Editing Mole fraction (section)

==Properties==
Mole fraction is used very frequently in the construction of [[phase diagram]]s. It has a number of advantages:
* it is not temperature dependent (as is [[molar concentration]]) and does not require knowledge of the densities of the phase(s) involved
* a mixture of known mole fraction can be prepared by weighing off the appropriate masses of the constituents
* the measure is ''symmetric'': in the mole fractions ''x''&nbsp;=&nbsp;0.1 and ''x''&nbsp;=&nbsp;0.9, the roles of 'solvent' and 'solute' are reversed.
* In a mixture of [[ideal gas]]es, the mole fraction can be expressed as the ratio of [[partial pressure]] to total [[pressure]] of the mixture
* In a ternary mixture one can express mole fractions of a component as functions of other components mole fraction and binary mole ratios:
*: <math>\begin{align}
  x_1 &= \frac{1 - x_2}{1 + \frac{x_3}{x_1}} \\[2pt]
  x_3 &= \frac{1 - x_2}{1 + \frac{x_1}{x_3}}
\end{align}</math>

Differential quotients can be formed at constant ratios like those above:
: <math>\left(\frac{\partial x_1}{\partial x_2}\right)_{\frac{x_1}{x_3}} = -\frac{x_1}{1 - x_2}</math>

or 
: <math>\left(\frac{\partial x_3}{\partial x_2}\right)_{\frac{x_1}{x_3}} = -\frac{x_3}{1 - x_2}</math>

The ratios ''X'', ''Y'', and ''Z'' of mole fractions can be written for ternary and multicomponent systems:
: <math>\begin{align}
  X &= \frac{x_3}{x_1 + x_3} \\[2pt]
  Y &= \frac{x_3}{x_2 + x_3} \\[2pt]
  Z &= \frac{x_2}{x_1 + x_2}
\end{align}</math>

These can be used for solving PDEs like:
: <math>
  \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}
</math>

or 
: <math>
  \left(\frac{\partial\mu_2}{\partial n_1}\right)_{n_2, n_3, n_4, \ldots, n_i} =
  \left(\frac{\partial\mu_1}{\partial n_2}\right)_{n_1, n_3, n_4, \ldots, n_i}
</math>

This equality can be rearranged to have differential quotient of mole amounts or fractions on one side.
: <math>
   \left(\frac{\partial\mu_2}{\partial\mu_1}\right)_{n_2, n_3} =
  -\left(\frac{\partial n_1}{\partial n_2}\right)_{\mu_1, n_3} =
  -\left(\frac{\partial x_1}{\partial x_2}\right)_{\mu_1, n_3}
</math>

or 
: <math>
   \left(\frac{\partial\mu_2}{\partial\mu_1}\right)_{n_2, n_3, n_4, \ldots, n_i} =
  -\left(\frac{\partial n_1}{\partial n_2}\right)_{\mu_1, n_2, n_4, \ldots, n_i}
</math>

Mole amounts can be eliminated by forming ratios:
: <math>
  \left(\frac{\partial n_1}{{\partial n_2}}\right)_{n_3} =
  \left(\frac{\partial\frac{n_1}{n_3}}{\partial\frac{n_2}{n_3}}\right)_{n_3} =
  \left(\frac{\partial\frac{x_1}{x_3}}{\partial\frac{x_2}{x_3}}\right)_{n_3}
</math>

Thus the ratio of chemical potentials becomes:
: <math>
   \left(\frac{\partial\mu_2}{\partial\mu_1}\right)_{\frac{n_2}{n_3}} =
  -\left(\frac{\partial\frac{x_1}{x_3}}{\partial\frac{x_2}{x_3}}\right)_{\mu_1}
</math>

Similarly the ratio for the multicomponents system becomes
: <math>
   \left(\frac{\partial\mu_2}{\partial\mu_1}\right)_{\frac{n_2}{n_3}, \frac{n_3}{n_4}, \ldots, \frac{n_{i-1}}{n_i}} =
  -\left(\frac{\partial\frac{x_1}{x_3}}{\partial\frac{x_2}{x_3}}\right)_{\mu_1, \frac{n_3}{n_4}, \ldots, \frac{n_{i-1}}{n_i}}
</math>