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Internal energy (thermodynamics)

Internal energy is the total microscopic energy contained within a system—kinetic and potential at the molecular level—important for heat, work, phase changes, and thermodynamic relations.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: June 1, 2026

Overview

Internal energy, usually denoted by U (or sometimes E), is the total microscopic energy contained within a thermodynamic system. It comprises the kinetic energy of particles (translational, rotational and vibrational motion) and the potential energy associated with molecular interactions, chemical bonds and electronic configurations. Internal energy excludes bulk translational kinetic energy of the system as a whole and macroscopic external potential energy due to fields acting on the entire body.

Key characteristics

Internal energy is a state function: its value depends only on the thermodynamic state of the system, not on the history of how that state was reached. It is an extensive quantity (proportional to the system size); the corresponding intensive quantity is specific internal energy (energy per unit mass). The SI unit is the joule, although calories remain in common use for heat-related contexts.

In classical thermodynamics the first law expresses energy conservation for closed systems in terms of internal energy: the change in internal energy equals heat added to the system minus work done by the system. For many idealized models, such as an ideal monatomic gas, the internal energy depends primarily on temperature. For real substances, U also varies with volume and composition because of intermolecular forces and chemical bonding.

Microscopic origin and statistical view

From a microscopic perspective, internal energy is the expectation value of the system's Hamiltonian over its microscopic states. It therefore includes contributions from molecular motions, vibrational modes, electron energies in atoms and solids, and, where relevant, radiation energy (for example, blackbody radiation confined in a cavity). In solids and metals, free electron energy and lattice vibrations (phonons) are significant parts of U.

Applications and importance

Internal energy underlies heat capacity, phase transitions, chemical reaction energetics, and the performance of heat engines and refrigerators. Changes in internal energy are measured by calorimetry and inferred in many processes using thermodynamic relations. Enthalpy, another common thermodynamic quantity, is defined as H = U + pV and is useful for processes at constant pressure.

Equilibrium, constraints and notable facts

At equilibrium and for a closed system, thermodynamic potentials take extremal values under fixed constraints: at constant entropy and volume, the internal energy is minimized. Distinguishing internal energy from kinetic or potential energy associated with the system as a whole is important when formulating conservation laws. Radiation and electronic excitation can contribute significantly to U at high temperatures.

History and conceptual development

The concept of internal energy developed in the 19th century as scientists such as James Prescott Joule, Julius Robert Mayer and others clarified the equivalence of heat and mechanical work and established energy conservation for thermal processes. That work led to the formulation of the first law of thermodynamics and to modern treatments that connect macroscopic thermodynamic quantities to microscopic statistical mechanics.

Contributions to the internal energy

Which forms of energy are taken into account when considering the internal energy depends on the type of processes that take place within the system under consideration. Forms of energy which remain constant within the framework of the processes to be considered need not be taken into account, since in any case no absolute value for the internal energy can be determined experimentally which is independent of this selection.

In the simplest case, the system under consideration consists merely of a fixed number of invariant mass points without potential energy, corresponding, for example, to a dilute noble gas at a not too high temperature. Then its internal energy is given by the total kinetic energy of the disordered motion of the particles.
For polyatomic ideal gases, the kinetic rotational energy of the molecules (see molecular rotation) and the kinetic and potential energy of their internal vibrations are added.
In the case of real gases, liquids and solids, the mutual potential energy of the particles also counts towards the internal energy. In the presence of external fields (e.g. electric field, magnetic field, gravitational field), the potential energy of the particles relative to a fixed point in the system is often included.
If chemical reactions are possible, the internal energy is extended by the energy of the chemical bonds of the atomic species involved. The internal energy of matter in the plasma state also includes the ionization energies of the molecules and atoms.
When nuclear reactions such as radioactivity, nuclear fusion or nuclear fission are considered, the nuclear binding energy belongs to the internal energy. If particle creation and annihilation also occur, such as in the early universe shortly after the big bang, the internal energy also includes the rest energy of the particles and is thus the same as total rest energy $E_{0}=M\,c^{2}$ , where is the mass of the system.
The internal energy of a cavity is given by the radiant energy present in it.

The energy resulting from the motion or from the position of the overall system (e.g. kinetic energy, position energy) is not counted as internal energy and could therefore be compared to it as external energy.

