\[U\]
representing the internal energy of the gas, and that is function of any two of the pressure \[V\]
, the volume \[V\]
and the temperature \[T\]
. We only need two because they area related by the ideal gas equation \[\frac{pV}{T} = CONSTANT\]
.When heat is supplied to an ideal gas, the equation expressing the most general change that can take place is
\[Q(t_2)-Q(t_1) =\int^{t_2}_{t_1} (\frac{dU}{dt} + p \frac{dV}{dt}) dt\]
To show this we can take the First Law of Thermodynamics
\[dQ = dU + p dV\]
and write each physical quantity as a function of time. The we can rewrite the First Law of Thermodynamics \[\frac{dQ}{dt} = \frac{dU}{dt} + p \frac{dV}{dt}\]
.Integration then gives
\[Q(t_2)-Q(t_1) =\int^{t_2}_{t_1} (\frac{dU}{dt} + p \frac{dV}{dt}) dt\]