In Thermodynamic modern, the characteristic functions of state pure Substance can be expressed by means of many relations.

The functions of state are: the Pressure p , the Temperature T , the Volume V , the energy interns U , the Entropie S , the Enthalpie H , the free energy F and the free Enthalpie G .

Between these partial functions and their Derived S, one can establish tens of formulas: there exist 28 expressions of the functions of state, therefore 56 derivative partial.

Actually, only some have a scientific and practical interest.

Thermoelasticity

The first important formula is intrinsically related to the equation of state:

$f\; \backslash \; left\; (p,\; V,\; T\; \backslash \; right)\; =\; 0$

It can be interpreted like the equation of a Surface in a vector Space with three Dimension S.

The practice is to define the tangent plan in surface in a point has (called a “state” has). For example, if the easily measurable quantity is considered:

$\backslash \; ln\; \backslash \; frac\; \{V\}\; \{NR\}\; =\; F\; (p,\; T)$

One can deduce the 2 thermoelastic coefficients from them from the gradient:

• $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; p\}\; =\; -\; \backslash \; chi\_T\; (p,\; T)$

The number $\backslash \; chi\_T$, positive, is called “thermoelastic coefficient of isothermal compressibility”, homogeneous contrary to a Pression, connected in Physique statistics to the fluctuations in pressure, $1\; \backslash \; over\; \backslash \; chi\_T\; (p,\; T)\; \backslash ,$ (in Pascals).

• $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; T\}\; =\; \backslash \; alpha\; (p,\; T)$

The number $\backslash \; alpha$ is called “thermoelastic coefficient of isobar dilation”, which is often positive, with notable exceptions like the Eau liquid between 0 °C and 4 °C and the Bismuth.

One defines also the coefficient β:

$\backslash \; frac\; \{1\}\; \{P\}\; \backslash \; frac\; \{\backslash \; partial\; p\; (V,\; T)\}\{\backslash \; partial\; T\}\; =\; \backslash \; beta\; (V,\; T)$

called “thermoelastic coefficient of relative increase in isochoric pressure” (to be preferred with “isochoric dilation”).

There exists a relation between β, α and χ _T :

$\backslash \; alpha\; =\; \backslash \; beta\; \backslash \; cdot\; (p\; \backslash \; cdot\; \backslash \; chi\_T)\; \backslash ,$

• α is related to the Loi of Gay-Lussac;
• β is related to the Loi of Charles;
• χ _T is related on the Loi of Mariotte and the Expériences of Amagat.

Adiabatic compressibility is defined:

$\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; p\}\; =\; -\; \backslash \; chi\_S\; (p,\; S)$

with

$\backslash \; frac\; \{V\}\; \{NR\}\; =\; G\; (p,\; S)\; \backslash ,$

One then shows the Formule of Reech:

$\backslash \; frac\; \{\backslash \; chi\_T\}\; \{\backslash \; chi\_S\}\; =\; \backslash \; gamma\; >\; 1\; \backslash ,$

Relations of Mayer

One defines the Enthalpie:

H = U + statement

This relation suggests a bond between the thermoelastic C_p , C_V and coefficients. The Relation of Mayer establishes it .

Reciprocally, the formula of Reech suggests another relation of the same type:

$C\_p=\; \backslash \; alpha\; T\; \backslash \; cdot\; V\; \backslash \; frac\; \{\backslash \; partial\; p\; (S,\; T)\}\{\backslash \; partial\; T\}$

This relation utilizes P = ƒ ( S , T ), which is more difficult to obtain.

In the same way:

$C\_V\; =\; -\; \backslash \; beta\; T\; \backslash \; cdot\; p\; \backslash \; frac\; \{\backslash \; partial\; V\; (S,\; T)\}\{\backslash \; partial\; T\}$

With constant entropy, volume V can decrease when the temperature T increases (if α T is positive), which justifies the negative sign.

There exist relations similar to that of Mayer for the coefficients of answer to the pressure, i.e. for χ _T and χ _S .

Relations of Clapeyron

The “coefficient of latent heat of dilation”, noted ℓ, is worth:

$\backslash \; ell\; =\; T\; \backslash \; cdot\; \backslash \; frac\; \{\backslash \; partial\; p\; (V,\; T)\}\{\backslash \; partial\; T\}\; =\; p\; (\backslash \; beta\; \backslash \; cdot\; T)$

The “coefficient of latent heat of compression”, noted H , is worth:

$H\; =\; -\; T\; \backslash \; cdot\; \backslash \; frac\; \{\backslash \; partial\; V\; (p,\; T)\}\{\backslash \; partial\; T\}\; =\; -\; V\; (\backslash \; alpha\; \backslash \; cdot\; T)$

These relations were found very early. Their experimental checking by Thomson and Joule will constitute in fact the first test showing a contradiction in the “theory of heating”, calling the construction of a science thermodynamic, different, which is at the origin of that one knows today.

Demonstration

To establish ℓ, it is observed that the gradient of G ( p , T ) has a Rotationnel no one. To establish H , it is observed that the gradient of F ( V , T ) has rotational no one.

Formulas of Maxwell

The formulas of Maxwell relate to the derivative second functions of state, therefore the derivative first of the calorimetric coefficients.

One obtains thus that the derivative partial of C_p ( p , T ) compared to p is related to the equation of state. Therefore, only its value in extreme cases of the null pressures C_p (0, T ) = C ⁰ ( T ) is “free choice”: one thus studies in experiments much this area of the space of the states, to rebuild a characteristic Fonction real gas.

It is the same for C_V ( V , T ) for V tending towards the infinite one. One knows already by the relation of Mayer which this limit is worth C ⁰ ( T ) - R .

The experimental study of a gas pure substance is thus summarized with:

• To determine the equation of state;
• To determine C ⁰ ( T ).

One can then rebuild the free function of enthalpy G ( p , T ) which characterizes all the thermodynamic properties of real gas.

Other relations

There exists in Thermomagnétisme similar relations (one defines for example adiabatic demagnetization).

See too

• Thermodynamic;
• Heat;
• characteristic Function;
• Function of state;
• Equation of state.

References

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