• Temperature characterizes thermal equilibrium and predicts the direction of spontaneous heat transfer between ordinary systems: from hotter to colder.
• Internal energy is a state quantity encompassing microscopic kinetic and interaction energies, excluding the system's bulk motion and external potential energy in the chosen accounting.
• Heat is energy transferred because of a temperature difference. It describes a transfer, not a substance stored inside an object.

Two objects can have the same temperature but very different internal energies. Heating is also not the only way to raise internal energy: compression and electrical work can do so. The first law keeps these routes distinct.

Why a thermometer can work

The zeroth law says that if A and B are each in thermal equilibrium with C, they are in thermal equilibrium with each other. This lets a thermometer compare temperatures without needing direct contact between every pair of objects. A thermometer must first approach equilibrium with what it measures; an immediate reading need not be the object's original temperature.

Absolute temperature in kelvin belongs in gas-law ratios. A change of 1 K equals a change of 1 °C, but 20 °C is not twice the absolute temperature of 10 °C.

Deriving gas pressure from collisions

Assume many classical particles moving randomly in a box, negligible particle volume, negligible intermolecular forces except brief elastic collisions, and equilibrium with no preferred direction. A particle of mass m, with velocity component vx, transfers momentum 2mvx on reflecting from a wall. Its round-trip time to that wall is 2L/|vx|. Average force is therefore mvx²/L.

Sum over particles and divide by wall area. With box volume V and isotropy ⟨ vx²⟩=⟨ v²⟩/3,

The ideal-gas equation of state is PV=Nk(B)T=nRT, where N counts particles, n counts moles, and k(B) is Boltzmann’s constant. Combining the two gives

Thus temperature measures mean translational kinetic energy for this model. It is not universally a synonym for “all particle motion”: solids, molecular rotations and vibrations, and interaction energies require a richer description. At equal temperature, lighter gas particles have greater root-mean-square speeds because v(rms)=√(3k(B)T/m).

A monatomic ideal gas has U=(3/2)nRT, up to a reference constant. Diatomic molecules can store rotational and, at sufficiently high temperatures, vibrational energy. Their heat capacities depend on which modes are thermally accessible.

A pressure example that forces assumptions into view

A fixed amount of ideal gas begins at 100 kPa and 300 K. Its volume is halved and its final temperature is 360 K. Then

so P₂=240 kPa absolute. The same volume reduction would only double pressure if temperature stayed constant. Gauge pressure would need conversion to absolute pressure before taking this ratio. Nothing in the ideal-gas equation alone tells us the required heat or work; that depends on the path.

Heat capacity is a response, not a temperature definition

When heat capacity is approximately constant and no phase change or other relevant work complicates the process, Q=mcΔT. For two isolated water samples of masses 0.20 and 0.30 kg, initially 80 and 20 °C, equal specific heats give

The common specific heat cancels. This assumes negligible container heat capacity, evaporation, and heat loss. During a phase change, appreciable energy can change molecular arrangement without changing temperature, so the same formula cannot be stretched through melting or boiling.

The ideal-gas model is strongest for sufficiently dilute gases away from condensation. High density and low-temperature quantum behavior expose its limits. A good model earns its usefulness by stating where it stops.