Internal energy

The internal energy is the total energy available for thermodynamic transformation processes of a physical system that is at rest and in thermodynamic equilibrium. The internal energy is composed of a variety of other forms of energy (see below), it is constant according to the first law of thermodynamics in a closed system.

The internal energy changes when the system exchanges heat or work with its environment. The change Δ $\Delta U$ of the internal energy is then equal to the sum of the heat supplied to the system and the work which is done on the system, but leaves it as a whole in a state of rest:

$\Delta U=Q+W$

The internal energy is an extensive state variable and a thermodynamic potential of the system. From the caloric equation of state of the system it follows how the internal energy is to be calculated from other state variables (e.g. pressure, temperature, number of particles, entropy, volume).

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.