Enthalpy

H = E + p V

and satisfies

d H = T d S + V d p

for a simple compressible system.

Table of contents

Joule-Thomson coefficient ( $μ = (\partial T / \partial p)_{H}$ )

The temperature change per pressure change at fixed enthalpy is

μ = (\partial T / \partial p)_{H} = [T (\partial V / \partial T)_{p} - V] / C_{p}

. A positive coefficient means cooling under an adiabatic throttle pressure drop.

Virial criterion for Joule-Thomson cooling

For

p = k_{B} T [n + B_{2} (T) n^{2}]

, the coefficient is

μ = N [T B_{2}^{'} - B_{2}] / [C_{p} (1 + 2 B_{2} n)]

. On the stable dilute branch it is positive precisely when

d (B_{2} / T) / d T > 0

.

Specific enthalpy ( $h$ )

Specific enthalpy is enthalpy per unit mass. For a simple compressible fluid of fixed composition, the first law of thermodynamics gives

d h = T d s + \frac{d p}{ρ},

where

T

is temperature,

s

is specific entropy,

p

is pressure, and

ρ

is mass density. Along an isentropic flow,

d h = d p / ρ

. For a perfect gas with constant specific-heat ratio

γ

,

h = γ p / [(γ - 1) ρ]

.

Polytropic enthalpy ( $Q = (n + 1) p / ρ$ )

For

p = K ρ^{1 + 1/ n}

with

K, n > 0

, the specific enthalpy relative to vacuum is

Q = (n + 1) K ρ^{1/ n} = (n + 1) p / ρ

. It obeys

\nabla Q = \nabla p / ρ

and

n D_{t} Q = - Q \nabla \cdot u

. The vacuum reference is fixed when these equations are combined with the polytropic equation of state.

Enthalpy Maxwell relation

Since

T = (\partial H / \partial S)_{p}

and

V = (\partial H / \partial p)_{S}

, equality of mixed derivatives gives

(\partial T / \partial p)_{S} = (\partial V / \partial S)_{p}

.

Heat capacity at constant pressure

At fixed pressure, supplied heat equals enthalpy change; for constant

C_{p}

,

Q = C_{p} Δ T

.

Difference between constant-pressure and constant-volume heat capacities ( $C_{P} - C_{V}$ )

For a simple compressible system,

C_{P} - C_{V} = T (\frac{\partial V}{\partial T})_{P} (\frac{\partial P}{\partial T})_{V} .

This follows by expressing the entropy as a function of temperature and volume and using a Maxwell relation.

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Wikipedia Bot  1

Enthalpy is a thermodynamic property that represents the total heat content of a system. It is denoted by the symbol \( H \) and is a state function, meaning its value depends only on the current state of the system, not on how it got there.

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Ciro Santilli  40 Updated 2025-07-16

Adds up chemical energy and kinetic energy.

Wikipedia mentions however that the kinetic energy is often negligible, even for gases.

The sum is of interest when thinking about reactions because chemical reactions can change the number of molecules involved, and therefore the pressure.

To predict if a reaction is spontaneous or not, negative enthalpy is not enough, we must also consider entropy via Gibbs free energy.

Bibliography:

chemistry.stackexchange.com/questions/7696/how-do-i-distinguish-between-internal-energy-and-enthalpy

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