# Can Le Chatelier's principle be derived?

Le Chatelier's principle says :

If a constraint (such as a change in pressure, temperature, or concentration of a reactant) is applied to a system in equilibrium, the equilibrium will shift so as to tend to counteract the effect of the constraint.

This seems as an analogue of Newton's first law of motion where we talk of inertia resisting a change in the state of rest or motion of a body. Now in physics we have General Relativity whence we can derive Newton's laws. Is there a corresponding general theory in the domain of chemical kinetics too, based on sounder mathematical principles rather than just empirical observations, whence we can derive the Le Chatelier's principle as a special case?

Any suggestions are thankfully invited.

• What kind of derivation are you looking for, exactly? Nov 11 '18 at 9:57
• One based on mathematical abstractions, like we do in the quantum theory. Nov 11 '18 at 10:04
• The equilibrium constant is derived from the chemical potential and with expressions such as the Van't Hoff isochore (temperature dependence of equilibrium constant) explains Le Chatelier. Nov 11 '18 at 10:06

Yes. To prove Le Chatelier's principle doesn't require a general theory of chemical kinetics, just an understanding of (thermodynamic!) fluctuations and response. As usual, Callen's Thermodynamics and an Introduction to Thermostatistics is a good reference. Ch. 8.5 is relevant, and I shall simply recapitulate Callen's development below.

(Incidentally, I think this principle is closer to Newton's third law of action and reaction, but in general thermodynamic laws have a very different flavor from mechanical laws because of their underlying structure. Thermodynamics deals with nebulous relationships; mechanics with definite equations of motion.)

In fact, we will prove an extra statement, the Le Chatelier-Braun principle. This principle states that the secondary effects induced by a perturbation also serve to curtail it. This augments the Le Chatelier principle, that the primary effect induced by a perturbation serves to curtail it.

I will steal Callen's example of a system immersed within a pressure and temperature reservoir, with diathermal walls and a movable domain wall with which to control its volume. The wall is moved slightly outward, causing a positive volume change $$\text{d}V$$. The primary effect is the decrease in pressure of the system, which leads to a driving force to decrease the system's volume. A secondary effect is the change in temperature $$\text{d}T$$ resulting from this change in volume, $$\text{d}T = \left(\frac{\partial T}{\partial V}\right)_S\text{d}V = -\frac{T\alpha}{Nc_v\kappa_T}\text{d}V.$$ The prefactor is unimportant; we care only about the sign of the result. All variables are positive except for $$\alpha$$, which is of variable sign. For now, we can assume it positive. The reduction of temperature in the system then drives heat flow into it, which will itself affect the pressure of the system: $$\text{d}P = \left(\frac{\partial P}{\partial S}\right)_V\frac{\text{d}Q}{T} = \frac{\alpha}{NT^2c_v\kappa_T}\text{d}Q.$$ This change in pressure is positive and diminishes the effect of the original perturbation. The same result is obtained if one takes $$\alpha$$ negative.

Let us consider a thermodynamic system with first law $$\text{d}U = f_1\,\text{d}X_1 + f_2\,\text{d}X_2.$$ It is coupled to a reservoir, itself with first law $$\text{d}U' = f_1'\,\text{d}X_1' + f_2'\,\text{d}X_2' = -f_1'\,\text{d}X_1 - f_2'\,\text{d}X_2,$$ noting that $$\text{d}X_i' = -\text{d}X_i$$, because $$X_i+X_i'$$ is assumed fixed. A perturbation $$\text{d}X_1^\text{pert}$$ drives fluctuations $$\text{d}f_1^\text{fluc} = \frac{\partial f_1}{\partial X_1}\text{d}X_1^\text{pert} \quad \text{and} \quad \text{d}f_2^\text{fluc} = \frac{\partial f_2}{\partial X_1}\text{d}X_1^\text{pert}.$$ These fluctuations in intensive quantities themselves lead to responses $$\text{d}X_1^\text{resp}$$ and $$\text{d}X_2^\text{resp}$$. The signs of these responses can be determined by minimizing the total energy of the system and reservoir given the initial perturbation, $$\text{d}(U+U') = (f_1-f_1')\,\text{d}X_1^\text{resp} + (f_2-f_2')\,\text{d}X_2^\text{resp} = \text{d}f_1^\text{fluc}\text{d}X_1^\text{resp} + \text{d}f_2^\text{fluc}\text{d}X_2^\text{resp} \leq 0.$$ Because $$X_1$$ and $$X_2$$ are independent variables, each term in the sum must be negative, and we have $$\text{d}f_1^\text{fluc}\text{d}X_1^\text{resp} \leq 0 \quad \text{and} \quad \text{d}f_2^\text{fluc}\text{d}X_2^\text{resp} \leq 0.$$ The first equation yields $$0 \geq \text{d}f_1^\text{fluc}\text{d}X_1^\text{resp} \quad \Longleftrightarrow \quad 0 \geq \text{d}f_1^\text{fluc}\frac{\partial f_1}{\partial X_1}\text{d}X_1^\text{resp} = \text{d}f_1^\text{fluc}\text{d}f_1^{\text{resp}(1)},$$ where we have multiplied both sides of the inequality by $$\partial f_1/\partial X_1$$, which must be positive by stability. $$\text{d}f_1^{\text{resp}(1)}$$ is the response of $$f_1$$ to the fluctuation due to $$X_1$$ only, and so we have the Le Chatelier principle. The second equation yields $$0 \geq \text{d}f_2^\text{fluc}\text{d}X_2^\text{resp} = \frac{\partial f_2}{\partial X_1}\text{d}X_1^\text{pert}\text{d}X_2^\text{resp},$$ and hence that $$0 \geq \frac{\partial f_1}{\partial X_1}\text{d}X_1^\text{pert}\frac{\partial f_1}{\partial X_2}\text{d}X_2^\text{resp} = \text{d}f_1^\text{fluc}\text{d}f_1^{\text{resp}(2)},$$ where we have both used a Maxwell relation and multiplied both sides of the inequality by $$\partial f_1/\partial X_1$$. This is the Le Chatelier-Braun principle.

• Note that both dT and dP as written cannot be correct since $(∂T∂V)_S = -(∂P∂S)_V$ Nov 11 '18 at 18:14