Combining the two processes by which the internal energy of a thermodynamic
system can change we obtain the first law of thermodynamics:
Remember the sign convention here: dQ is the heat that flows into the system,
dW is the mechanical work performed by the system.
We shall now study a number of different thermodynamic processes
mainly involving the ideal gas
by determining expression for 
, 
and 
.
As mentioned previously two of
the thermodynamic variables p, V, and T are sufficient to
fully specify the thermodynamic equilibrium state of a thermodynamic system since the
third variable may be obtained from the equation of state. (Note here
that we will mainly consider thermodynamic systems consisting
of a fixed amount of atoms). We can therefore
specify the state of a thermodynamic system as a point in a
p-V, p-T, or T-V diagram. We choose here to use the P-V
diagram because it is most easily to visualize the mechanical work
in a thermodynamic process when the process is represented in such a
diagram. It is important to note that for each point in this diagram
corresponds a specific value of T which we can obtain from the
equation of state. Specifically for the ideal gas we have
Thus the lines of constant temperature correspond to constant
values of pV or to the functional form
Each point in the p-V diagramm also corresponds to a specific value for the internal energy function. In the special case of the ideal gas the lines of constant internal energy coincide with the lines of constant temperature because the internal energy depends only on T. This feature however is not true for gases and liquids where interactions between molecules are significant.
The p-V diagram only can describe the state of systems in thermodynamic equilibrium. When external conditions change the thermal equilibrium state of changes. If the changes occur so slowly that the system remains in thermodynamic equilibrium at all times we have a reversible thermodynamic process. The name stems from the fact that such a process can always be reversed by reversing the external conditions that gave rise to the modified state of thermodynamic equilibrium. Processes which proceed through non-equilibrium states are only represented in the p-V by their initial and final states. Such processes are irreversible in the sense that it is not possible to bring the system back to where it came from by reversing the sequence of time dependent external conditions which induced the process.
Work performed by the system when it is brought from point A to B in the
p-V diagram is
We see that 
is simply
the area under the curve representing the
thermodynamic process. If the process increase the volume then
is positive if it decreases the volume
then the work done by the system is negative. These remarks hold
for any reversible process.
We now consider changes in internal energy, 
,
the heat flow, 
, and the work done
, for various thermodynamic processes
which bring the system from point A to point B in the
p-V phase diagram.
