Does average molar mass equal sum of mole fractions times species masses?

[Big-Daddy]

This is a more general question: is it always true for a mixture that if we define M(average) such that m(total) = n(total) * M(average), where n(total) is the total number of moles of species in the mixture (summed over all species) and m(total) is the total mass of the mixture, then M(average) = Ʃ (x(Species) * M(species)) where M(species) is the relative molecular mass of that species and x(Species) is the mole fraction of that species in the mixture, and the summation occurs over all species in the mixture? Is this true for any mixture, including solid phase substances?

 

[mfb]

This is just a weighted average.
Multiply both sides of your second equation with n(total) and you get the first equation, both definitions are equivalent.

It is mathematics, so it true independent of the substances you have.