9702 · 16.1Internal energy

What internal energy means

0:00Take a cup of hot tea and put it on a table. It isn't moving and it isn't falling, so it has no kinetic energy and no change in gravitational potential energy that you could see. And yet, if you pour it on your hand, it can clearly do something. The tea is holding energy somewhere. Now drop an ice cube into a glass of water. The ice melts, the water around it cools, and yet the ice itself sits at zero degrees Celsius the whole time it's melting.

0:23Energy is going into the ice, but its temperature isn't going up. So where is that energy going? Both puzzles have the same answer: internal energy. In this video we'll build up exactly what internal energy is, what it's made of at the level of molecules, why it depends only on the state a system is in, and how it's connected to temperature. That last link turns out to be more subtle than most people expect.

0:44From the kinetic model of matter, you already know two key things. First, all matter is made of particles, atoms or molecules, and those particles are always moving. In a gas they fly around freely. In a liquid they slide past each other. In a solid they vibrate about fixed positions. Second, there are forces between those particles. Mostly they attract each other when they're a little way apart, and they repel strongly if you push them too close together.

1:09Moving particles have kinetic energy. And particles with forces between them have potential energy, which depends on how far apart they are, in the same way that a stretched spring stores energy depending on its extension. Hold on to those two ideas, because internal energy is built entirely from them.

Defining internal energy

1:25Here's the definition. The internal energy of a system is the sum of a random distribution of kinetic and potential energies of all the molecules in that system. Let's unpack each part of that sentence, because every word is doing work. "Kinetic energies": each molecule is moving, so each one has its own kinetic energy. In a gas that's mostly the energy of flying from place to place. In a solid it's the energy of vibrating back and forth.

1:48"Potential energies": because of the forces between molecules, each pair of molecules has some potential energy that depends on their separation. Pull molecules further apart against their attraction, and their potential energy goes up, just like lifting a book against gravity. "Sum": internal energy is the total of all of these, for every single molecule in the system. For a glass of water that's something like ten to the power twenty-five molecules, so it's a very big sum.

2:12And the word "random" is the one students most often skip, but it matters. The molecules don't all have the same speed. At any instant some are fast, some are slow, and they're moving in every possible direction. Collisions keep reshuffling who has how much energy. So the energy is spread out in a random distribution across the molecules. That randomness is what separates internal energy from the ordinary, organised energy of the object as a whole.

2:36Imagine throwing a tennis ball across a room. Every molecule in the ball is now moving forward together at the same speed as the ball. That shared, organised motion is the ball's kinetic energy as a whole object. It is not part of its internal energy. Similarly, if you lift the ball onto a high shelf, its gravitational potential energy increases, but that's energy of the whole ball in Earth's gravitational field, not energy stored in the forces between its own molecules.

3:00That's not internal energy either. Internal energy only counts the random jiggling of the molecules relative to each other, and the potential energy of the molecules due to their positions relative to each other. A tennis ball at the same temperature has the same internal energy whether it's resting on the floor, sitting on a shelf, or flying across the room.

Internal energy as a state function

3:18The syllabus says internal energy is determined by the state of the system. What does "state" mean here? The state of a system is its condition described by measurable quantities like its pressure, volume, and temperature, and whether it's solid, liquid or gas. If you know those, you know the state. The key claim is this: if two samples of the same substance are in exactly the same state, they have exactly the same internal energy.

3:41It doesn't matter what happened to them before. Here's an analogy. Think about your height above sea level when you're standing on top of a mountain. It doesn't matter whether you took the steep path, the winding path, or a cable car. Your height at the top is the same. It depends only on where you are, not on how you got there. Internal energy works the same way. Take a gas from state A to state B.

4:01You could heat it first and then compress it, or compress it first and then heat it. The route is different, but the change in internal energy is exactly the same, because it depends only on the starting state and the finishing state. This is also why, if you take a system through a complete cycle and bring it back to exactly its starting state, its internal energy ends up exactly where it began.

4:23The total change in internal energy over a full cycle is zero. You'll need that idea in the next video, on the first law of thermodynamics.

Ideal gases and internal energy

4:31For a real substance, both kinetic and potential energy contribute. But there's one important case where the potential energy part disappears completely: an ideal gas. One of the assumptions of the kinetic theory of gases is that there are no forces between the molecules except during collisions. No forces means no potential energy stored between the molecules. So for an ideal gas, the internal energy is simply the total random kinetic energy of its molecules.

4:56Nothing else. And from the previous topic, you know that the mean kinetic energy of a gas molecule is directly proportional to the thermodynamic temperature. The average kinetic energy of one molecule is three-halves k T, where k is the Boltzmann constant and T is the temperature in kelvin. Multiply that by the total number of molecules, and for an ideal monatomic gas the internal energy is three-halves N k T, which is the same as three-halves n R T, where n is the number of moles and R is the molar gas constant.

5:24That gives a very clean conclusion: for an ideal gas, internal energy is directly proportional to thermodynamic temperature. Double the temperature in kelvin, and you double the internal energy. The volume and pressure don't matter on their own. Only the temperature does. Let's make that concrete. A container holds 0.50 moles of helium, which we'll treat as an ideal monatomic gas, at a temperature of 300 kelvin. The gas is heated to 400 kelvin.

