Thermodynamics in one page

Equation of state, internal energy, adiabatic processes, polytropic envelope, enthalpy.

Thermodynamics is the bookkeeping of energy, heat, and work in equilibrium systems. The Sound book uses a thin slice of it — enough to derive the speed of sound and the adiabatic equation of state for a gas — but the same machinery underlies open-system flows, free energy, and phase change. This chapter takes the slice slowly, with an interactive per central equation.

State variables and the equation of state

A fluid at equilibrium is described by a handful of state variables: pressure $p$, density $\rho$ (or specific volume $v = 1/\rho$), temperature $T$, internal energy $U$, entropy $S$. Not all are independent — any two of them determine the rest via an equation of state.

For an ideal gas the equation of state is

where $R = 8.314\,\text{J/(mol·K)}$ is the universal gas constant, $M$ the molar mass, $n = N/V$ the number density, and $k_B = R/N_A$ Boltzmann’s constant.

The simulation is a 2-D ideal gas at adjustable temperature, number density, and volume. Slide $T$, $N$, $V$ and watch the computed pressure track $PV = N k_B T$. The ideal-gas law is the macroscopic shadow of the kinetic-theory picture: pressure as molecular momentum delivery, temperature as mean translational kinetic energy.

The first law of thermodynamics

Conservation of energy for a closed fluid element:

The internal energy $U$ changes by heat in ($T\, dS$) minus work done ($p\, dV$). For a closed fluid element (no mass exchange across its boundary), this is the operative form. Three classical paths trade $Q$, $W$, and $\Delta U$ differently:

• Isothermal ($T$ fixed). For an ideal gas, $\Delta U = 0$, so $Q = W$ — every joule of heat in becomes a joule of work out.
• Adiabatic ($Q = 0$). $\Delta U = -W$ — work comes at the expense of internal energy (cooling on expansion).
• Isobaric ($p$ fixed). $W = p\,\Delta V$ — work is volume change at constant pressure.
• Isochoric ($V$ fixed). $W = 0$, so $Q = \Delta U$ — heat in raises internal energy directly.

Pick a process and traverse it. The shaded area is the work; the readouts show cumulative $W$, $\Delta U$, and $Q$ at each step. For an ideal gas the relations among the three are completely fixed by the path type and the equation of state.

Adiabatic vs. isothermal — the two limits that matter for sound

The interactive plots both the isothermal curve $PV = \text{const}$ and the adiabatic curve $PV^\gamma = \text{const}$ through the current state point. The adiabatic curve is steeper than the isothermal at the same state — by exactly the factor $\gamma$ in slope. For air, $\gamma = 1.4$; this 40% extra steepness is what corrects Newton’s isothermal sound-speed calculation to Laplace’s adiabatic one.

• Isothermal: perfect thermal contact, heat flows freely. $pV = \text{const}$.
• Adiabatic: no heat exchanged, entropy fixed. $pV^\gamma = \text{const}$ with $\gamma = c_p/c_v$.

For air at audible frequencies, sound oscillations are too fast for the heat conducted in one cycle to matter ($\sim 5\,\mu\text{m}$ thermal-diffusion length per millisecond, versus a wavelength of $\sim 30\,\text{cm}$). The compressions are effectively adiabatic — Laplace’s correction. The viscosity & diffusion chapter develops this scale comparison.

The polytropic envelope

The isothermal and adiabatic processes are the two limits of a more general family: a polytropic process satisfies $pV^\kappa = \text{const}$ for some exponent $\kappa$ in $[1, \gamma]$. Real processes choose $\kappa$ by the ratio of the process timescale to the thermal-diffusion time across the fluid element.

Slide $\kappa$ from 1 (isothermal) to $\gamma$ (adiabatic). The curve sweeps continuously between the two limits. Two physical instances:

• A bubble of gas oscillating inside a liquid at frequency $\omega$ has a thermal-diffusion time across the bubble of $\sim R^2/\alpha_g$. If $\omega R^2/\alpha_g \ll 1$ the gas is isothermal; if $\gg 1$ it’s adiabatic. Real bubbles at MHz frequencies typically sit at $\kappa \approx 1.1$–$1.3$. See Cavitation Ch 3.2.
• The atmospheric lapse rate — the temperature drop with altitude in a rising parcel of air — is approximately adiabatic ($\kappa = \gamma$) on short timescales but tends toward isothermal in slowly mixing layers.

The speed of sound, thermodynamically

For any fluid, the speed of small-amplitude pressure waves is

with the subscript $s$ meaning at constant entropy — the adiabatic derivative. For an ideal gas with $p = \text{const}\cdot\rho^\gamma$,

The square-root $T$-dependence is universal across gases; the ratio $c/v_\text{rms} = \sqrt{\gamma/3}$ is gas-independent for each $\gamma$ (0.68 for diatomic, 0.75 for monatomic). Lighter molecular mass — H₂, He — gives faster sound at the same temperature. The Sound book’s four-route derivation lives at Sound Ch 4; this is the cleanest of them.

Enthalpy and the first law for open systems

A closed fluid element exchanges no mass with its surroundings; the first law reads $dU = T\,dS - p\,dV$. A control volume through which mass flows in and out — a slab in a sound wave, a turbine, a Rayleigh–Plesset bubble — needs an extra term. The in-flowing fluid does work against the pressure at the boundary to push itself across; the out-flowing fluid is pushed out by pressure work done by the system.

The clean way to absorb that flow-work is to define the enthalpy

per unit mass. The first law for an open, steady-state stream then reads

a statement of energy conservation along a streamline that includes the flow-work bookkeeping. Bernoulli’s equation is the incompressible limit (where $u$ is constant so $\Delta h = \Delta p/\rho$); the Rankine–Hugoniot energy condition across a shock front is another instance.

Enthalpy is “the natural energy variable for things that flow through control volumes.” Whenever you see $h$ in the bookshelf, the underlying argument is open-system energy conservation.

Entropy and the second law (briefly)

The first law alone does not forbid heat flowing from cold to hot. The second law does, in two equivalent statements:

• (Clausius) No process whose sole result is heat transfer from a cold to a hot reservoir is possible.
• (Mathematical) The entropy of an isolated system never decreases: $dS_\text{universe} \ge 0$.

For our purposes the second law enters in two places: (i) it picks out the adiabatic isentrope as the relevant compression curve for sound, and (ii) it provides the foundation for the free-energy chapter — the right thermodynamic potential to minimise at fixed external conditions follows from “total entropy ≥ 0” applied to system + reservoir.

Equipartition and γ from molecular structure

For a classical gas in equilibrium at temperature $T$, every quadratic degree of freedom carries average energy $\tfrac12 k_B T$ (see kinetic theory). A monatomic gas has three translational degrees of freedom, $U = \tfrac32 N k_B T$, $\gamma = 5/3$. A diatomic gas at room temperature has three translational + two rotational, $U = \tfrac52 N k_B T$, $\gamma = 7/5$. The thermal speed $\sqrt{3 k_B T / m} \sim 500\,\text{m/s}$ for air at 20°C is the same scale as the speed of sound — not a coincidence.

For the cross-book applications — adiabatic speed of sound, polytropic bubble interior, enthalpy in Rankine–Hugoniot shock relations, equipartition setting γ — see the key examples sub-page.