Isothermal compressibility ( $κ_{T} = - V^{- 1} (\partial V / \partial P)_{T, N}$ )

Isothermal compressibility is the compressibility measured at fixed temperature, particle number and composition. A slowly compressed sealed piston in contact with a thermal reservoir can approach this constraint. For an ideal gas, differentiating

V = N k_{B} T / P

at fixed temperature gives

(\partial V / \partial P)_{T, N} = - V / P

, hence

κ_{T} = 1/ P

. A small pressure increase

Δ P

therefore gives

Δ V / V ≃ - Δ P / P

. For a finite change, use the whole isothermal process curve

V_{2} / V_{1} = P_{1} / P_{2}

instead.

Take one mole of classical monatomic gas, with stipulated molar gas constant

R = 8.314 J mo l^{- 1} K^{- 1}

, initially at

T_{0} = 300 K

and

P_{0} = 100 kPa

. The initial volume is

V_{0} = n R T_{0} / P_{0} = 0.024942 m^{3}

. Compressing to

V_{0} /2

gives

Isothermal: Reversible adiabatic: P_{2} = 2 P_{0} = 200 kPa, T_{2} = 300 K, P_{2} = 2^{5/3} P_{0} ≃ 317.48 kPa, T_{2} = 2^{2/3} T_{0} ≃ 476.22 K .

The isothermal compression releases heat to the thermal reservoir; the reversible adiabatic compression retains the input work as internal energy. Reversing either curve gives the corresponding expansion, with adiabatic expansion cooling the gas.

Figure 1.
An ideal gas resists reversible adiabatic compression more strongly than isothermal compression
. Left: two pressure-volume curves through the same initial state; arrows point toward compression. Right: the corresponding temperature changes, including cooling during reversible adiabatic expansion. The dotted line marks half the initial volume. One mole of classical monatomic gas, with $γ = 5/3$ and the stipulated $R$ above. Details.

The equilibrium curvature of Gibbs free energy provides another measurement of isothermal compressibility:

κ_{T} = - \frac{1}{V} (\frac{\partial ^{2} G}{\partial P ^{2}})_{T, N} .

This follows from

V = (\partial G / \partial P)_{T, N}

, with

G

denoting Gibbs free energy. Within a stable single thermodynamic phase, increasing pressure reduces volume and makes this curvature negative. At phase coexistence, changing the phase fraction can change the volume at constant temperature and pressure; the usual finite single-phase isothermal compressibility then need not exist. Near a critical point, it can diverge. ^[ref]

Table of contents
- Critical-density compressibility of a cubic gas Isothermal compressibility
- Isothermal compressibility from chemical-potential response Isothermal compressibility

Critical-density compressibility of a cubic gas

For the cubic-density model,

κ_{T} = (n \partial p / \partial n)^{- 1}

gives

κ_{T} (n_{c}, T) = 1/ [n_{c} k_{B} (T - T_{c})]

. Divergence from above is a mean-field critical prediction. The negative value below

T_{c}

signals an unstable homogeneous branch rather than an equilibrium negative compressibility.

Isothermal compressibility from chemical-potential response ( $κ_{T} = n^{- 2} (\partial n / \partial μ)_{T}$ )

For a homogeneous extensive single-component thermodynamic phase, the Gibbs-Duhem equation at fixed temperature gives

d p = n d μ

. With number density

n = N / V

, fixed

N

gives

d n = - (n / V) d V

. The chain rule therefore gives

κ_{T} = \frac{1}{n} (\frac{\partial n}{\partial p})_{T} = \frac{1}{n ^{2}} (\frac{\partial n}{\partial μ})_{T} .

The derivatives are evaluated within the same smooth equation of state; an interface contribution or phase coexistence can invalidate this simple homogeneous derivation.

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