# Inner energy

The internal energy is the total energy available for thermodynamic conversion processes of a physical system that is at rest and in thermodynamic equilibrium . The internal energy is made up of a multitude of other forms of energy ( see below ); according to the first law of thermodynamics , it is constant in a closed system . ${\ displaystyle U}$ The internal energy changes when the system exchanges heat or work with its surroundings . The change in internal energy is then equal to the sum of the heat supplied to the system and the work that is done on the system, but leaves it as a whole in a state of rest: ${\ displaystyle \ Delta _ {\ text {r}} U}$ ${\ displaystyle Q}$ ${\ displaystyle W}$ ${\ displaystyle \ Delta _ {\ text {r}} U = Q + W}$ The internal energy is an extensive state variable and a thermodynamic potential of the system. The caloric equation of state of the system shows how the internal energy is to be calculated from other state variables (e.g. pressure , temperature , number of particles, entropy , volume).

## Contributions to internal energy

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

• In the simplest case, the system under consideration consists only of a fixed number of unchangeable mass points without potential energy , corresponding to a dilute noble gas at a temperature that is not too high. Then its internal energy is given by the total kinetic energy of the disordered movement of the particles.
• In the case of polyatomic ideal gases, the kinetic energy of rotation 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 as internal energy. In the presence of external fields ( e.g. electric field , magnetic field , gravitational field ), that potential energy is often included that the particles have in relation to a point that is firmly defined relative to the system.
• If chemical reactions are possible, the internal energy is expanded by the energy of the chemical bonds of the types of atoms involved. The internal energy of matter in the plasma state also includes the ionization energies of molecules and atoms.
• When considering nuclear reactions such as radioactivity , nuclear fusion or nuclear fission , the nuclear binding energy belongs to the internal energy. Come particle creation and annihilation against such. B. in the early universe shortly after the Big Bang , the internal energy also contains the rest energy of the particles and is therefore the same as total rest energy , where is the mass of the system.${\ displaystyle E_ {0} = M \, c ^ {2}}$ ${\ displaystyle M}$ • The internal energy of a cavity is given by the radiation energy present in it.

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

## Change in thermodynamic processes

### One type of fabric ( K = 1)

The first law of thermodynamics describes a change in the internal energy as the sum of the heat input and extraction as well as the work performed on the corresponding (closed) system :

${\ displaystyle \ mathrm {d} U = \ delta Q + \ delta W = T \ cdot \ mathrm {d} Sp \ cdot \ mathrm {d} V}$ With

• the absolute temperature ${\ displaystyle T}$ • of entropy ${\ displaystyle S}$ • the pressure and${\ displaystyle p}$ • the volume .${\ displaystyle V}$ In each case and is written instead because it is not a question of total differentials of a state function, as is the case with the state variable, but rather infinitesimal changes in process variables . The last term has a negative sign because an increase in volume is associated with a loss of work. ${\ displaystyle Q}$ ${\ displaystyle W}$ ${\ displaystyle \ delta}$ ${\ displaystyle \ mathrm {d}}$ ${\ displaystyle U}$ Integrated:

${\ displaystyle \ mathrm {\ Delta} U = Q + W = \ int {T \ cdot \ mathrm {d} S} - \ int {p \ cdot \ mathrm {d} V}.}$ The following applies to every closed path : ${\ displaystyle c}$ ${\ displaystyle \ oint \ limits _ {c} {\ mathrm {d}} U = 0,}$ however you choose the differentials and . ${\ displaystyle \ mathrm {d} S}$ ${\ displaystyle \ mathrm {d} V}$ Therefore the following applies to stationary cycle processes :

{\ displaystyle {\ begin {aligned} \ mathrm {\ Delta} U & = 0 \\\ Leftrightarrow Q_ {1} - \ left | Q_ {2} \ right | + W_ {1} - \ left | W_ {2} \ right | & = 0, \ end {aligned}}} whereby the energies indicated with 1 are supplied (positive) and those indicated with 2 are discharged (negative) (see energy balance for circular processes ).

With a variable amount of substance or number of particles , the chemical potential also belongs to the total differential ( fundamental equation ): ${\ displaystyle n}$ ${\ displaystyle N}$ ${\ displaystyle \ mu}$ ${\ displaystyle \ mathrm {d} U = T \ cdot \ mathrm {d} Sp \ cdot \ mathrm {d} V + \ mu \ cdot \ mathrm {d} N.}$ ### Several types of fabric ( K > 1)

Internal energy and its natural variables (entropy  , volume  and amount of matter  ) are all extensive state variables. When the thermodynamic system is scaled, the internal energy changes proportionally to the corresponding state variable (S, V) with the proportionality factor  : ${\ displaystyle U}$ ${\ displaystyle S}$ ${\ displaystyle V}$ ${\ displaystyle N}$ ${\ displaystyle \ alpha}$ ${\ displaystyle U (\ alpha \ cdot S, \ alpha \ cdot V, \ alpha \ cdot N_ {1}, \ dots, \ alpha \ cdot N_ {K}) = \ alpha \ cdot U (S, V, N_ {1}, \ dots, N_ {K})}$ with ( ): Amount of substance of the particles of the type . ${\ displaystyle N_ {i}}$ ${\ displaystyle i = 1, \ dots, K}$ ${\ displaystyle i}$ Such a function is called a homogeneous function of the first degree.

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

${\ displaystyle U = TS-pV + {\ sum _ {i = 1} ^ {K}} \, \ mu _ {i} N_ {i}.}$ ### Equal distribution theorem for ideal gas

For an ideal gas , the uniform distribution law applies (internal energy distributed to each degree of freedom with each ). ${\ displaystyle {\ tfrac {1} {2}} \, k _ {\ mathrm {B}} T}$ For an ideal gas with three degrees of freedom and particles we get: ${\ displaystyle N}$ ${\ displaystyle U = {\ frac {3} {2}} \ Nk _ {\ mathrm {B}} T}$ or for moles of an ideal gas with degrees of freedom: ${\ displaystyle n}$ ${\ displaystyle f}$ ${\ displaystyle U = {\ frac {f} {2}} \ nRT.}$ each with

• ${\ displaystyle k _ {\ mathrm {B}}}$ - Boltzmann's constant
• ${\ displaystyle R}$ - ideal gas constant .