# Thermodynamic equilibrium

A system is in thermodynamic equilibrium when it is in a steady state in which all macroscopic flows of matter and energy within the system disappear. Several systems are in equilibrium when the macroscopic flows between the systems disappear.

The thermodynamic equilibrium can be divided into three components. For a system to be in thermodynamic equilibrium, all conditions of thermal, mechanical and chemical equilibrium must be met.

• The thermal equilibrium assumes that there is no macroscopic heat flow in the system . This is especially the case when the temperature of the system is the same everywhere.${\ displaystyle T}$
• The mechanical equilibrium assumes that a macroscopic subsystem does no work on another subsystem. This is especially the case when the pressure of the system is the same everywhere and there are no external fields. In the case of an adjacent external field z. B. Gravitation, the pressure is not constant, but dependent on location: Barometric height formula${\ displaystyle p}$
• the chemical equilibrium assumes that the composition of the system from different phases remains the same. This is especially the case when the chemical potentials are the same (and no external field is applied). For example, if there is an electric field, the electrochemical potential must be constant instead of the chemical one so that the system is in electrochemical equilibrium.${\ displaystyle \ mu}$

## Equilibrium conditions

### Closed system

A closed system is in thermodynamic equilibrium when its entropy is at its maximum. The same applies to the differential ${\ displaystyle S}$

${\ displaystyle \ mathrm {d} S = 0}$

### Fixed volume and temperature system

For a system in which a constant volume and a constant temperature are specified from the outside ( heat bath ), the free energy is minimal; stands for the internal energy and for the number of particles . With the differential ${\ displaystyle V}$${\ displaystyle T}$ ${\ displaystyle F (T, V, N) = U (S, V, N) -TS}$${\ displaystyle U}$${\ displaystyle N}$

${\ displaystyle \ mathrm {d} F = -S \ mathrm {d} Tp \ mathrm {d} V + \ mu \ mathrm {d} N = 0}$

follows because of that , or, in the case of a mixture of several substances, the sum is. ${\ displaystyle \ mathrm {d} T = \ mathrm {d} V = 0}$${\ displaystyle \ mathrm {d} N = 0}$${\ displaystyle i}$${\ displaystyle \ textstyle \ sum _ {i} \ mu _ {i} \ mathrm {d} N_ {i} = 0}$

### Fixed pressure and fixed temperature system

For a system in which a constant pressure and a constant temperature are specified from the outside , the Gibbs free enthalpy is minimal. With the differential ${\ displaystyle p}$${\ displaystyle T}$ ${\ displaystyle G (T, p, N) = U (S, V, N) + pV-TS}$

${\ displaystyle \ mathrm {d} G = -S \ mathrm {d} T + V \ mathrm {d} p + \ mu \ mathrm {d} N = 0}$

follows because of that , or, in the case of a mixture of several substances, the sum is. ${\ displaystyle \ mathrm {d} T = \ mathrm {d} p = 0}$${\ displaystyle \ mathrm {d} N = 0}$${\ displaystyle i}$${\ displaystyle \ textstyle \ sum _ {i} \ mu _ {i} \ mathrm {d} N_ {i} = 0}$

## Thermal equilibrium

The term thermal equilibrium is used in two different contexts.

• On the one hand, in the sense used above, as the state of an individual thermodynamic system:
it is in thermal equilibrium if it can be described by a few state variables and these do not change over time.
An object in the refrigerator is e.g. B. in thermal equilibrium, because its state is clearly determined by mass , temperature , pressure and composition and remains constant over a long period of time. Boiling water, on the other hand, is not in thermal equilibrium, because a lot of information is required to describe its turbulent flow movement and it is therefore not a thermodynamic system in the strict sense.
• On the other hand, as a relationship between several systems:
two bodies that are in thermal contact with one another are in thermal equilibrium with one another if they have the same temperatures. The property of systems to be in equilibrium is an equivalence relation .
If a system A is in thermal equilibrium with both a system B and a system C, then systems B and C are also in thermal equilibrium with one another ( transitivity ). This statement forms an important basic assumption of thermodynamics and is sometimes referred to as the zeroth law of thermodynamics .

For systems in dynamic equilibrium, the virial theorem applies in the respective sub-area of ​​physics. Explicit knowledge of orbits is not required for this. A proportion of externally added energy can be compensated for by the external virial, in contrast to the internal virial of the system. But ultimately the inner is responsible for the stationarity.

## Local thermodynamic equilibrium

In thermal equilibrium, all processes are in equilibrium, u. a. also the rates of emission and absorption of radiation ( cavity radiation ).

In many cases, the emission and absorption rate but is selective: the radiation of gases and liquids over a wide wavelength range optically thin , since only certain energy states in accordance with the quantum numbers are allowed; for the radiation whose energy not to a excitation of the particles can result, gases or liquids are transparent .

With the local thermodynamic equilibrium (engl. Local thermodynamic equilibrium - abbreviation LTE ) the ratio of excited becomes non-excited molecules described, the temperature and the radiation intensity depends. In the isothermal equilibrium of radiation and molecular excitation, this relationship is described by the Boltzmann statistics . Deviations from the Boltzmann statistics are reduced by multiple impacts ; 'Hot' particles, which are not continuously supplied with energy, thermalize .

LTE is e.g. B. in the largest area of ​​the earth's atmosphere . Only at very high altitudes, where the frequency of impacts is very low due to the low pressure, do the deviations from the Boltzmann statistics become significant, and LTE is no longer available.

## Individual evidence

1. ^ Thermodynamics: Fundamentals and technical applications, Hans Dieter Baehr, Stephan Kabelac, Springer DE, 2012, ISBN 3-642-24160-3 , p. 32, Google Books