Ehrenfest equations (named after
Paul Ehrenfest
Paul Ehrenfest (18 January 1880 – 25 September 1933) was an Austrian theoretical physicist, who made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition a ...
) are equations which describe changes in specific
heat capacity
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity i ...
and derivatives of
specific volume
In thermodynamics, the specific volume of a substance (symbol: , nu) is an intrinsic property of the substance, defined as the ratio of the substance's volume () to its mass (). It is the reciprocal of density (rho) and it is related to the mol ...
in second-order
phase transitions
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
. The
Clausius–Clapeyron relation
The Clausius–Clapeyron relation, named after Rudolf Clausius and Benoît Paul Émile Clapeyron, specifies the temperature dependence of pressure, most importantly vapor pressure, at a discontinuous phase transition between two phases of matter ...
does not make sense for second-order phase transitions,
[Sivuhin D.V. General physics course. V.2. ''Thermodynamics and molecular physics''. 2005] as both specific
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
and
specific volume
In thermodynamics, the specific volume of a substance (symbol: , nu) is an intrinsic property of the substance, defined as the ratio of the substance's volume () to its mass (). It is the reciprocal of density (rho) and it is related to the mol ...
do not change in second-order phase transitions.
Quantitative consideration
Ehrenfest equations are the consequence of continuity of specific entropy
and specific volume
, which are first derivatives of specific
Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and pr ...
– in second-order phase transitions. If one considers specific entropy
as a function of
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
and
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
, then its
differential is:
.
As
, then the differential of specific entropy also is:
,
where
and
are the two phases which transit one into other. Due to continuity of specific entropy, the following holds in second-order phase transitions:
. So,
Therefore, the first Ehrenfest equation is:
.
The second Ehrenfest equation is got in a like manner, but specific entropy is considered as a function of temperature and specific volume:
The third Ehrenfest equation is got in a like manner, but specific entropy is considered as a function of
and
:
.
Continuity of specific volume as a function of
and
gives the fourth Ehrenfest equation:
.
Limitations
Derivatives of
Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and pr ...
are not always finite. Transitions between different magnetic states of metals can't be described by Ehrenfest equations.
See also
*
Paul Ehrenfest
Paul Ehrenfest (18 January 1880 – 25 September 1933) was an Austrian theoretical physicist, who made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition a ...
*
Clausius–Clapeyron relation
The Clausius–Clapeyron relation, named after Rudolf Clausius and Benoît Paul Émile Clapeyron, specifies the temperature dependence of pressure, most importantly vapor pressure, at a discontinuous phase transition between two phases of matter ...
*
Phase transition
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
References
Thermodynamic equations