So, let me define a state function. A state function is a property that
depends only upon the state of the system.
It does not depend on how the system got to that state.
That is, it is independent of the path. And so, what defines a state, well we've
seen in some instances for instance, the specification of particular variables.
So we've worked with partition functions up 'til now for example that have
specified number of particles, temperature and volume.
And so a state function within that [UNKNOWN] would be something that depends
indeed only on number of particles, temperature and volume.
Not how I got there, not how I might have changed the temperature until I got to
the current temperature. A key property of a state function is
that its differential can be integrated as a mathematical quantity in a normal
path independent way. And so, in particular, energy is a state
function, internal energy. That is, I can think of integrating the
differential of the energy from state one to state two, that will be equal to the
difference between the energy of state two and state one, and I can write that
as just delta u. So, the energy difference, and it's
important to emphasize that in thermodynamics, we're almost always
interested in differences in quantities. It's quite rare we calculate something
absolutely, indeed, we often have to take conventions to define where zero is, and
that makes quite clear that we're usually interested in differences.
But, in any case we can integrate a differential for a state function to get
a change in a simple and path independent way.