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Thermodynamics, quantum mechanics and mathematical physics

Partial derivatives appear in thermodynamic equations like Gibbs-Duhem equation, in quantum mechanics as in Schrödinger wave equation, as well as in other equations from mathematical physics. The variables being held constant in partial derivatives here can be ratios of simple variables like mole fractions x in the following example involving the Gibbs energies in a ternary mixture system:

G 2 ¯ = G + (1 − x 2) (∂ G ∂ x 2) x 1 x 3 {\displaystyle {\bar {G_{2}}}=G+(1-x_{2})\left({\frac {\partial G}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}}

Express mole fractions of a component as functions of other components' mole fraction and binary mole ratios:

x 1 = 1 − x 2 1 + x 3 x 1 x 3 = 1 − x 2 1 + x 1 x 3 {\textstyle {\begin{aligned}x_{1}&={\frac {1-x_{2}}{1+{\frac {x_{3}}{x_{1}}}}}\\x_{3}&={\frac {1-x_{2}}{1+{\frac {x_{1}}{x_{3}}}}}\end{aligned}}}

Differential quotients can be formed at constant ratios like those above:

(∂ x 1 ∂ x 2) x 1 x 3 = − x 1 1 − x 2 (∂ x 3 ∂ x 2) x 1 x 3 = − x 3 1 − x 2 {\displaystyle {\begin{aligned}\left({\frac {\partial x_{1}}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}&=-{\frac {x_{1}}{1-x_{2}}}\\\left({\frac {\partial x_{3}}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}&=-{\frac {x_{3}}{1-x_{2}}}\end{aligned}}}

Ratios X, Y, Z of mole fractions can be written for ternary and multicomponent systems:

X = x 3 x 1 + x 3 Y = x 3 x 2 + x 3 Z = x 2 x 1 + x 2 {\displaystyle {\begin{aligned}X&={\frac {x_{3}}{x_{1}+x_{3}}}\\Y&={\frac {x_{3}}{x_{2}+x_{3}}}\\Z&={\frac {x_{2}}{x_{1}+x_{2}}}\end{aligned}}}

which can be used for solving partial differential equations like:

(∂ μ 2 ∂ n 1) n 2, n 3 = (∂ μ 1 ∂ n 2) n 1, n 3 {\displaystyle \left({\frac {\partial \mu _{2}}{\partial n_{1}}}\right)_{n_{2},n_{3}}=\left({\frac {\partial \mu _{1}}{\partial n_{2}}}\right)_{n_{1},n_{3}}}

This equality can be rearranged to have differential quotient of mole fractions on one side.