Understanding Internal Energy in Thermodynamics: A Comprehensive Guide

In the last video we defined internal energy as literally all the energy that's in a system. That's kind of the most inclusive version,

at least in my head. So that's my system, it's some type of container. And I have a bunch of particles in here.

It's literally the sum of the kinetic energies of all these particles. If they had potential energies, we'd

throw that in there. If they have electrical potential, we'd throw that in there.

If they have bonds with each other, the energies in that bond, we would throw it in there. It's everything.

It's all inclusive. And, I told you in the last video it's unintuitive that U stands for internal energy.

But I kind of think that, you know, U, it contains the universe of energy. You know, that's just for me to memorize it.

Don't think too deeply into what I just said. But this is internal energy. If you show me a system, it has some internal energy.

It's a function of its state. I don't care how it got to that state, but if you told me a system at a certain state, maybe with a certain pressure,

or a certain volume, or at a certain temperature-- although if you give me pressure and volume that should be enough-- I can tell you what its internal energy is.

Especially if you tell me the type of molecule I have, and things like that. Now we also said in the last video that because internal

energy is all the energy in the system, it can't be randomly created or destroyed. It can just be transformed from one form to another.

So if I have a change in internal energy, it can only be due to-- well, it can be due to more than what I'm describing.

But in our simple world where all of the energy in a system-- and maybe we're dealing with gases, because that's normally what you deal with in a first-year chemistry

course-- it's going to be the heat that could be added to the system, plus the work done on the system. And like I said in the last video, sometimes they'll say,

instead of plus the work done on the system, sometimes they'll say, minus the work that the system does. Either way.

And here I just want to make another side discussion here, because I decided to write it without the little deltas here.

And the reason why I did that is if you were to write this equation, which you will see, you'll see it in textbooks, teachers will do it-- nothing inherently criminal about

that-- but I just do that because it clears in my mind what heat and what work are relative to internal energy. If I were to write delta U is equal to heat delta Q plus

delta W to me this implies that at some point I had some amount of heat in my system, and then I have a different amount of heat in my system, and I took the difference

between the two and I got a change in heat. So this implies that heat is somehow an inherent macrostate of the system, and that's not the case.

I can tell you what the internal energy of this system is. I can tell you its pressure.

I can tell you its volume. I can tell you its temperature. These are all macrostates of the system.

I don't know how it got to this situation, but I can tell you about it. I cannot tell you what the heat is of this system.

And that might be a little unintuitive, because if I ask you, hey I have a cup of coffee, what's the heat of it? You might say, oh, it's, you know, you might give me the

temperature. Because in our everyday language we use things like heat and temperature interchangeably.

But in thermodynamics, heat is a transfer of energy. A way to think about it is, if internal energy is your bank account.

So you could say change in bank account. And it really is like the energy bank account of a system.

If you have a change in a bank account, you had some deposits or withdrawals. And heat and work are really just deposits or withdrawals

into our energy bank account. Heat is one kind. Maybe heat is like a wire transfer.

So you could say it's wire transfers to your account. Transfers to your account plus check deposits. Now it makes a lot of sense to say, you could say, what is my

value of my bank account? Or you could say what is my change in my bank account? You take two snapshots of your bank account at

two different times. But would it make sense to say-- you know, I could say I wire transferred $10, right?

So I could say, this statement would be plus $10. And I could say that I wrote $20 in checks, minus $20 in checks.

In which case, my change in my bank account would be minus $10. Now would it make sense for me to say

change in wire transfer? That implies that when I started off maybe my bank account had $100 in it and now it has $90.

When my bank account had $100 in it, did it have any wire transfer amount? No, wire transfer was how money was deposited or taken

away from my account. Likewise, it didn't have a checking deposit account, so I can't really-- it just seems weird to me to say change in

wire transfer is $10, or change in check deposits is $20 or minus $20. Would you say, I made a $10 wire transfer and

I paid $20 in checks. So I had a net change in my bank account of $10. Likewise, I say how much work was done to me or I did, which

is essentially a deposit or withdrawal of energy. Or I could say much heat was given to me or how much heat was released, which is another way of depositing or

withdrawing energy from my energy bank account. So that's why I like to stick to this. And I like to stay away from this.

And just like I said, you can't say how much heat is in the system. So someone will say, oh, how much heat is in this?

You cannot say that. There's no heat state variable for that system. You have internal energy.

The closest thing to heat, we'll talk about it in a future video, is enthalpy. Enthalpy is essentially a way of measuring how much heat is

in a system, but we won't talk about that just now. And you can't say how much work is in a system. You can't say, oh I have x amount of work in a system.

