Internal Energy

Internal energy, usually denoted $U$, is the energy associated with the internal structure, microscopic motion, and interactions of a thermodynamic system. In thermodynamics, it is a property of the system’s state, rather than a measure of energy passing across its boundary. It includes microscopic kinetic and interaction energies but excludes the kinetic energy of the system’s overall motion and its potential energy arising from position in an external field. Changes in internal energy connect microscopic descriptions of matter with macroscopic measurements of heat and work. (web.mit.edu)

Microscopic meaning

For ordinary matter, internal energy includes the kinetic energy of the random motion of atoms and molecules, together with potential energy associated with their interactions. Molecular rotation and vibration, electronic excitation, and changes in chemical bonding can contribute. Which contributions matter depends on the substance, temperature range, and physical model. Internal energy is therefore broader than energy associated solely with random translational motion. (openstax.org)

In statistical mechanics, internal energy is the expectation value of the system’s microscopic energy. If microstate $r$ has energy $E_r$ and probability $p_r$, then

A quantum-mechanical description uses the energy levels of the system in this average. The macroscopic quantity does not require every constituent to possess the same energy: it represents an average over the statistical distribution of possible microscopic states. (web.mit.edu)

State dependence

Internal energy is a state function. Its change between two specified states,

is independent of the process connecting them. Heating, compression, or another sequence of operations can produce the same change in $U$, even when the amounts of heat transferred and work performed differ. For a cycle returning to the initial state, $\Delta U=0$. (openstax.org)

At thermodynamic equilibrium, internal energy can be expressed in terms of appropriate state variables and composition. For a simple compressible system, a fundamental representation is $U(S,V,\{N_i\})$, where $S$ is entropy, $V$ is volume, and $N_i$ is the number of particles of component $i$. Knowing internal energy alone generally does not specify the complete state; other independent information is required. (ocw.mit.edu)

Heat, work, and the first law

The first law of thermodynamics expresses conservation of energy. For a closed system whose overall kinetic and external potential energies do not change,

where $Q$ is net heat supplied to the system and $W$ is net work done by it. Thus heating increases $U$, while work performed by the system removes energy, unless another transfer compensates. An alternative convention takes work done on the system as positive and writes $\Delta U=Q+W_{\mathrm{on}}$. (openstax.org)

Heat and work describe transfers of energy, not separate substances or quantities stored inside matter. Heat transfer occurs because of a temperature difference; work transfers energy through other mechanisms, such as compression or mechanical stirring. Once transferred, the energy contributes to the system’s internal energy without retaining a thermodynamic label identifying how it entered. (openstax.org)

In an adiabatic process, $Q=0$, but internal energy can still change through work. An isolated system exchanges neither heat, work, nor matter, so its internal energy remains constant. Systems admitting matter require an energy balance that also accounts for energy carried across the boundary by material flows. (openstax.org)

Temperature and heat capacity

Internal energy and temperature are not interchangeable. For a fixed amount and composition of an ideal gas, internal energy depends only on temperature. For a classical monatomic ideal gas, with the energy reference chosen conventionally,

where $N$ is particle number, $n$ is amount in moles, $k_{\mathrm B}$ is the Boltzmann constant, and $R$ is the molar gas constant. Additional molecular degrees of freedom alter the temperature dependence for more complex gases. (openstax.org)

The constant-volume heat capacity of a fixed-composition system is

For an ideal gas, this gives

regardless of the path taken between the temperatures. Here $C_V$ denotes the heat capacity of the entire sample, not its molar value. At constant volume, supplied heat equals $\Delta U$ only when no other work is performed. At constant pressure, some supplied energy can instead be used in expansion work. (openstax.org)

Fundamental relation and related potentials

For equilibrium states of a simple compressible multicomponent system,

where $p$ is pressure and $\mu_i$ is the chemical potential of component $i$. Additional terms are needed when other work coordinates are relevant. This relation identifies temperature, pressure, and chemical potentials as derivatives of internal energy with respect to its natural variables. It relates neighboring equilibrium states; it does not imply that every actual process is reversible. (ocw.mit.edu)

Other thermodynamic potentials incorporate internal energy:

These are enthalpy, Helmholtz free energy, and Gibbs free energy, respectively. Their different natural variables make them useful under different constraints, particularly prescribed pressure or temperature. (old.iupac.org)

Statistical calculation

For a system in thermal equilibrium with a heat reservoir at fixed volume and particle numbers, the canonical partition function is

The microscopic probabilities are $p_r=e^{-\beta E_r}/Z$, giving

This provides a practical connection between microscopic energy levels and observable thermodynamic properties: calculating $Z$ determines internal energy and, through further differentiation, its temperature dependence and heat capacity. (ocw.mit.edu)