Transformation of Legendre

The transformation of Legendre is an operation which makes it possible to pass from a function of state of a system describes by a whole of variables to another function of state describing the same system using an other set of variables. In fact, this transformation makes it possible to create starting from a function of state given unsuited to the particular conditions of the system another function of state adapted better to the effective configuration of the system.

Principle

The principle is the following:

That is to say a Function of state E of the variables B, D. G whose Différentielle is written for example in the vicinity of a state defined by $B=B\_0,\; D=D\_0,\dots ,\; G=G\_0$:

$of\; =\; has\; (B\_0,\; D\_0.\; G\_0)\; dB\; +\; C\; (...)\; dD\; +\dots \; +\; F\; (...)\; dG\; ~$

that one writes formally:

$of\; =\; has\; dB\; +\; C\; dD\; +\dots \; +\; F\; dG\; ~$

If F is nonnull around the point considered $B=B\_0,\; D=D\_0,\dots ,\; G=G\_0$, then for $B,\; D.$ fixed the restriction of the function E at the variable G is locally bijective around $G=G\_0$ for $B=B\_0,\; D=D\_0,\dots$. One can then create a new function of E' state depending this time on the set of variables $(B,\; D\dots \; F)$ having a extremum for $B,\; D.$ fixed around $F=F\; (B\_0,\; D\_0,\; G\_0)$ by considering the combined Variables F and G and while posing:

$E\text{'}\; =\; E\; -\; FG\; ~$

(If one had had:

$of\; =\; AdB\; +\; CdD\; +\dots \; -\; FdG\; ~$

One would have posed:

$E\text{'}\; =\; E\; +\; FG\; ~$)

The differential of E' will be then:

$dE\text{'}\; =\; AdB\; +\; CdD\; +\dots \; +\; FdG\; -\; FdG\; -\; GDF\; ~$

Who is simplified in the form:

$dE\text{'}\; =\; AdB\; +\; CdD\; +\dots \; -\; GDF\; ~$

And we see that the new function of E' state has the same number of independent variables as the function of state E. the difference lies in the fact that E was based on the variable G whereas E' is based on the variable F.

This will be particularly useful in the case or one has the average materials to impose on the system since the outside the value of F, which is not an original variable of the system. One thus creates a function of state in which F becomes a variable of the system, that one can then fix; For example in thermodynamics, the rigid container imposes the volume of gas contained independently of its pressure (if the container is sufficiently solid!). We see with the following lines that one creates starting from the function of state " energy interns " a function of état" Enthalpy " who deals with this problem. On the contrary, in the case or the container is not a bottle but for example a deflated balloon, it is not any more the volume which is imposed outside but well the pressure. The function of state energy interns is thus adapted to deal with this problem.

On the whole, the transformation of Legendre allows in fact to modify the whole of the independent variables to have a whole of variables better adapted to the problem considered.

Example of the thermodynamic functions

Let us consider, as example, the energy interns of a system of which one of the principal differentials can be written:

$of\; =\; -\; pdV\; +\; TdS\; ~$

Who would be well adapted to a context or one controls well the Volume V and the Entropie S as independent variables

• Taking into account the presence of the term - pdV, we can make a transformation of Legendre by adding statement to the function of state U.

• Taking into account the presence of the term TdS, we can make a transformation of Legendre by removing TS with the function of state U.

By adding statement

If statement is added, one obtains a new function of state H which one calls Enthalpie.

$H\; =\; U\; +\; statement\; ~$

Its differential is then:

$dH\; =\; of\; +\; D\; (statement)\; =\; -\; pdV\; +\; TdS\; +\; pdV\; +\; VdP\; =\; Vdp\; +\; TdS\; ~$

And we see that one obtained a function of state H adapted well to a context or one controls the Pression p and the Entropie S as independent variables.

By cutting off TS

If TS is cut off, one obtains a new function of state F which one calls free energy.

$F\; =\; U\; -\; TS\; ~$

Its differential is then:

$dF\; =\; of\; -\; D\; (TS)\; =\; -\; the\; pdV\; +\; TdS\; -\; TdS\; -\; SdT\; =\; -\; pdV\; -\; SdT\; ~$

And we see that one obtained a function of state F adapted well to a context or one controls the Volume V and the Température T as independent variables like the chemical reactions with constant volume and constant temperature. It is the case if one causes a chemical reaction between several gases in a calorimetric Bombe with gas products and if one brings back the temperature of the products to the temperature which the reagents had before the reaction.

By adding statement and by cutting off TS

One can, of course, make the two operations simultaneously. This transformation of Legendre enables us to obtain another function of state G which one calls free Enthalpie.

$G\; =\; U\; +\; statement\; -\; TS\; ~$

Its differential is then:

$dG\; =\; of\; +\; D\; (statement)\; -\; D\; (TS)\; =\; -\; pdV\; +\; TdS\; +\; pdV\; +\; Vdp\; -\; TdS\; -\; SdT\; =\; Vdp\; -\; SdT\; ~$

And we see that one obtained a function of state G adapted well to a context or one controls the Pression p and the Température T as independent variables like the chemical reactions with constant pressure and constant temperature. It is the case if one causes a chemical reaction with the free air, i.e. subjected to the atmospheric pressure and if one brings back the temperature of the products at the temperature which the reagents had before the reaction.

The case of the chemical Potential

The exact differential of internal energy in its original variables is: $of\; =\; -\; pdV\; +\; TdS\; +\; \backslash \; driven\; DNN\; ~$ or NR is the number of particles of the system, which one can possibly vary; The resultant of the transformation of Legendre compared to all the natural extensive variables $V,\; S,\; NR\; ~$ are the function Grand potential defined by:

$\backslash \; Omega\; =\; U\; +\; statement\; -\; TS\; -\; \backslash \; driven\; N~$

where driven chemical Potentiel of the species considered is the ;

Additional variables

One can define generalized free enthali relating to the electric properties (Diélectricité, Ferroélectricité.), magnetic (Diamagnetism, Paramagnetism, Ferromagnetism,…) and of the chemical potentials generalized with several different chemical species exactly in the same way…

Case of the Hamiltonian formalism in Mechanical traditional

The relationship between the Lagrangian formalism and the formalism Hamiltonien in traditional mechanics evokes ways immediate the transform of Legendre. Let us leave the Lagrangian one:
$L=T-V$
and let us define $p\_i$ the combined moment of $x\_i$, which is the combined variable of $\backslash \; dowry\; x\_i$. We write then:
$=\; p\_i$
From what one defines the Hamiltonian:
$H\; =\; \backslash \; sum\_i\; \{\backslash \; dowry\; x\_i\}\; p\_i\; -\; L$
who is the opposite of the transformation of Legendre of the Lagrangian one.

Category: Thermodynamics

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