Understanding Gas Molecules and Internal Energy

What are the different kinds of energy that a gas molecule can have? Well, since the gas molecule can move around, we know it can have just regular kinetic energy. Sometimes we call that translational kinetic energy. And if it's a molecule like this, look it, a diatomic molecule, it can rotate about some center point. And so if we had a molecule built out of multiple atoms, it can rotate, and because of this, it can also have rotational kinetic energy. This would not be true if it was just monatomic. In other words, if it was just a single atom, the rotation of a single atom, it turns out is not a meaningful, significant contribution to the energy that a gas can have. But if it's diatomic or triatomic or any multiatomic atom, it can have a rotational kinetic energy as well. And you might think that's it. It can rotate, it can move around. What other kind of energy can it have? Turns out it can have one more. Again, if it's a diatomic atom like this, the atoms that make up the molecule are bonded. They can oscillate, kind of like two masses on a spring, and because of this, you can get an oscillation form of energy. This degree of freedom, we call it, is another place that the energy can go. So if you add energy to a gas, Those gas molecules are going to either start moving around faster, start rotating faster, or start oscillating faster, or some combination of all of those. Those are the three ways that energy shows up when it gets added to a gas, the three ways that energy manifests itself when you put energy into a gas. And physicists created a name for all that energy. If you add up all that energy for a gas, we call it the internal energy, and we give it the letter U. So U is the internal energy of the gas. And by internal energy of a gas, we mean all the energy, the kinetic, the rotational kinetic, the vibrational energy, all that energy added up is what we mean by the internal energy of a gas. But why am I telling you all this? I'm telling you this because I want to talk to you about the first law of thermodynamics, And the first law of thermodynamics is really an answer to the question, how do you change the internal energy of a gas? How do you increase the internal energy or decrease the internal energy? We know what's gonna happen once you increase or decrease it. The gas molecules are either gonna speed up or slow down, rotate faster, rotate slower, and so on. But how do you get the energy in there? It's usually formulated, this first law of thermodynamics is usually formulated in the context of a gas that's contained in an enclosed container, usually some sort of cylinder is the way it's shown. And in equation form, the first law looks like this. We want to know how you change the internal energy of a gas, so it looks like this delta, which represents the change in, the final value minus the initial value of the internal energy. equals, so this is the equation representation of the first law, what's gonna go on the right hand side? All the ways you can change the internal energy of a gas. One way you can do that, well, just stick a fire underneath this container. Let's say this container's closed up, and you put a fire underneath. That heat's gonna enter into the gas, and that gas will start moving around faster, start vibrating faster, rotating faster, depending on the temperature. And this heat, Heat is the first way you can change the internal energy of a gas. So if heat enters this enclosed container, the internal energy will go up. So if you add heat, and that's represented with the letter Q. Q is the letter we choose to represent the heat energy. If I add 100 joules of heat energy, that's 100 joules that can go into increasing the internal energy of a gas. But it doesn't have to be fire that you use to add heat energy to a gas. You can imagine just submerging This enclosed container in some sort of heat reservoir, maybe some boiling hot water or just warm water, and that would also add heat to the gas. You might object, you might say, wait, it's enclosed. That means nothing can get in or out. How can heat get in? Well, by enclosed we mean no particles, no molecules can get in or out. But heat's just a form of energy, so what's really happening is this heat is causing the sides of this container, the atoms and molecules that make up the container. to start vibrating faster back and forth. What happens is when this molecule collides with that faster vibrating side, it gets a kick, a boost. So every time it hits one of those faster moving molecules, it gets a boost. That's how energy's entering, but just energy's entering. There's no molecules actually entering into this gas, so it really is enclosed. Okay, so heat is the first way we talk about Internal energy of a gas changing. But there's another common way to add internal energy to a gas. So we need another term over here. And the additional way to add internal energy to a gas is imagine instead of having a container where none of the sides can move, right, where this container's completely rigid, nothing can move, imagine the top of this container being such that it has a tightly fitted piston. And this piston, imagine, can move up and down. So this piston can change the volume in which this gas gets to play in here. So since this piston can move up or down, well, what can happen? How can we add energy? You can just exert a big force downward and compress the gas into a smaller and smaller region. We said that the gas, when it hits the faster moving molecules in the wall, gets a kick. Well, the same is true here. If we take this gas molecule and imagine we're pushing the piston