The system can do work or have work done to it, but there's not a certain amount of work, because that energy in the system could be all used for work, it could all be used for

heat, it could all be used for a ton of different things. So you can't say those things. And that's why I don't like treating them like state

variables, or state functions. So with that said, this is our definition. Let's do a couple of simple problems,

just to give you intuition. And I really want to make you comfortable. My real goal is to make you comfortable with, when to know

to use plus or minus on the work. And the best way to do it is not to memorize a formula, but just to kind of think about what's happening.

So let's say that I have some system here and, I don't know, it's a balloon. And let's say that I have no change in internal energy.

Internal energy is 0. And for our purposes we can kind of view it as the kinetic energy of the particles haven't changed.

And let's say by expanding a bit, by my balloon expanding a bit-- I did some work. I'll do this in more detail in the next video.

So the system does 10 joules of work. My question is, how much heat was added or taken away from the system?

So the way I can think about it-- you don't even have to write the formula down, or you can write it-- you could say, look, the internal energy, the amount of energy in the system

hasn't changed. The system did 10 joules of work. So that's energy going out of the system.

It did 10 joules. So 10 joules went out in the form of work. If the internal energy did not change, then essentially 10

joules of energy had to go into the system. 10 joules had to go into the system. If it didn't, the internal energy would have gone down by

the amount of work we had. And the only way, if is this is the net work, the only way that we're talking about, that we can add energy outside of

work, is through heat. So 10 joules of heat must have been added. So we can write 10 joules of heat added to system.

Now let's look at that from the point of view of the actual formula. If we have delta U is equal to heat added, plus work done to

the system, then we would say, OK, this was 0. It's equal to the heat added to the system. Remember, in this way we're saying this is the heat done,

or the work done, to the system. W is work done. Now, the system did work to something else.

It didn't have work done to it. So if this is work done to the system, and it did work, then this is going to be a minus 10.

Minus 10 joules. And then you solve both sides of this. You add 10 to both sides and you get 10 is equal to Q,

which is exactly what we got up here. But this can get confusing sometimes, because you're like, oh, is this heat that the system did?

Is this heat that was added to the system or taken away? The convention tends to be that this is heat added. But then sometimes it's confusing.

Is this the work done to the system or work done by? And that's why I like just doing it this way. If the system does work it loses energy.

If the system has work done to it, it gains energy. So let's do another problem. I mean, I could have done this exact same thing using the

other formula that you'll sometimes see. Delta U is equal to Q minus the work that the system does by the system-- Work done by the system.

And in this case, once again, change in internal energy was 0. That is equal to heat added to the system,

minus the work done. So minus-- I told you at the beginning of the problem that the system did 10 joules of work-- so minus the work done.

Minus 10 joules. We get the same situation up here from two different formulas.

And we got 10 is equal to Q. Either way, the heat added to the system is 10 joules. Let's do one more.

Let's say that, I don't know, 5 joules of heat taken away from system. And let's say that 1 joule of work done on system.

So maybe we're compressing the balloon on the system. What is our change in internal energy? Let's just figure out our change.

So the way I think of it is 5 joules of heat taken away from the system, that's going to reduce our internal energy by minus 5.

And if 1 joule work is done onto the system, we're putting energy into it so that'll be plus 1. So minus 5, plus 1, is going to be minus 4.

Or enter our change in internal energy is minus 4 joules. Now we could have done that a little bit more formally with

the formula, change in internal energy is equal to heat added to the system, plus work done on the system. So it's equal to heat added to the system.

We had 5 joules heat taken away, so this is minus 5, plus work done on the system. We have 1 joule of work done on the system, and there we

get, once again, minus 4. Now, I could have written that same formula the other way. I could have said change in internal energy is equal to

heat added to the system minus the work that the system does. I want to do this both ways, because I don't want you to get confused, either way you see it on problems or in

school, or maybe you take one class that does it one way and one class that does it the other. If you use it this way, what was the heat

added to the system? We had heat taken away, so it's minus 5. So minus the work that the system does, how much work

does the system do? Well, the system had 1 joule of work done to it. So it did itself minus 1 joules of work.

So it's minus minus 1. I want to be clear, when I use this formula, this is work done by system, done by.

This is work done to. The system had work done to it. So it had minus 1 joules of work done by the system, so

these become pluses, and you get back to a minus 4. Do whatever's intuitive for you. For me, this tends to be the most intuitive, where you

don't even use a formula. We just think, if I'm doing work, I'm using up energy. If I get work done to me, I'm being handed over energy.

If I have heat taken away from me, I'm giving away energy. If I have heat added to me, I'm getting energy. See you in the next video.