A Simple Explanation of Why the Commonly Used Equation for Alloy Density is Wrong

Folks,

It still surprises me that my most popular blog topic in the 15 years (yikes!) that I have been blogging is how to calculate solder alloy density having the mass fractions and densities of the alloy metals. Some years ago, I received the following comment on a blog post on this topic. The names and companies have been changed to protect the innocent (guilty?):

Dr. Lasky,

My name is John Smith, a PhD metallurgist. I read with interest your article on calculating densities of alloys. Surely your formula: 1/rho_Total = x/rho_x + y/rho_ycan’t be correct. I have been at ACME metals for over 30 years and have always used:rho_Total = xrho_x + yrho_y.

Please tell me it is not wrong!

Best,
John Smith

Sorry, Dr. Smith;it is wrong. The correct derivation for 1/rho_Total = x/rho_x + y/rho_y is given here. For years, I havethought of ways to explain why rho_Total = xrho_x + yrho_{y ­}is not correct. One obvious point is that density has the units of mass/volume, so it is not obvious that rho_Total = xrho_x + yrho_y should work since x and y are mass fractions. But, in spite of this point, this wrong equation is even in textbooks.

So, I decided to calculate terms that when multiplied by the metal densities dogive the total density. I call these terms “density fractions.” I put them in quotes because if you look at the derivation, you will see that they are not quite in the form of a mass fraction.

Also, these density fractions are not useful in that you have to know the density of the alloy to calculate them. Note that the incorrect equation rho_Total = xrho_x + yrho_y­would give a density for tin-lead eutectic solder of 8.79 g/cc, an error of 4.625%.

The derivation of the “density fraction” follows.

Cheers,

Dr. Ron