Internal energy and thermal equilibrium

Choose which matter belongs to the system. Its internal energy describes its microscopic energy, while heating and work describe energy crossing its boundary.

Define the system and its state

A thermodynamic system is the matter selected for an energy account. Its boundary separates it from the surroundings. For example, a gas inside a cylinder can be the system, with the piston and the room outside it. In a calorimeter, the selected system might instead include the sample, heater and vessel together.

Internal energy U is associated with the distribution of microscopic kinetic and potential energies of the system's particles. It excludes the kinetic energy of the whole system moving together and its gravitational potential energy relative to the surroundings.

At equilibrium, the macroscopic state determines U. Specify the amount, composition and phase as well as the relevant pressure, volume and temperature. A temperature reading alone does not determine the total internal energy of every possible sample.

Internal energy is a state quantity: its change between the same initial and final states does not depend on how the change occurred. Heating Q and work W are transfers during a process. Their separate values can depend on the route, even when the initial and final states are the same. A hot object has internal energy; it does not contain a stored quantity of heat Q.

Temperature describes a particle average

In the classical particle model, thermodynamic temperature is proportional to mean microscopic kinetic energy. For an ideal gas, the precise translational result is:

T is in kelvin. The particle-energy derivation concerns an average per particle, not equal energy for every particle. Increasing the number of particles at the same temperature does not change this average, but can increase the total energy.

For a monatomic ideal gas in this model, internal energy is the total translational kinetic energy:

N is the particle number and n the amount in mol. At the same T, doubling N doubles this U. Two such samples can therefore have the same temperature and different internal energies.

Keep the monatomic ideal-gas condition beside this total-energy equation. Molecular gases and other phases can have additional microscopic energy contributions, so U = 3NkT/2 is not a universal formula for every material.

Thermal contact and equilibrium

When systems at different temperatures are put in thermal contact without a competing imposed transfer, net energy moves by heating from higher to lower temperature. They approach a common temperature. At thermal equilibrium, there is no net heating between them.

The direction of net heating follows temperature, not which object has more total internal energy. At equal temperature, microscopic exchanges may continue in both directions while their net effect is zero.

A system can also receive energy without a temperature rise. During a phase change under the stated conditions, its microscopic potential-energy contribution can change while temperature remains constant.

Use the zeroth law

If systems A and B are each in thermal equilibrium with a third system C, then A and B are in thermal equilibrium with each other. This is the zeroth law of thermodynamics.

This makes a calibrated thermometer useful for comparing temperatures. Allow it to settle in good thermal contact. An early reading may describe the sensor while the sample is still at a different temperature.

A thermometer with substantial heat capacity can itself change a small sample's temperature. It then measures their changed equilibrium. A sensor with small heat capacity, suitable contact and a stable reading reduces this disturbance; equal readings do not imply that the thermometer exchanged equal amounts of energy with different samples.