Internal energy
The hidden energy inside
- A still cup of hot tea isn't moving — yet it holds lots of energy.
- That energy is in its jiggling molecules, not in the cup's motion.
- We call it the internal energy.
What internal energy is
- $U$ = the random kinetic energy of all the molecules + the potential energy from forces between them.
- It is the energy of the random motion — not the object moving as a whole.
Internal energy is the sum of which energies?
Internal energy = random molecular KE + intermolecular PE. The object's overall motion (bulk KE) is separate.
It depends only on the state
- $U$ is fixed by the state (temperature, pressure, volume, amount) — not the path taken.
- A train rushing along has bulk kinetic energy, but that is separate from $U$.
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.
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 $U$.
Internal energy of an ideal gas
- An ideal gas has no intermolecular PE, so $U$ is purely kinetic: $U = \tfrac{3}{2}nRT$.
- So $U$ is proportional to temperature — double $T$, double $U$ (ideal gas only).
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 $U$.
During a phase change
- When ice melts or water boils, the temperature stays constant.
- But $U$ still rises: the energy breaks the bonds, raising the molecular potential energy.
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.
You've got it
- internal energy $U$ = random molecular KE + PE (not the object's bulk motion)
- $U$ depends only on the state, not the path
- ideal gas: $U = \tfrac{3}{2}nRT$ (∝ $T$); a phase change raises $U$ at constant $T$