$ — the error Newton made and Laplace corrected.