# For a binary solution, why is the sum of the mole fractions, chi_n, of each component ALWAYS equal to ONE?

Mar 3, 2017

Because of the way we define the mole fraction..........

#### Explanation:

Let's make it simple: consider a 2 component mixture, with ${n}_{A}$ and ${n}_{B}$ moles of $A$ and $B$.

Now the total number of moles of stuff is ${n}_{A} + {n}_{B}$, but the mole fraction of $A = {\chi}_{A} = {n}_{A} / \left({n}_{A} + {n}_{B}\right)$, and likewise, ${\chi}_{B} = {n}_{B} / \left({n}_{A} + {n}_{B}\right)$,

Each $\chi$ value is dimensionless (why? because we have units of $\text{moles"/"moles}$). But for the sum of the moles fractions,

${\chi}_{A} + {\chi}_{B} = {n}_{A} / \left({n}_{A} + {n}_{B}\right) + {n}_{B} / \left({n}_{A} + {n}_{B}\right) = \frac{{n}_{A} + {n}_{B}}{{n}_{A} + {n}_{B}} = 1$ clearly.

I could do the same for ternary mixtures, or however many species there are in the mixtures. $\Sigma {\chi}_{n} = 1$ for whatever value we have for $n$.

Capisce?