Internal Energy and Specific Heat Capacity

The $\text{internal energy}$ of a substance is the sum of the randomly distributed kinetic and potential energies of all its particles.

The kinetic part comes from particles moving and vibrating. The potential part comes from the forces between them, and depends on how far apart they are.

$\text{Temperature}$ is a measure of the average kinetic energy of the particles, not of the total internal energy. A bath of warm water has more internal energy than a cup of boiling water, though it is at a lower temperature.

Energy flows by heating from a region of higher temperature to one of lower temperature, until they reach $\text{thermal equilibrium}$.

$\text{Absolute zero}$ is the temperature at which particles have the minimum possible internal energy. It is $0\mathrm{K}$, or $-273.15^{\circ }\mathrm{C}$.

The kelvin scale starts at absolute zero, and one kelvin is the same size as one degree Celsius. So a temperature change is numerically the same on both scales.

Any equation involving gases or particle energy needs temperatures in kelvin. Heating equations only use $\Delta \theta$, so Celsius is fine there.

The $\text{specific heat capacity}$ $c$ of a substance is the energy needed to raise the temperature of 1 kg by 1 K, with no change of state. Its unit is $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$.

Water has a very high value, about $4200\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$, which is why it is used in heating systems and why coastal climates are mild.

To measure it, heat a known mass with an electrical heater in an insulated block, measuring the energy supplied with a joulemeter or from $E=VIt$, and plot temperature against time. The gradient $\Delta \theta /\Delta t$ gives $c=\frac{P}{m(\Delta \theta /\Delta t)}$. Using the gradient avoids errors from the initial lag while the heater warms up.

During melting or boiling, energy is supplied but the temperature stays constant. The energy increases the $\text{potential}$ energy of the particles as bonds are broken or weakened; their average kinetic energy does not change.

The $\text{specific latent heat}$ $L$ is the energy needed to change the state of 1 kg without changing temperature. Fusion refers to melting; vaporisation to boiling.

Latent heat of vaporisation is much larger than of fusion. Melting only loosens the structure, but boiling separates particles completely and does work pushing back the atmosphere as the gas expands.