Friday Night State Functions

Happy Friday Night! I’m going to postpone the discussion about work in thermodynamic systems to discuss something that I found very exciting in lecture today, namely state functions.

To quote my professor, “If I was to throw a ball to you, you would catch it. It’s intuitive. You don’t need to understand classical mechanics. Cavemen didn’t need classical mechanics to throw their spears. Now, you can’t catch an electron. It’s all over the place, and too small. So in quantum mechanics you need to follow the mathematical equations to gain insight.”

This was quite a nice view, I thought. But sooner or later, he is waving his hands about at these equations, with another memorable quote that went thus:

“Does everybody follow what I am doing?”

A dissatisfactory howl went up through the lecture theatre.

“Well, just follow what I’m doing!”

Yep. That pretty much covered today’s lecture. To be fair, he was using mathematical concepts from calculus 3 to illustrate his point, disregarding the students present had only taken up to calculus 2, and were not familiar with multivariable equations.

Now, what was he talking about? I shall try to illuminate his words a bit more clearly, because the proof brought me great joy, and once you get past the intimidating looking format perhaps it will bring you glad tidings and such too.

So, state functions, you will recall, are independent from the path taken. They also are a total differentials, and have an exact value. Recall that the internal energy is a state function. The First Law of Thermodynamics states that 

Now, another way to represent this equation is as follows:

Oh no! What is this foreign beast? 

Well, the total derivative dU is defined by two variables, the pressure and the volume and is being expressed as a sum of its partial derivatives. I believe proving the two equations are equivalent is beyond the scope of the course. But we can prove a well-known equation to be equal to the total differential, which I shall outline below.

Let's take a state function, say pressure. Let's show that the total differential, dP, is the same as the ideal gas law up to a constant. Bear with me, and fasten your seat belts. 

Remember that 

We let n=1 for simplicity's sake.

Now, we express dP like this: 

Now we work towards solving this equation. We look at the first partial derivative in the sum, which expresses pressure as a function of temperature at constant volume. 

 (a)

In step (a) we remembered that 

and made a substitution for pressure in the partial derivative. 

Then, we know that R is the gas constant, and that volume is a constant as specified, so we can pull them out of the differential equation giving 

(b)

Now, 

So then

 (c)

 And now we multiply the result in (c) by

giving

 (d) 

Substituting back in for pressure we find that 

(e)

Doing a similar procedure for the second partial derivative, which expresses pressure as a function of volume at constant temperature in the summation of the total differential, we find that: 

(f)

Applying 

We simplify to 

 (g)

And substituting pressure back into the equation find that

(h)

So returning to the original equation and substituting the simplified versions of our two partial derivatives, we have 

(i)

Recall that

So that (i) can be expressed as

(j)

Bringing everything over to one side and applying rules of logs gives

(k)

Which tells us something very important. Taking the derivative of a constant equates to zero, so this implies that is a constant. So what? 

Well, we know that PV is proportional to T by a constant R, and that constant is implied above. Sadly, it's not a rigorous proof, but it certainly gives a flavour for state functions and it is pretty beautiful all the same! :)