I continue to receive regular requests for the Excel spread I developed that calculates alloy densities. We have a new improved (looks more attractive) version I can send to folks that want it. Send me a note at firstname.lastname@example.org if you would like a copy.
Recently someone requested the derivation. Here tis:
We want to find the density of an alloy composed of 3 metals. Assume the mass of the alloy is M. Metal A has a mass ma and a density da, Metal B has a mass mb and a density db and Metal C has a mass mc and a density dc. The total volume, V, of the 3 metals is va + vb+ vc, however since v = m/d, the total volume can be expressed:
1) V = ma/da + mb/db +mc/dc The density of the resulting alloy is D = M/V, hence 1/D = V/M, therefore:
2) 1/D = V/M = (ma/M)/da + (mb/M)/db +(mc/M)/dc
Now ma/M is the mass fraction of a, which we will call Xa, and similarly Xb and Xc for metals B and C.
Equation 2 then becomes:
1/D = Xa/da + Xb/db +Xc/dc which is our solution.
This principle can be applied to alloys of more than 3 metals. It assumes no chemical interaction between the metals and no formation of interstitials. It works well for solder alloys.