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

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Answer 1 · 2017-03-03T19:12:40

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

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 $A=chi_A=n_A/(n_A+n_B)$ , and likewise, $chi_B=n_B/(n_A+n_B)$ ,

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

$chi_A+chi_B=n_A/(n_A+n_B)+n_B/(n_A+n_B)=(n_A+n_B)/(n_A+n_B)=1$ clearly.

I could do the same for ternary mixtures, or however many species there are in the mixtures. $Sigmachi_n=1$ for whatever value we have for $n$ .

Capisce?

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