\[U\]

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

\[Q(t_2)-Q(t_1) =\int^{t_2}_{t_1} (\frac{dU}{dt} + p \frac{dV}{dt}) dt\]