Heat capacity

Heat capacity is the heat required per unit temperature change under a specified constraint.

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Entropy from heat capacity

Along a reversible path at fixed volume,

d S = C_{V} (T) d T / T

. Hence entropy differences can be measured macroscopically from

S (T_{2}) - S (T_{1}) = \int_{T_{1}}^{T_{2}} \frac{C _{V} ( T )}{T} d T,

with an additional

L / T_{c}

across a first-order transition of latent heat

L

at

T_{c}

.

Entropy concavity and positive heat capacity

For a thermodynamically stable system, the entropy is concave in energy, so

\frac{\partial ^{2} S}{\partial E ^{2}} = - \frac{1}{T ^{2}} \frac{\partial T}{\partial E} < 0.

Consequently

\partial E / \partial T > 0

: energy increases with temperature, and the corresponding heat capacity is positive.

Specific heat capacity ( $c_{p}$ )

Specific heat capacity is heat capacity per unit mass. The volumetric heat capacity is

ρ c_{p}

, and

k / (ρ c_{p})

is the thermal diffusivity when

k

is the thermal conductivity.

Specific heat capacity at constant volume ( $c_{V} = (\partial u / \partial T)_{h} o$ )

The specific heat capacity at constant volume is the temperature derivative of specific internal energy at fixed density and composition:

c_{V} = (\partial u / \partial T)_{ρ}

. It is the corresponding extensive heat capacity at constant volume divided by material mass.

Specific heat capacity at constant pressure ( $c_{P}$ )

The specific heat capacity at constant pressure is

c_{P} = (\partial h / \partial T)_{P}

, where

h

is specific enthalpy. It includes expansion work as well as the change in internal energy.

Pure-radiation constant-pressure heat-capacity singularity

For pure blackbody radiation,

P = a T^{4} /3

depends only on temperature. Constant pressure therefore fixes temperature, so a usual finite constant-pressure temperature derivative is unavailable. In the material gas-radiation limit the specific-heat ratio diverges, although all three stellar adiabatic exponents approach

4/3

. The radiation index

4/3

is an adiabatic pressure-volume response, not a finite heat-capacity ratio.

Heat capacity at constant volume ( $C_{V}$ )

The heat capacity at constant volume is

C_{V} = (\partial E / \partial T)_{V}

when particle number and other conserved quantities are also fixed.

Dulong-Petit law

In the classical high-temperature limit, each independent harmonic mode contributes

k_{B}

to the constant-volume heat capacity.

Heat capacity at constant area ( $C_{A}$ )

For a two-dimensional system of fixed area

A

, the heat capacity at constant area is

C_{A} = (\partial E / \partial T)_{N, A}

when the particle number and other conserved quantities are also fixed. It is the two-dimensional analogue of heat capacity at constant volume.

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Heat capacity by

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Heat capacity is a physical property of a substance that measures the amount of heat energy required to change its temperature by a certain amount. It quantifies how much heat is needed to raise the temperature of a material based on its mass and specific heat capacity. There are two key concepts related to heat capacity: 1. **Specific Heat Capacity**: This is the amount of heat required to raise the temperature of one unit mass of a substance by one degree Celsius (or one Kelvin).

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