Internal energy

Written $U$, measured in joules. Essential knowledge 9.4.A.1 gives the general version in two terms: kinetic, plus the potential energy of the configuration.

For an ideal gas the second term disappears. 9.4.A.1.i says the atoms do not interact through conservative forces and their internal structure is not considered, so an ideal gas has no internal potential energy. What survives is the total kinetic energy of the atoms:

The CED attaches that expression to an ideal monatomic gas, at 9.4.A.1.ii, while the sheet prints it with no qualifier. Diatomic molecules also store energy in rotation and vibration, so the factor of $\frac{3}{2}$ does not describe them.

Three consequences:

• $U$ depends only on $T$ for an ideal gas. Not on volume, not on pressure separately, not on the route taken. Same temperature, same internal energy.
• So zero temperature change gives zero internal energy change, which is the whole of the isothermal case.
• $U$ is extensive. Halve the sample and $U$ halves while temperature holds still. That is why two systems at equilibrium have equal temperatures and unequal internal energies.

Combining $U = \frac{3}{2}nRT$ with $PV = nRT$ gives $U = \frac{3}{2}PV$, so a single point on a pressure-volume diagram fixes the internal energy.