down. When it collides up here with this piston that's moving downward, again, it gets a kick and it starts moving faster, it gains energy, and that can manifest itself as translational kinetic energy, rotational kinetic energy, vibrational, regardless, it's all internal energy. And so, well, we're exerting a force. This force is exerted over a certain amount of distance if we're pushing this thing down, and we know what force times distance is. We're doing work. That's how we're adding energy into this gas, is work is being done. by pushing the piston down, and if we do work on the gas, that means we're adding internal energy to the gas, and the value of the work is the amount of energy we're trying to add to the internal energy of the gas. So these are the two common ways that you can add energy, internal energy, to a gas, and this is the formula version of the first law of thermodynamics. So this is it. These are the two ways you can add energy. Q is the heat, W is the work done on the gas. Now, important point here. This is the work done on the gas. And that's why I put a plus sign here, because by doing work on the gas, we're adding energy to the internal energy. We're adding energy. The gas is gaining energy because we're doing work on it. Now, some textbooks and other resources, you'll see the first law written like this, except instead of a plus sign, you'll see a minus sign, and there'll be a minus sign here instead of a plus sign because they'll be talking about the work done, instead of on the gas, they'll be talking about the work done by the gas, which is to say, if the gas were to push the piston up, it would be losing energy, it'd be doing work on its surrounding environment out here, losing that energy. that would be energy that's getting taken away from the internal energy of the gas, because we're talking about the gas in the piston here, inside the cylinder. So you've got to be careful. I've formulated it in the version with the plus sign, but you can also see this with the minus sign. And in problems with questions, you've got to be really careful. You've got to look at, is it asking you for the work done on the gas? Or is it asking you for the work done by the gas? You might think, oh my gosh, this is gonna be really hard. How do I figure out one if I know the work done on the gas? How do I figure out the work done by the gas? Well, that's really easy. The work done on the gas is equal to negative the work done by the gas. And vice versa, the work done by the gas is equal to negative the work done on the gas because if the gas does 100 joules of work, that's like someone doing negative 100 joules of work on the gas. Because if you do negative work on an object, you're actually taking energy away from it. So that's the first law of thermodynamics. It tells you how to add internal energy to a gas. Now if you're clever, you might object at this point and say, wait a minute, this isn't a new law of physics. This is just conservation of energy. It says the energy you add into a gas shows up as the energy in the gas. And what I have to say to that is, you're right. This is just conservation of energy. It's not really a new law at all. Why do we give it a special name? Well, the reason is when physicists were developing this law, it wasn't clear what heat was. It wasn't clear that heat was just another form of energy. It took a while to realize that heat is just another form of energy. For a while, there was talk about heat being some sort of weird fluid that could actually enter a substance and... Now we laugh at that, we're like, ha ha ha, what a bunch of dopes. But this is a macr, or a microscopic property, and it's hard to see, unless you have an ability to talk about these things microscopically, which took a while to do. That's not obvious what we should think about this material of heat as being. So that's why I got a special name, and the name stuck, we like it. It's the first law of thermodynamics. Now the way you use it in problems, you gotta be careful, heat can't just enter, it could also exit. So if heat enters, Q is a positive, is a positive number. So if 100 joules of heat enters, you'd plug in a positive 100 joules of heat for Q. But if heat leaves, if you stick this whole thing in an ice bath, and a certain amount of heat leaves, maybe 100 joules of heat leaves, you'd have to plug in negative 100, because this Q represents the heat added to the gas. And if you're taking heat away, well that's like negative heat being added to the gas. So if heat leaves your gas, that's a negative Q. If heat's added, That's a positive Q. How about work? This is work done on the gas. How do you know if this is positive or negative? Well, if you're pushing the piston down, well, it doesn't matter. If the piston moves down, whether you're pushing it or not, work is being done on the gas. So this is positive work done. So if the piston's moving down, positive work is being done on the gas. That means energy is entering. That's a positive value for W. Now if the piston expands, that is to say if the gas expands and the piston moves up, That's gas doing work on the piston in the environment. That's negative work done on the gas. Remember, we're talking about work done on the gas. So if piston moves down, positive work done on the gas. Gotta plug in a positive value here. If the piston moves up, that's negative value of the work done on the gas. You'd have to plug in a negative value for the work here. So be very careful. This is a way to calculate the internal energy. It's the first law of thermodynamics and one of the most fundamental and most often used equations. and all of thermodynamics.

