delta w reversible, what does this mean? It's an infinitesimal change in the
energy levels, with probability remaining the same times a dv.
That's why I want to equate this with work.
It's because it's multiplying a dv term And the differential work is associated
with a pressure times a differential volume.
Meanwhile over here, the other piece of the term, just by what's left in the
first law, must be associated with differential heat.
And what is that? Well, it's infinitesimal changes in the
probabilities of the levels with the energy of the levels remaining the same.
So it might not be obvious what, how do these really correspond, one to another?
It's actually a little helpful to take a look at it graphically, to help make this
more clear. And in particular, let me, let me think
about this first term. What's it really saying?
Work. So differential work is, small changes in
energy levels with constant probabilities.
So, that would be like, so here might be a distribution within energy space, so
every dot, the more dots the more probable.
And, so as I go up in energy, I've got less probability.
So, increasing the energy would be like pushing these levels further and further
apart. And, in a sense, what pressure does in,
say, reducing a volume in a box, for example, that would split the energy
levels apart, they would become wider. And if I don't change the probabilities,
the net energy is raised, and that's really sort of what this term is doing,
that's the work term. What about the heat term?
The heat term might even be a, a little bit more sensible in, in a way.
If I keep the energy levels the same, so these lines are spaced just the same as
these lines but I changed the probability of being in those levels.
So, to increase the energy, I'd move probabilities out of low levels, up into
high levels. That's like increasing the temperature,
it's adding heat. Decreasing the energy, obviously the
opposite direction. So, pressure then, I mean, given this
relationship of work, differential work is this probability times differential
energy. And the more common differential work is
just minus pdv. I'll get that the pressure is this
expression here, proceeding dv. It's the probability weighted, the
ensemble weighted values of partial e, partial v.
Or, if I want to compute pressure, it is minus.
So I've just been carrying this negative sign through.
Partial derivative of energy with respect to volume.