Heat bath

A heat bath is the idealized idea of ​​a system environment that provides a constant temperature . For this, an exchange of heat between the system and the environment must be guaranteed. This is also called thermal coupling .

In reality, a heat bath can only be implemented approximately. To do this, the surroundings must be much larger than the system itself. In addition, a coupling must be established through a suitable choice of materials for the system walls: metal e.g. B. is a good conductor of heat .

In the practice of chemistry , biochemistry and biophysics it is assumed that the constant temperature is naturally given by the surrounding medium, usually the room temperature from the atmosphere or the temperature of the living cell from an aqueous solution .

thermodynamics

In the theory of thermodynamics , the provision of a constant temperature plays an extremely important role: ${\displaystyle T}$

• If a closed system (volume and particle number constant) is thermally coupled to the heat bath, the system can be described by the canonical ensemble with the associated Boltzmann statistics . The thermodynamic potential that is minimized in this case is the free energy . ${\displaystyle N}$ ${\displaystyle F(T,V,N)}$
• If a closed system, which can change its volume (pressure and number of particles constant), is thermally coupled to the heat bath, the thermodynamic potential associated with this preparation is the free enthalpy . ${\displaystyle G(T,p,N)}$
• If a system that can exchange particles with its environment (volume and chemical potential constant) is thermally coupled to the heat bath, the thermodynamic potential associated with this preparation is the grand canonical potential . ${\displaystyle \mu }$ ${\displaystyle \Omega (T,V,\mu )}$

Statistical Physics

The statistical physics provides the following claims to a heat bath:

• the system is prepared microcanonically (the meaning of this is the definition of the temperature as a heat bath characteristic that can be calculated ab initio )
• the heat bath is infinitely large (much larger than the system that is coupled to the heat bath), so that the supply of a finite amount of energy does not change the temperature of the heat bath
• the leading term of entropy in the limit of large numbers of particles is extensive (for this reason alone the system must be very large; for small systems the entropy does not have to be extensive) ${\displaystyle S(E,V,N)}$
• the energy spectrum of the system is open to the top (otherwise negative temperatures would be possible)
• the entropy increases with increasing energy (so that the temperature increases with increasing energy)

The partial derivatives of extensive entropy according to its three extensive natural variables energy , volume and number of particles result in three intensive quantities that are characteristic of the heat bath, namely temperature , pressure and chemical potential : ${\displaystyle E}$ ${\displaystyle V}$ ${\displaystyle N}$ ${\displaystyle T}$ ${\displaystyle p}$ ${\displaystyle \mu }$

${\displaystyle {\begin{aligned}{\frac {1}{T}}&:={\frac {\partial }{\partial E}}S(E,V,N)\\{\frac {p}{T}}&:={\frac {\partial }{\partial V}}S(E,V,N)\\-{\frac {\mu }{T}}&:={\frac {\partial }{\partial N}}S(E,V,N)\end{aligned}}}$

The idealized concept of a heat bath plays a fundamental role for other ensembles of statistical mechanics, e.g. B. the canonical ensemble . The temperature of a heating bath is constant, and the first equation can be integrated at constant and trivially. With the relation to the micro - canonical partition function , this leads directly to the Boltzmann factor of the canonical ensemble. ${\displaystyle V}$ ${\displaystyle N}$ ${\displaystyle S(E,V,N)=k_{B}\ln Z(E,V,N)}$ ${\displaystyle Z}$

See also

• Normal condition
• water bath

literature

• H. Schulz: Statistical Physics. Based on quantum theory . Publisher Harri Deutsch, ISBN 3817117450