Extensive thermodynamic properties are those which are additive for independent and non interactive subsystems.
Consider the thermodynamic function is an extensive property of multiple sub systems as temperature pressure and number of moles.
So, the extensive property will be explained as:
As, this is an extensive property, so, we can write this as below:
Now, consider a system in which temperature and pressure is unchanged. So, and.
So, the equation will be:
Now, we can sayas this is a molar quantity.
So, we can say:
Now, we choose a factorsuch a way that
So, for the extensive properties:
So, we get:
Integrating in the limit, then we get:
Integrating we get:
Here we have takenand that is are constant
So, if we takeindividually as a set, then we can say:
So, comparing we get:
Hence, this is proved that the extensive thermodynamic quantities are homogeneous functions of their extensive variables.