This would not be true if it was just monatomic. In other words, if it was just a single atom, the rotation of a single atom, it turns out is not a meaningful, significant contribution to the energy that a gas can have. But if it's diatomic or triatomic or any multiatomic atom, it can have a rotational kinetic energy as well.

And you might think that's it. It can rotate, it can move around. What other kind of energy can it have?

Turns out it can have one more. Again, if it's a diatomic atom like this, the atoms that make up the molecule are bonded. They can oscillate, kind of like two masses on a spring, and because of this, you can get an oscillation form of energy. This degree of freedom, we call it, is another place that the energy can go.

So if you add energy to a gas, Those gas molecules are going to either start moving around faster, start rotating faster, or start oscillating faster, or some combination of all of those. Those are the three ways that energy shows up when it gets added to a gas, the three ways that energy manifests itself when you put energy into a gas. And physicists created a name for all that energy. If you add up all that energy for a gas, we call it the internal energy, and we give it the letter U.

So U is the internal energy of the gas. And by internal energy of a gas, we mean all the energy, the kinetic, the rotational kinetic, the vibrational energy, all that energy added up is what we mean by the internal energy of a gas. But why am I telling you all this? I'm telling you this because I want to talk to you about the first law of thermodynamics, And the first law of thermodynamics is really an answer to the question, how do you change the internal energy of a gas?

How do you increase the internal energy or decrease the internal energy? We know what's gonna happen once you increase or decrease it. The gas molecules are either gonna speed up or slow down, rotate faster, rotate slower, and so on. But how do you get the energy in there?

It's usually formulated, this first law of thermodynamics is usually formulated in the context of a gas that's contained in an enclosed container, usually some sort of cylinder is the way it's shown. And in equation form, the first law looks like this. We want to know how you change the internal energy of a gas, so it looks like this delta, which represents the change in, the final value minus the initial value of the internal energy. equals, so this is the equation representation of the first law, what's gonna go on the right hand side?

All the ways you can change the internal energy of a gas. One way you can do that, well, just stick a fire underneath this container. Let's say this container's closed up, and you put a fire underneath.

That heat's gonna enter into the gas, and that gas will start moving around faster, start vibrating faster, rotating faster, depending on the temperature. And this heat, Heat is the first way you can change the internal energy of a gas. So if heat enters this enclosed container, the internal energy will go up.

So if you add heat, and that's represented with the letter Q. Q is the letter we choose to represent the heat energy. If I add 100 joules of heat energy, that's 100 joules that can go into increasing the internal energy of a gas. But it doesn't have to be fire that you use to add heat energy to a gas.

You can imagine just submerging This enclosed container in some sort of heat reservoir, maybe some boiling hot water or just warm water, and that would also add heat to the gas. You might object, you might say, wait, it's enclosed. That means nothing can get in or out.

How can heat get in? Well, by enclosed we mean no particles, no molecules can get in or out. But heat's just a form of energy, so what's really happening is this heat is causing the sides of this container, the atoms and molecules that make up the container. to start vibrating faster back and forth. What happens is when this molecule collides with that faster vibrating side, it gets a kick, a boost.

So every time it hits one of those faster moving molecules, it gets a boost. That's how energy's entering, but just energy's entering. There's no molecules actually entering into this gas, so it really is enclosed. Okay, so heat is the first way we talk about Internal energy of a gas changing. But there's another common way to add internal energy to a gas.

So we need another term over here. And the additional way to add internal energy to a gas is imagine instead of having a container where none of the sides can move, right, where this container's completely rigid, nothing can move, imagine the top of this container being such that it has a tightly fitted piston. And this piston, imagine, can move up and down. So this piston can change the volume in which this gas gets to play in here. So since this piston can move up or down, well, what can happen?

How can we add energy? You can just exert a big force downward and compress the gas into a smaller and smaller region. We said that the gas, when it hits the faster moving molecules in the wall, gets a kick.

Well, the same is true here. If we take this gas molecule and imagine we're pushing the piston down. When it collides up here with this piston that's moving downward, again, it gets a kick and it starts moving faster, it gains energy, and that can manifest itself as translational kinetic energy, rotational kinetic energy, vibrational, regardless, it's all internal energy.

