Internal energy

A body contains particles that move randomly and interact with one another. Each particle has kinetic energy because of its motion. The particles also have potential energy associated with their positions relative to one another and the interactions between them.

The internal energy of the body is the sum of these microscopic energies: the kinetic energies of its particles due to their random motion and their potential energies due to their relative positions and interactions.

Internal energy is a property of the body as a whole, even though it arises from the energies of its individual particles. It does not include kinetic energy due to the motion of the whole body.

First law of thermodynamics

A system can gain or lose energy through its interactions with its surroundings. Its internal energy is the energy stored in its particles, including their kinetic and potential energy. The internal energy changes when energy is transferred by heating or by work.

Heating the system transfers energy into it. Conversely, energy transferred from the system by heating reduces its internal energy. Work transfers energy too: work done on the system increases its internal energy, while work done by the system transfers energy to the surroundings and decreases its internal energy.

The first law as an energy balance

The first law of thermodynamics accounts for all these transfers. Using the convention that energy transferred into the system is positive:

Here, $$\Delta U$$ is the change in internal energy, $$Q$$ is the net energy transferred to the system by heating, and $$W$$ is the net work done on the system. $$Q$$ is negative if there is a net transfer from the system by heating; $$W$$ is negative if the system does net work on its surroundings.

For example, if a system is heated while doing work on its surroundings, heating increases its internal energy but the work decreases it. The overall change is the balance of both transfers. This expresses conservation of energy: energy is transferred between the system and its surroundings, not created or destroyed.

Specific heat capacity

When a substance is heated without changing state, its temperature rises. The energy needed depends on the substance, its mass and the temperature rise.

The specific heat capacity $$c$$ is the energy transferred by heating per unit mass per unit temperature change, when there is no change of state. A substance with a larger specific heat capacity needs more energy to raise the temperature of the same mass by the same amount.

Energy transferred and temperature change

For a mass $$m$$ whose temperature changes by $$\Delta\theta$$, the energy transferred by heating is:

Here $$Q$$ is the energy transferred in joules, $$m$$ is the mass in kilograms and $$\Delta\theta$$ is the temperature change. The SI unit of specific heat capacity is $$\mathrm{J,kg^{-1},K^{-1}}$$. For example, a specific heat capacity of $$500\ \mathrm{J,kg^{-1},K^{-1}}$$ means that $$500\ \mathrm{J}$$ is needed to raise the temperature of $$1\ \mathrm{kg}$$ of the substance by $$1\ \mathrm{K}$$.

A temperature difference has the same numerical value in kelvin as in degrees Celsius: a rise of $$1\ \mathrm{K}$$ is a rise of $$1\ ^\circ\mathrm{C}$$. For cooling, the magnitude of the temperature decrease can be used in the equation to find the energy transferred out of the substance. The equation assumes that $$c$$ is constant over the temperature range considered.

Specific heat capacity in steady flow

For a substance flowing steadily through a heater, each portion of mass enters at one temperature and leaves at a higher temperature. The energy transferred per second depends on the mass flowing through per second, called the mass flow rate $$\dot{m}$$:

Here $$P$$ is the power transferred to the substance in watts, $$\dot{m}$$ is the mass flow rate in $$\mathrm{kg,s^{-1}}$$, and $$\Delta\theta$$ is the difference between the outflow and inflow temperatures. The relationship assumes no change of state and that the stated power is transferred to the flowing substance, with negligible energy losses.

Specific latent heat and change of state

A substance can transfer energy while changing state without changing temperature. For example, energy must be supplied to melt a solid even though its temperature remains constant during the change.

Temperature is related to the mean kinetic energy of the particles. When the temperature stays constant during a change of state, their mean kinetic energy does not change. Instead, their arrangement and interactions change, so the potential energy of the particle ensemble changes.

Energy is supplied for a change to a higher-energy state, such as melting or boiling. In the reverse change, such as freezing or condensation, energy is released and the potential energy of the particle ensemble decreases.

Specific latent heat and energy transfer

The specific latent heat, $$l$$, is the energy transferred per unit mass when a substance changes state without a temperature change. The energy transferred for a mass $$m$$ is:

Here, $$Q$$ is the amount of energy transferred in joules, $$m$$ is the mass in kilograms, and $$l$$ is the specific latent heat in $$\mathrm{J,kg^{-1}}$$. The equation gives the amount of energy supplied for a change to a higher-energy state or released in the reverse change.