Most General Change of State for an Ideal Gas

Each ideal gas has associated with it a scalar function  
  representing the internal energy of the gas, and that is function of any two of the pressure  
, the volume  
and the temperature  
. 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  
\[Q(t_2)-Q(t_1) =\int^{t_2}_{t_1} (\frac{dU}{dt} + p \frac{dV}{dt}) dt\]