A reversible adiabatic process is a reversible thermodynamic process in which no heat or mass transfer occurs.
A reversible adiabatic expansion follows the path from B to C, while a reversible adiabatic compression follows the path from D to A. The curves describing the paths of adiabatic processes are called adiabats (see diagram above, where we have included two isotherms, AB and CD, for comparison).
A reversible adiabatic compression is carried out in a frictionless, insulated piston-cylinder device containing a gas (see diagram above). The gas undergoes infinitesimal steps of compression, resulting in an increase in the internal energy of the system and therefore the temperature of the system. Since no heat enters or leaves the system, and according to the first law of thermodynamics,
For an ideal gas, we have shown in a previous article that and hence, work done on the ideal gas system for a reversible adiabatic compression is:
Substituting in eq84 and integrating throughout,
We have assumed that the heat capacity of the ideal gas is independent of temperature over the temperature range of interest and therefore a constant. Solving the above equation gives:
where . Similarly, for an adiabatic expansion from B to C, we have
Equating eq85 and eq86,
Question
Show that internal energy of a system containing a perfect gas is a state function using the diagram above where
AC: reversible isothermal process
AD: reversible adiabatic process
AB: reversible isobaric process
BC & CD: reversible isochoric processes
Answer
Consider the change in internal energy of a system containing one mole of ideal gas from point A to point C. The change can be brought about by three different paths, with the first being a reversible isothermal expansion (AC), where .
The second path involves a reversible adiabatic expansion (AD) followed by a reversible isochoric heating (DC). The changes in internal energy for paths AD and DC are:
The change in internal energy for path ADC is:
Since
The third path consists of a reversible isobaric expansion (AB) followed by a reversible isochoric cooling (BC). The change in internal energy for path AB is:
From a previous article, , and so
Substituting and
in the above equation,
The change in internal energy for path BC is:
The change in internal energy for path ABC is:
Regardless of the path taken, the change in internal energy from point A to point C is the same. Therefore, is a state function.