# 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.

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