Internal energy
| English | Chinese | Pinyin |
|---|---|---|
| internal energy | 内能 | nèi néng |
| translational | 平动 | píng dòng |
| rotational | 转动 | zhuǎn dòng |
| vibrational | 振动 | zhèn dòng |
| ideal gas | 理想气体 | lǐ xiǎng qì tǐ |
| intermolecular | 分子间 | fèn zǐ jiān |
A cup of tea that is going nowhere, fast
- A cup of tea sits still on the table. Nothing about it is moving, and yet its molecules are travelling at hundreds of metres per second, in every direction at once.
- The velocities cancel, which is why the cup stays put. The energies do not cancel, because kinetic energy has no direction to cancel with.
- That energy, invisible from outside, is what makes the tea hot and what a kettle had to pay for.
- This lesson is internal energy 内能: what it is made of, what it depends on, and what happens to it when a temperature does not change.
What internal energy is made of
- The internal energy $U$ of a system is the sum of the random distribution of the kinetic and potential energies of its molecules.
- The kinetic part: molecules fly through space (translational 平动), and unless they are single atoms they also spin (rotational 转动) and shake (vibrational 振动).
- The potential part: the energy stored in the forces between molecules, which depends on how far apart they are.
- Both marks of the two-mark definition need "sum of kinetic and potential energies" and the word random. Leaving out "random" describes the energy of a moving object instead.
The spread of molecular energies
Internal energy is the total random kinetic + potential energy of the molecules. Heat the gas and the whole speed distribution shifts to higher energy.
Internal energy is the sum of which energies?
Internal energy = random molecular KE + intermolecular PE. The object's overall motion (bulk KE) is separate.
Which is the full two-mark definition of internal energy?
Both marks need "kinetic and potential" and the word "random". Without "random" the phrase describes a moving object's energy instead.
Random, not bulk
- $U$ is a sum over the molecules, not the kinetic energy of the object moving as a whole.
- A tanker of gas driving down a motorway has a great deal of bulk kinetic energy. Its internal energy is exactly the same as when it was parked, because the random molecular motion has not changed.
- Stop the tanker and the bulk kinetic energy becomes heat, which then does raise $U$. The two are separate quantities that can be converted into one another.
One of these has internal energy; both have energy
A moving train's bulk kinetic energy counts as part of its internal energy.
No — internal energy is the energy of the random molecular motion, not the whole object moving along.
A tanker of gas driving along a motorway has more internal energy than the same tanker parked, at the same temperature.
Bulk motion is not internal energy. The random molecular motion is unchanged, so U is the same; the lorry simply also has kinetic energy.
A state function
- $U$ is determined by the state of the system: its temperature, pressure, volume and amount of substance.
- It does not depend on the path taken to reach that state. Compress a gas and then heat it, or heat it and then compress it: if the final state is the same, $U$ is the same.
- That is what makes the first law useful in the next lesson: $\Delta U$ can be found from the endpoints alone, however complicated the journey.
Internal energy depends only on the state of the system, not on the path taken to reach it.
Yes — $U$ is a function of state (T, p, V, amount); two routes to the same state give the same $U$.
Temperature and internal energy
- Raising an object's temperature raises the random kinetic energy of its molecules, and so raises its internal energy.
- For an ideal gas 理想气体 the intermolecular 分子间 forces are ignored, so the molecular potential energy is zero and the internal energy is purely kinetic. With $\tfrac32 kT$ per molecule:
- So for an ideal gas $U$ is directly proportional to the thermodynamic temperature. Double $T$ and you double $U$. This is exact only for an ideal gas.
For an ideal gas, the internal energy is:
No intermolecular PE, so $U$ is all kinetic: $U = \tfrac{3}{2}NkT = \tfrac{3}{2}nRT$.
For an ideal gas, if the absolute temperature doubles, the internal energy multiplies by:
$U = \tfrac{3}{2}nRT \propto T$, so doubling $T$ doubles $U$.
Worked example: a gas heated at constant pressure
- Sketch how the internal energy of a fixed mass of ideal gas varies with its volume as it is heated at constant pressure.
- At constant pressure, Charles's law gives $V \propto T$. For an ideal gas, $U \propto T$.
- Therefore $U \propto V$: a straight line through the origin.
- The origin is on the line because at absolute zero both the extrapolated volume and the internal energy are zero. Say why the line passes through the origin; that is usually the second mark.
A fixed mass of ideal gas is heated at constant pressure. What does a graph of internal energy against volume look like?
At constant pressure V is proportional to T, and for an ideal gas U is proportional to T, so U is proportional to V. It passes through the origin because both extrapolate to zero at absolute zero.
Phase changes
- When ice melts or water boils, the temperature does not change, yet energy is still being supplied.
- The kinetic energy is unchanged, because the temperature is unchanged. The potential energy rises, as the bonds between molecules are broken and their separation grows.
- So $U$ increases at constant temperature, by the latent heat supplied. This is the clearest case where internal energy and temperature come apart.
While water boils at constant temperature, its internal energy:
The temperature (and so KE) is unchanged, but energy goes into breaking bonds — raising the molecular PE, so $U$ rises.
Worked example: name both energies, every time
- A three-mark "describe and explain, with reference to molecular kinetic and potential energies" answer says what happens to each:
- A gas heated at constant volume: the kinetic energy increases, because temperature measures mean molecular kinetic energy; the potential energy is unchanged, since the separation of the molecules does not change; so $U$ increases.
- A wire stretched within its elastic limit at constant temperature: the kinetic energy is unchanged, same temperature; the potential energy increases, because the atoms are pulled further apart against the interatomic forces; so $U$ increases.
- Ice melting at $0\ ^\circ\text{C}$: the kinetic energy is unchanged; the potential energy increases as bonds break and separations grow; so $U$ increases by the latent heat supplied.
Match each change to what happens to the molecular kinetic and potential energies.
Temperature changes the kinetic part; changing the separation of the molecules changes the potential part. Name both in every answer.
Marks that slip away
- The definition needs random and both energies. "The energy of the molecules" is not enough.
- Bulk motion is not internal energy. A moving object's kinetic energy is separate from $U$.
- $U = \tfrac32 nRT$ holds for an ideal gas only, because only there is the molecular potential energy zero.
- At a phase change the temperature is constant but $U$ rises, through the potential term. Say which energy changes and which does not.
You've got it
- internal energy is the sum of the random distribution of the kinetic and potential energies of a system's molecules
- it is a property of the state, not of the path, and it is separate from the bulk kinetic energy of a moving object
- for an ideal gas the potential term is zero, so $U = \tfrac32 nRT$ and $U \propto T$ exactly
- a phase change raises $U$ at constant temperature by raising the potential energy as bonds break