The_Cambridge_Handbook_of_Physics_Formulas

108 Thermodynamics
Expansion processes
Joule
expansion a
Joule–Kelvin
expansion b
η = ∂T
∂V
∣
U
= − T 2
= − 1
C V
(
T ∂p
∂T
µ = ∂T
∂p
∣
H
= T 2
= 1 C p
(
T ∂V
∂T
∂(p/T )
∣ (5.24)
C V ∂T V )
∣ −p
(5.25)
V
∂(V/T)
∣ (5.26)
C p ∂T p )
∣ −V
(5.27)
p
η
T
p
U
C V
Joule coefficient
temperature
pressure
internal energy
heat capacity, V constant
µ Joule–Kelvin coefficient
V volume
H enthalpy
C p heat capacity, p constant
a Expansion with no change in internal energy.
b Expansion with no change in enthalpy. Also known as a “Joule–Thomson expansion” or “throttling” process.
Thermodynamic potentials a
Internal energy dU = T dS −pdV +µdN (5.28)
H = U +pV (5.29)
Enthalpy
dH = T dS +V dp+µdN (5.30)
U internal energy
T temperature
S entropy
µ chemical potential
N number of particles
H enthalpy
p pressure
V volume
Helmholtz free F = U −TS (5.31)
energy b dF = −S dT −pdV +µdN (5.32)
F
Helmholtz free energy
Gibbs free energy c = F +pV = H −TS (5.34)
G = U −TS+pV (5.33)
dG = −S dT +V dp+µdN (5.35)
G
Gibbs free energy
Grand potential
Φ=F −µN (5.36)
dΦ = −S dT −pdV −N dµ (5.37)
Φ
grand potential
Gibbs–Duhem
relation
Availability
a dN=0 for a closed system.
b Sometimes called the “work function.”
c Sometimes called the “thermodynamic potential.”
−S dT +V dp−N dµ = 0 (5.38)
A = U −T 0 S +p 0 V (5.39)
dA =(T −T 0 )dS −(p−p 0 )dV (5.40)
A
T 0
p 0
availability
temperature of
surroundings
pressure of surroundings