Thermal expansion

Thermal expansion is a change in a body's dimensions as its temperature changes under a specified mechanical constraint. A heated gas under a freely moving loaded piston usually increases its volume at fixed pressure; heating the same gas in a rigid vessel instead raises its pressure. The coefficient of thermal expansion quantifies the first response. Expansion can be negative, so the name does not assert that every substance grows on heating.

For an ideal gas at fixed pressure and particle number,

V = N k_{B} T / P

gives

V_{2} / V_{1} = T_{2} / T_{1}

. Heating from

300 K

330 K

therefore increases the volume by exactly

10%

within the ideal gas model. For a solid, specify whether the measurement concerns length or volume: for uniform expansion in all three directions, the volumetric coefficient is three times the linear thermal expansion coefficient, because

V \propto L^{3}

. Direction-dependent expansion requires the separate dimensional responses. ^[ref]

Table of contents

Thermal expansion tensor ( $α$ )

 0  0

The tensor relating a small unconstrained thermal strain to a temperature change:

ε^{th} = α Δ T

. A crystal orientation rotates this tensor into laboratory coordinates.

Anisotropic thermal expansion

 0  0

Direction-dependent thermal expansion. Different crystal grain orientations can demand incompatible strains even under uniform macroscopic heating.

Coefficient of thermal expansion ( $α = V^{- 1} (\partial V / \partial T)_{P, N}$ )

 0  0

The volumetric coefficient of thermal expansion is the fractional volume increase per temperature increase at fixed pressure and composition:

α = \frac{1}{V} (\frac{\partial V}{\partial T})_{P, N} = - \frac{1}{ρ} (\frac{\partial ρ}{\partial T})_{P, N}, ρ = \frac{m}{V} .

(1)

The mass density form requires fixed mass

m

. Its units are

K^{- 1}

;

α_{T}

is another notation for this coefficient. For small temperature changes

θ

at fixed pressure,

V (T_{0} + θ) ≃ V (T_{0}) (1 + α (T_{0}) θ), ρ (T_{0} + θ) ≃ ρ (T_{0}) (1 - α (T_{0}) θ) .

(2)

For a finite change in a single thermodynamic phase, integrate the possibly varying coefficient:

ln (V_{2} / V_{1}) = \int_{T_{1}}^{T_{2}} α (T, P, N) d T

. A constant-coefficient approximation gives an exponential, while the displayed linear formula additionally requires

∣ α θ ∣ ≪ 1

For an ideal gas at fixed particle number,

α = \frac{1}{T}, α (300 K) = \frac{1}{300} K^{- 1} ≃ 3.333 \times 1 0^{- 3} K^{- 1} .

(3)

Integrating

1/ T

recovers the exact fixed-pressure thermal expansion ratio

V_{2} / V_{1} = T_{2} / T_{1}

. Combining expansion with isothermal compressibility gives the local response of any smooth simple compressible system of fixed composition,

\frac{d V}{V} = α d T - κ_{T} d P, (\frac{\partial P}{\partial T})_{V, N} = \frac{α}{κ _{T}} .

(4)

The ratio has units

Pa K^{- 1}

. For the one-mole ideal gas state in isothermal compressibility, it is

P_{0} / T_{0} = 333.33 Pa K^{- 1}

: a

1 K

increase at fixed volume raises the pressure by about

333.33 Pa

. Maxwell relations identify this same ratio as the entropy slope with respect to volume and give

C_{P} - C_{V} = T V α^{2} / κ_{T}

Under the Boussinesq approximation, the small fractional mass density change produces buoyancy while mass density is otherwise treated as constant. A positive temperature excess

θ

gives reduced gravity

g^{'} = g α_{T} θ

in the convention used in natural ventilation. The coefficient therefore converts a heat budget into a buoyancy budget. The same measurable expansion response supplies entropy changes and heat capacity differences through Maxwell relations, connecting fluid motion to equilibrium thermodynamics.

Reference-volume thermal expansion coefficient ( $α_{0} = V_{0}^{- 1} d V / d T$ )

 0  0

The derivative

V_{0}^{- 1} d V / d T

uses a fixed reference volume

V_{0}

. It differs from the instantaneous fractional coefficient

V^{- 1} d V / d T

by the factor

V / V_{0}

. The volume convention must be specified before comparing coefficients.

Linear thermal expansion coefficient ( $α_{L} = L^{- 1} (\partial L / \partial T)_{P}$ )

 0  0

The linear thermal expansion coefficient is the fractional change in a specified length per change in temperature, at fixed external pressure for a sample free to expand. For small changes,

Δ L / L ≃ α_{L} Δ T

. An isotropic sample with the same response in all directions has volumetric coefficient of thermal expansion

α = 3 α_{L}

, since differentiating

V \propto L^{3}

gives

d V / V = 3 d L / L

. Direction-dependent materials require a specified measurement direction. ^[ref].

Thermal expansion of a weakly anharmonic bond ( $α = - γ k_{B} / (2 a κ^{2})$ )

 0  0

For a bond free of applied tension, use a local potential energy

V (a + x) = V (a) + κ x^{2} /2 + γ x^{3} /6 + \dots

with

κ > 0

. Classical equipartition gives

⟨ x^{2} ⟩_{0} = k_{B} T / κ

; vanishing mean force then yields

⟨ x ⟩ = - γ k_{B} T / (2 κ^{2})

to first order in anharmonicity. Thus the linear thermal expansion coefficient is

α = - γ k_{B} / (2 a κ^{2})

. For a Morse potential,

γ = - 3 κ / ξ

, giving positive expansion

3 k_{B} / (2 aκ ξ)

. This is a local bound-state approximation; the isolated cubic truncation is not a globally stable equilibrium potential.

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Thermal expansion refers to the tendency of matter to change its shape, area, and volume in response to changes in temperature. As the temperature of a substance increases, its particles move more rapidly, leading to an increase in the average distance between them. This results in the expansion of the material. Thermal expansion occurs in all states of matter—solids, liquids, and gases—but the degree of expansion can vary significantly among different materials.

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