Adiabatic Process

Equivalent forms

T V^{\gamma-1} = \text{const}

T^{\gamma} P^{1-\gamma} = \text{const}

\delta Q = 0

One exponent γ turns the gentle isotherm into a steep adiabat — and explains the speed of sound, cloud formation, and diesel ignition at once.

Symbol	Name	SI unit	Dimension	Range
$P$	Pressureoutput Gas pressure	Pa	M L⁻¹ T⁻²	0.1 – 100
$V$	Volume Gas volume	m³	L³	1 – 5
$γ$	Heat capacity ratio Cp/Cv; 5/3 monatomic, 7/5 diatomic	dimensionless	1	1.1 – 1.67

Unit systems

SI:: P in Pa, V in $m^{3}$
natural:: PV in energy units
CGS:: P in $dyn/cm^{2}$

Where it holds

Reversible, quasi-static, no heat exchange; real fast processes are approximately adiabatic but produce some entropy.

Discovery

Siméon Denis Poisson / Pierre-Simon Laplace · 1823

Laplace used the adiabatic (not isothermal) relation to correct Newton's speed-of-sound calculation in 1816; Poisson published the PV^γ law in 1823, fixing the famous 20% error.

Try this

Why does a bicycle pump get hot when you compress fast?

Compress a gas faster than heat can escape and all the work goes into the gas itself — temperature shoots up with no flame in sight. The same law cools the air that makes clouds.

Research status: stable

Common misconceptions

'Adiabatic' means no heat transfer, not constant temperature — temperature changes a lot. It also requires the process be fast OR insulated, not necessarily both.

Derivation

First Law with

\delta Q = 0

:

dU = -P dV

.

For an ideal

gas dU = C_v dT

and

PV = nRT

, so

C_v dT = -(nRT/V) dV

.

Using

C_p - C_v = nR

and \gamma = C_p/C_v

gives

dT/T = -(\gamma -1) dV/V

, which integrates to

TV^{\gamma -1}

= const, equivalently

PV^\gamma

= const.

Limiting cases

\gamma \to 1

⟶ Approaches the isotherm

PV =

const (heat keeps T fixed)

Rapid compression⟶

Q \approx 0

holds even without insulation; temperature spikes (diesel ignition)

Rapid expansion⟶ Gas cools — adiabatic cooling that forms clouds and fog

Ideal Gas Law→Heat Capacity at Constant Pressure→Heat Capacity at Constant Volume→Carnot Efficiency→