Alright, so obviously the degeneracy g1 has no dependence on temperature, and

hence it doesn't appear in a derivative. This term on the other hand, the second

term in the overall partition function expansion, does, and if you adopt the

chain rule, you'll get a 1 over q electronic and then this is the piece I

asked you to work on, N, degeneracy of the second, of the first excited state,

so the second term in the partition function, epsilon two, e to the minus

beta epsilon two, and had there been more terms we would have had to be continue

with this differentiation. Now generally that contribution from the

electronic partition function is small, right this depends on the second term

actually having some relevance. Had we really been working with something

entirely in the ground stage. It doesn't depend on temperature.

There would be no contribution overall to the internal energy.

And in that circumstance when that term is indeed small we would have that the

internal energy is simply this first term.

3 halves NkT. So internal energy dominated by the

translational contribution because the fraction in excited electronic states is

usually very small at low at every day temperatures is what I mean by low.

Is a room temperature here we are at a temperature that is not all that hot.

Roughly 300 kelvin. So we may as well actually put some

numbers on that though just to get a feel for the importance of that electronic

term, so let me take two example gases, monatomic ideal gases.

One would be a gas of lithium atoms and one would be a gas of fluorine atoms, so

those are mildly unusual gases. You might have trouble actually

assembling a flask full of lithium atom gas and fluorine atom gas, but that's

okay, this is a demonstration that's a thought experiment.

So we'll just work with the data, which are known for these atoms.

And in particular, what's known for lithium is that its first excited state

is quite high in energy relative to its ground state.

14,900 let's round to 4, reciprocal centimeters.

And that excited state has a degenerisity of 2.

For fluorine on the other hand, there's actually a rather low line first excited

state. It's only 404 reciprocal centimeters

above the ground state. And it too has a degenerisity of 2.

And in order to put specific numbers on things.

Let's actually pick values for the number of particles, and the temperature.

And in particular, let's take Avagadro's number of particles.

So we'll be working with a mole. And, let's use standard room temperature,

298.15 Kelvin. And I'll express my energy in kilojoules.

And so this low lying electronic state for the Florine atom means that at room

temperature, so if, if you think of room temperature multiplied times the

universal gas constant, gives you a feel for sort of what's the ambient energy

that's just floating around in the room available for harvesting.

If you were to express that in wave numbers, it would be about 200 wave

numbers, more or less. And so here we have a excited state, a

first excited state, that's only about double that up in energy.

So it's relatively accessible given the thermal bath in which these fluorine

atoms acting like an ideal gas would be residing.

And, the electronic partition function instead of being one, just remember one

refers to everything being in the ground state is got some population of the first

exited state, so it's 1.142. So, if we now go down we evaluate the

contribution of the internal energy from translation, that's 3 halves, N is

Avogadro's number times Boltzmann's constant, will be r.

So we'll get 3 halves, rT, and in kilojoules, 3 halves rT for this

temperature is 3.719. And for fluorine, of course, it doesn't

matter what the atom is here, there's no dependence on the atom, it's just a

constant. So it's 3.719 for fluorine as well.