In thermodynamics, an isobaric process is a type of thermodynamic process in which the pressure of the system stays constant: ΔP = 0. The heat transferred to the system does work, but also changes the internal energy (U) of the system. This article uses the physics sign convention for work, where positive work is work done by the system. Using this convention, by the first law of thermodynamics,
Q
=
Δ
U
+
W
{\displaystyle Q=\Delta U+W\,}
where W is work, U is internal energy, and Q is heat. Pressure-volume work by the closed system is defined as:
W
=
∫
p
d
V
{\displaystyle W=\int \!p\,dV\,}
where Δ means change over the whole process, whereas d denotes a differential. Since pressure is constant, this means that
W
=
p
Δ
V
{\displaystyle W=p\Delta V\,}
.Applying the ideal gas law, this becomes
W
=
n
R
Δ
T
{\displaystyle W=n\,R\,\Delta T}
with R representing the gas constant, and n representing the amount of substance, which is assumed to remain constant (e.g., there is no phase transition during a chemical reaction). According to the equipartition theorem, the change in internal energy is related to the temperature of the system by
Δ
U
=
n
c
V
,
m
Δ
T
{\displaystyle \Delta U=n\,c_{V,m}\,\Delta T}
,where cV, m is molar heat capacity at a constant volume.
Substituting the last two equations into the first equation produces:
Q
=
n
c
V
,
m
Δ
T
+
n
R
Δ
T
Q
=
n
Δ
T
(
c
V
,
m
+
R
)
Q
=
n
Δ
T
c
P
,
m
{\displaystyle {\begin{aligned}Q&=n\,c_{V,m}\,\Delta T+n\,R\,\Delta T\\Q&=n\Delta T(c_{V,m}+R)\\Q&=n\Delta Tc_{P,m}\end{aligned}}}
where cP is molar heat capacity at a constant pressure.