There is of energy per degree of freedom of a molecule
in a system. A monoatomic gas has three degrees of freedom (three
dimensions of kinetic energy). A diatomic gas typically has five (three
translational degrees, two rotational degrees, skip vibration and the
third rotation). A solid typically has six (three translational
degrees, three vibrational degrees). The internal energy of
molecules of a monoatomic gas are thus:
(231) |
At constant volume, no work is done and all heat that goes into a system
increases its internal energy. At constant pressure, heat going into a
system can both do work and increase internal energy and typically does
both. We define:
(232) | |||
(233) |
(234) |
(235) |
(236) |
(237) |
(238) |
(239) |
If we take the full derivative of the ideal gas law:
(240) |
(241) |
Collecting terms, rearranging, and dividing by we get:
(242) | |||
(243) | |||
(244) | |||
(245) | |||
(246) |
(247) |
(248) |
This is the equation for the curve of an adiabatic process. There are lots of ways to manipulate this algebraically by e.g. combining it with the ideal gas law to eliminate or in favor of and the remaining one.
We will need the result in order to solve problems involving adiabatic processes in cyclic heat engines, so learn it well.