5:50Calculate the internal energy at 300 kelvin, and the increase in internal energy. Step one: since helium is being treated as an ideal gas, there's no intermolecular potential energy. So the internal energy is just the total kinetic energy of the molecules, U equals three-halves n R T. Step two: substitute. U equals 1.5 times 0.50 times 8.31 times 300. That gives about 1,870 joules. Step three: the change. Because U is proportional to T, the change in U is three-halves n R times the change in T.

6:26The temperature rises by 100 kelvin. So delta U equals 1.5 times 0.50 times 8.31 times 100, which is about 623 joules. Notice a neat check: the temperature went from 300 to 400 kelvin, an increase of one third. And the internal energy increased by 623 out of 1,870 joules, which is also one third. That's exactly what proportionality predicts. One warning: this formula only applies to an ideal monatomic gas.

6:56You can't use it for water or for a block of metal, because there the potential energy between molecules is not zero.

Temperature, internal energy and melting

7:03The second syllabus point asks you to relate a rise in temperature of an object to an increase in its internal energy. Let's be precise about what that link is. Temperature is a measure of the average random kinetic energy of the molecules. When an object's temperature goes up, its molecules are, on average, moving or vibrating faster. More random kinetic energy means more internal energy. So, for a fixed amount of a substance: a rise in temperature always means an increase in internal energy.

7:30For a solid or a liquid, heating it usually increases the potential energy part too. As the molecules vibrate harder, their average separation increases slightly. That's why most things expand when they get hotter. So both components of internal energy tend to rise together. But here's the subtle part, and it's our ice cube from the introduction. The reverse is not always true. An increase in internal energy does not always mean a rise in temperature.

7:54When ice is melting at zero degrees Celsius, energy is flowing into it, so its internal energy is increasing. But its temperature stays fixed. The energy isn't going into making the molecules move faster on average. Instead, it's being used to pull the molecules apart against the forces holding them in the rigid solid structure. The random kinetic energy stays the same, but the potential energy goes up. The same thing happens when water boils at 100 degrees Celsius.

8:20That's why you met specific latent heat in the temperature topic: latent heat is exactly the energy that increases the potential energy part of internal energy during a change of state, without any change in temperature. So the precise summary is this. Temperature rise means the random kinetic energy has increased, which means internal energy has increased. But internal energy can also increase through potential energy alone, with no temperature change at all, during melting or boiling.

Common mistakes and exam answers

8:46Let's go through the mistakes that come up most often. Mistake one: treating heat, temperature, and internal energy as the same thing. They're three different ideas. Internal energy is energy stored in the system. Temperature measures the average random kinetic energy of its molecules. And heating is a transfer of energy from a hotter region to a colder region. A system doesn't "contain heat". It contains internal energy, and heating is one way of changing it.

9:11Mistake two: thinking a hotter object always has more internal energy than a cooler one. Temperature is about the average energy per molecule, but internal energy is the total. A bath of warm water at 40 degrees has far more internal energy than a cup of tea at 80 degrees, simply because it has vastly more molecules. Mistake three: including the kinetic energy or gravitational potential energy of the whole object.

9:34Only the random, microscopic energies count. A moving car doesn't gain internal energy just from moving. It gains internal energy if its brakes or tyres get hotter. Mistake four: forgetting that for an ideal gas the potential energy term is zero. If an exam question says "ideal gas", internal energy depends on temperature only. That's a very useful shortcut, and it becomes essential in the next video. Here's a typical exam-style question.

10:00"Explain why the internal energy of an ideal gas is directly proportional to its thermodynamic temperature." How do you approach it? Build the chain of reasoning one link at a time. First link: internal energy is the sum of the random kinetic and potential energies of the molecules. Second link: in an ideal gas there are no intermolecular forces, so there's no potential energy. The internal energy is just the total random kinetic energy.

10:25Third link: the mean kinetic energy of a molecule is proportional to thermodynamic temperature. So the total kinetic energy, and therefore the internal energy, is proportional to thermodynamic temperature. Each of those links is typically worth a mark, and the second one, "no intermolecular forces so no potential energy", is the one most often missed. Another common question describes a substance melting or boiling and asks what happens to the kinetic and potential energy of the molecules.

10:52The answer: the mean kinetic energy is unchanged, because the temperature is constant. The potential energy increases, because the molecules are being separated. So the internal energy increases. Let's pull it all together. Internal energy is the sum of a random distribution of kinetic and potential energies of all the molecules in a system. The kinetic part comes from their random motion. The potential part comes from the forces between them.

11:15It does not include the kinetic or potential energy of the object as a whole. Internal energy is determined by the state of the system, so the change in internal energy between two states is the same whatever route you take, and it's zero over a complete cycle. For an ideal gas there are no intermolecular forces, so internal energy is purely random kinetic energy, and it's directly proportional to thermodynamic temperature.

11:37A rise in temperature always means an increase in internal energy, because the molecules' random kinetic energy has increased. But during a change of state, internal energy increases through potential energy alone, with the temperature staying constant. In the next video, we'll look at the two ways internal energy can actually be changed, heating and doing work, and put them together into the first law of thermodynamics.