Cheat sheets/Thermodynamics in differential form
Heat: dQ
Work: dW
Entropy: dS = dQ / T
Internal energy
U
=
U
(
T
,
V
)
{\displaystyle U=U(T,V)}
d
U
=
(
∂
U
∂
T
)
V
d
T
+
(
∂
U
∂
V
)
T
d
V
{\displaystyle dU=\left({\frac {\partial U}{\partial T}}\right)_{V}dT+\left({\frac {\partial U}{\partial V}}\right)_{T}dV}
d
U
=
d
Q
+
d
W
{\displaystyle dU=dQ+dW}
d
U
=
C
V
d
T
+
(
∂
U
∂
V
)
T
d
V
{\displaystyle dU=C_{V}dT+\left({\frac {\partial U}{\partial V}}\right)_{T}dV}
Enthalpy
H
=
H
(
T
,
P
)
{\displaystyle H=H(T,P)}
d
H
=
(
∂
H
∂
T
)
P
d
T
+
(
∂
H
∂
P
)
T
d
P
{\displaystyle dH=\left({\frac {\partial H}{\partial T}}\right)_{P}dT+\left({\frac {\partial H}{\partial P}}\right)_{T}dP}
d
H
=
C
P
d
T
+
(
∂
H
∂
P
)
T
d
P
{\displaystyle dH=C_{P}dT+\left({\frac {\partial H}{\partial P}}\right)_{T}dP}
Combined first and second law
d
U
=
T
d
S
−
P
d
V
{\displaystyle dU=TdS-PdV}
Other thermodynamic potentials
d
U
=
T
d
S
−
P
d
V
{\displaystyle dU=TdS-PdV}
d
H
=
T
d
S
+
V
d
P
{\displaystyle dH=TdS+VdP}
d
A
=
−
S
d
T
−
P
d
V
{\displaystyle dA=-SdT-PdV}
d
G
=
−
S
d
T
+
V
d
P
{\displaystyle dG=-SdT+VdP}
Maxwell relations
(
∂
T
∂
V
)
S
=
−
(
∂
P
∂
S
)
V
{\displaystyle \left({\frac {\partial T}{\partial V}}\right)_{S}=-\left({\frac {\partial P}{\partial S}}\right)_{V}}
(
∂
T
∂
P
)
S
=
(
∂
V
∂
S
)
P
{\displaystyle \left({\frac {\partial T}{\partial P}}\right)_{S}=\left({\frac {\partial V}{\partial S}}\right)_{P}}
(
∂
S
∂
V
)
T
=
(
∂
P
∂
T
)
V
{\displaystyle \left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial P}{\partial T}}\right)_{V}}
(
∂
S
∂
P
)
T
=
−
(
∂
V
∂
T
)
P
{\displaystyle \left({\frac {\partial S}{\partial P}}\right)_{T}=-\left({\frac {\partial V}{\partial T}}\right)_{P}}