Change in thermodynamic processes

One substance type (K = 1)

The First Law of Thermodynamics describes a change in internal energy as the sum of the heat inputs and outputs and the work done on the corresponding (closed) system:

$\mathrm {d} U=\delta Q+\delta W=T\cdot \mathrm {d} S-p\cdot \mathrm {d} V$

with

the absolute temperature
the entropy
the pressure and
the volume .

For and one writes δ {\displaystyleinstead of $\delta$ each case. $\mathrm {d}$ , because they are not total differentials of a state function as in the case of the state variable , but infinitesimal changes of process variables. The last term has a negative sign because an increase in volume is associated with a release of work.

Integrated:

$\mathrm {\Delta } U=Q+W=\int {T\cdot \mathrm {d} S}-\int {p\cdot \mathrm {d} V}.$

On any closed path holds:

$\oint \limits _{c}{\mathrm {d} }U=0,$

however one $\mathrm {d} V$ chooses the differentials $\mathrm {d} S$ and

Therefore, for stationary circular processes:

${\begin{aligned}\mathrm {\Delta } U&=0\\\Leftrightarrow Q_{1}-\left|Q_{2}\right|+W_{1}-\left|W_{2}\right|&=0,\end{aligned}}$

where the energies indicated by 1 are supplied (positive) and those indicated by 2 are dissipated (negative) (cf. energy balance for cyclic processes).

For variable amount of substance or number of particles chemical potential μ also belongs to $\mu$ the total differential (fundamental equation):

$\mathrm {d} U=T\cdot \mathrm {d} S-p\cdot \mathrm {d} V+\mu \cdot \mathrm {d} N.$

Several substance types (K > 1)

Internal energy and its natural variables (entropy , volume and amount of substance ) are all extensive state variables. The internal energy changes proportionally to the corresponding state variable (S,V) when the thermodynamic system is scaled by the proportionality factor α $\alpha$ :

$U(\alpha \cdot S,\alpha \cdot V,\alpha \cdot N_{1},\dots ,\alpha \cdot N_{K})=\alpha \cdot U(S,V,N_{1},\dots ,N_{K})$

with $N_{i}$ ( $i=1,\dots ,K$ ) : substance set of particles of type .

Such a function is called a homogeneous first degree function.

Using Euler's theorem and the first law, the Euler equation for the internal energy follows:

$U=TS-pV+{\sum _{i=1}^{K}}\,\mu _{i}N_{i}.$

Equal distribution theorem for ideal gas

For an ideal gas, the equal distribution theorem holds (internal energy distributed to each degree of freedom with each ${\tfrac {1}{2}}\,k_{\mathrm {B} }T$ ).

For an ideal gas with three degrees of freedom and particles, we get:

$U={\frac {3}{2}}\ Nk_{\mathrm {B} }T$

or for moles of an ideal gas with degrees of freedom:

$U={\frac {f}{2}}\ nRT.$

each with

$k_{\mathrm {B} }$ - Boltzmann constant
- ideal gas constant.

Questions and Answers

Q: What is the symbol used to denote internal energy?

A: The symbol used to denote internal energy is U, or sometimes E.

Q: What type of energy does internal energy include?

A: Internal energy includes the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of atoms within molecules or crystals. It also includes the energy in all chemical bonds and free conduction electrons in metals.

Q: Is internal energy a state function?

A: Yes, internal energy is a thermodynamic potential and a state function of a system.

Q: What unit is used for measuring internal energy?

A: The SI unit for measuring internal energy is joules, although other historical units such as calories are still in use.

Q: How does entropy affect internal energies?

A: For a closed thermodynamic system held at constant entropy, its internal energies will be minimized.

Q: Can you calculate the internal energies of electromagnetic radiation or blackbody radiation?

A: Yes, it is possible to calculate the internal energies of electromagnetic radiation or blackbody radiation.

Q: Are food labels accurate when they list calories?

A: No, food labels are not accurate when they list calories because they actually refer to kilo-calories instead.

Author

AlegsaOnline.com - Internal energy - Leandro Alegsa

Creation date: November 13, 2022
Last updated: June 1, 2026

URL: https://en.alegsaonline.com/art/47620