## Most General Change of State for an Ideal Gas

Each ideal gas has associated with it a scalar function
$U$
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
$Q(t_2)-Q(t_1) =\int^{t_2}_{t_1} (\frac{dU}{dt} + p \frac{dV}{dt}) dt$