And so, well, we're exerting a force. This force is exerted over a certain amount of distance if we're pushing this thing down, and we know what force times distance is. We're doing work.

That's how we're adding energy into this gas, is work is being done. by pushing the piston down, and if we do work on the gas, that means we're adding internal energy to the gas, and the value of the work is the amount of energy we're trying to add to the internal energy of the gas. So these are the two common ways that you can add energy, internal energy, to a gas, and this is the formula version of the first law of thermodynamics.

So this is it. These are the two ways you can add energy. Q is the heat, W is the work done on the gas. Now, important point here. This is the work done on the gas.

And that's why I put a plus sign here, because by doing work on the gas, we're adding energy to the internal energy. We're adding energy. The gas is gaining energy because we're doing work on it.

Now, some textbooks and other resources, you'll see the first law written like this, except instead of a plus sign, you'll see a minus sign, and there'll be a minus sign here instead of a plus sign because they'll be talking about the work done, instead of on the gas, they'll be talking about the work done by the gas, which is to say, if the gas were to push the piston up, it would be losing energy, it'd be doing work on its surrounding environment out here, losing that energy. that would be energy that's getting taken away from the internal energy of the gas, because we're talking about the gas in the piston here, inside the cylinder. So you've got to be careful. I've formulated it in the version with the plus sign, but you can also see this with the minus sign. And in problems with questions, you've got to be really careful.

You've got to look at, is it asking you for the work done on the gas? Or is it asking you for the work done by the gas? You might think, oh my gosh, this is gonna be really hard.

How do I figure out one if I know the work done on the gas? How do I figure out the work done by the gas? Well, that's really easy.

The work done on the gas is equal to negative the work done by the gas. And vice versa, the work done by the gas is equal to negative the work done on the gas because if the gas does 100 joules of work, that's like someone doing negative 100 joules of work on the gas. Because if you do negative work on an object, you're actually taking energy away from it.

So that's the first law of thermodynamics. It tells you how to add internal energy to a gas. Now if you're clever, you might object at this point and say, wait a minute, this isn't a new law of physics. This is just conservation of energy. It says the energy you add into a gas shows up as the energy in the gas.

And what I have to say to that is, you're right. This is just conservation of energy. It's not really a new law at all. Why do we give it a special name?

Well, the reason is when physicists were developing this law, it wasn't clear what heat was. It wasn't clear that heat was just another form of energy. It took a while to realize that heat is just another form of energy.

For a while, there was talk about heat being some sort of weird fluid that could actually enter a substance and... Now we laugh at that, we're like, ha ha ha, what a bunch of dopes. But this is a macr, or a microscopic property, and it's hard to see, unless you have an ability to talk about these things microscopically, which took a while to do.

That's not obvious what we should think about this material of heat as being. So that's why I got a special name, and the name stuck, we like it. It's the first law of thermodynamics.

Now the way you use it in problems, you gotta be careful, heat can't just enter, it could also exit. So if heat enters, Q is a positive, is a positive number. So if 100 joules of heat enters, you'd plug in a positive 100 joules of heat for Q. But if heat leaves, if you stick this whole thing in an ice bath, and a certain amount of heat leaves, maybe 100 joules of heat leaves, you'd have to plug in negative 100, because this Q represents the heat added to the gas.

And if you're taking heat away, well that's like negative heat being added to the gas. So if heat leaves your gas, that's a negative Q. If heat's added, That's a positive Q. How about work? This is work done on the gas.

How do you know if this is positive or negative? Well, if you're pushing the piston down, well, it doesn't matter. If the piston moves down, whether you're pushing it or not, work is being done on the gas.

So this is positive work done. So if the piston's moving down, positive work is being done on the gas. That means energy is entering.

That's a positive value for W. Now if the piston expands, that is to say if the gas expands and the piston moves up, That's gas doing work on the piston in the environment. That's negative work done on the gas. Remember, we're talking about work done on the gas.

So if piston moves down, positive work done on the gas. Gotta plug in a positive value here. If the piston moves up, that's negative value of the work done on the gas.

You'd have to plug in a negative value for the work here. So be very careful. This is a way to calculate the internal energy.

It's the first law of thermodynamics and one of the most fundamental and most often used equations. and all of thermodynamics.

Transcript for:Understanding Gas Molecules and Internal Energy

Transcript for:
Understanding Gas Molecules and Internal Energy