Editing Le Chatelier's principle (section)

==Thermodynamic statement==
Le Chatelier–Braun principle analyzes the qualitative behaviour of a thermodynamic system when a particular one of its externally controlled state variables, say <math>L,</math> changes by an amount <math>\Delta L,</math> the 'driving change', causing a change <math>\delta_{\mathrm i} M,</math> the 'response of prime interest', in its conjugate state variable <math>M,</math> all other externally controlled state variables remaining constant. The response illustrates 'moderation' in ways evident in two related thermodynamic equilibria. Obviously, one of <math>L,</math> <math>M</math> has to be [[Intensive and extensive properties|intensive]], the other [[Intensive and extensive properties|extensive]]. Also as a necessary part of the scenario, there is some particular auxiliary 'moderating' state variable <math>X</math>, with its conjugate state variable <math>Y.</math> For this to be of interest, the 'moderating' variable <math>X</math> must undergo a change <math>\Delta X \ne 0</math> or <math>\delta X \ne 0</math> in some part of the experimental protocol; this can be either by imposition of a change <math>\Delta Y</math>, or with the holding of <math>Y</math> constant, written <math>\delta Y = 0.</math> For the principle to hold with full generality, <math>X</math> must be extensive or intensive accordingly as <math>M</math> is so. Obviously, to give this scenario physical meaning, the 'driving' variable and the 'moderating' variable must be subject to separate independent experimental controls and measurements.

===Explicit statement===

The principle can be stated in two ways, formally different, but substantially equivalent, and, in a sense, mutually 'reciprocal'. The two ways illustrate the Maxwell relations, and the stability of thermodynamic equilibrium according to the second law of thermodynamics, evident as the [[Entropy (energy dispersal)|spread of energy]] amongst the state variables of the system in response to an imposed change.

The two ways of statement differ in their experimental protocols. They share an ''index protocol'' (denoted <math>\mathcal {P}_{\mathrm i}),</math> that may be described as 'changed driver, moderation permitted'. Along with the driver change <math>\Delta L,</math> it imposes a constant <math>Y,</math> with <math>\delta _{\mathrm i} Y = 0,</math> and allows the uncontrolled 'moderating' variable response <math>\delta _{\mathrm i} X,</math> along with the 'index' response of interest <math>\delta _{\mathrm i} M.</math>

The two ways of statement differ in their respective ''compared protocols''. One form of ''compared protocol'' posits 'changed driver, no moderation' (denoted <math>\mathcal {P}_{\mathrm n}).</math> The other form of ''compared protocol'' posits 'fixed driver, imposed moderation' (denoted <math>\mathcal {P}_{\mathrm f}.</math>)

====Forced 'driver' change, free or fixed 'moderation'====

This way compares <math>\mathcal {P}_{\mathrm i}</math> with <math>\mathcal {P}_{\mathrm n},</math> to compare the effects of the imposed the change <math>\Delta L</math> with and without moderation. The protocol <math>\mathcal {P}_{\mathrm n}</math> prevents 'moderation' by enforcing that <math>\Delta X = 0</math> through an adjustment <math>\Delta Y,</math> and it observes the 'no-moderation' response <math>\Delta M.</math>  Provided that the observed response is indeed that <math>\delta_{\mathrm i} X \ne 0,</math> then the principle states that <math>|\delta _{\mathrm i} M| < |\Delta M|</math>.

In other words, change in the 'moderating' state variable <math>X</math> moderates the effect of the driving change in <math>L</math> on the responding conjugate variable <math>M.</math><ref>Münster, A. (1970), pp. 173–176.</ref><ref>Bailyn, M. (1994), pp. 312–318.</ref>

====Forcedly changed or fixed 'driver', respectively free or forced 'moderation'====

This way also uses two experimental protocols, <math>\mathcal{P}_{\mathrm i}</math> and <math>\mathcal{P}_{\mathrm f}</math>, to compare the index effect <math>\delta _{\mathrm i} M</math> with the effect <math>\delta _{\mathrm f} M</math> of 'moderation' alone. The ''index protocol'' <math>\mathcal{P}_{\mathrm i}</math> is executed first; the response of prime interest, <math>\delta _{\mathrm i} M,</math> is observed, and the response <math>\Delta X</math> of the 'moderating' variable is also measured. With that knowledge, then the ''fixed driver, imposed moderation protocol'' <math>\mathcal{P}_{\mathrm f}</math> enforces that <math>\Delta L = 0,</math> with the driving variable <math>L</math> held fixed; the protocol also, through an adjustment <math>\Delta _{\mathrm f} Y,</math> imposes a change <math>\Delta X</math> (learnt from the just previous measurement) in the 'moderating' variable, and measures the change <math>\delta _{\mathrm f} M.</math> Provided that the 'moderated' response is indeed that <math>\Delta X \ne 0,</math> then the principle states that the signs of <math>\delta _{\mathrm i} M</math> and <math>\delta _{\mathrm f} M</math> are opposite.

Again, in other words, change in the 'moderating' state variable <math>X</math> opposes the effect of the driving change in <math>L</math> on the responding conjugate variable <math>M.</math><ref>Bailyn, M. (1994), p. 313.</ref>