I have occasionally written on calculating solder alloy density, as there is surprisingly more interest than I thought there would be in this topic. Recently, it occurred to me that it might be beneficial to compare the calculated densities to actual densities of a few alloys to see how accurate the correct formula is (for the derivation of the correct formula see below). The formula assumes “perfect mixing” (i.e., no interactions between the alloy elements). The alloys we investigated were tin-bismuth-silver, tin-silver, tin and tin-bismuth.
To measure the density, I obtained a few alloys from Indium. My student, Evan Zeitchick, determined that a good technique to measure density is to machine the alloy into a rectangular parallelepiped (see photo), weigh it, and calculate its volume from its dimensions. The results agree with the correct formula to about 1 to 2%. Some people would ask why there is any difference. The reason is that all alloys form different phases, and some form intermetallics. These phases and intermetallics would typically have different densities than that calculated for the alloy. I will have more detail on this work in a future post.
Here is a derivation of the correct density formula:
Many people incorrectly assume that if you have an alloy of x % tin by weight and y % silver, that the density of this alloy would be 0.x*Density tin +0.y*Density silver. This intuitive linear formula is incorrect however, as density has two units (mass and volume).
An easy way to understand the derivation of the correct formula (proposed by Indium engineer Bob Jarrett) is to consider a 96% tin, 4 % silver example.
Let’s assume I have 1 g of this alloy, 0.96 g is tin and 0.04 g is silver.
The volume of the tin is 0.96 g/7.31g/cc = 0.131327cc
The volume of the silver is 0.04g/10.5g/cc = 0.00381cc
So 1 g of the alloy has a volume of 0.131327 + 0.00381 cc = 0.135137 cc
Hence it’s density is 1g/0.135137cc = 7.39989g/cc
Hence, the general formula is:
1/Da = x/D1 + y/D2 + z/D3
Da = density of final alloy
D1 = density of metal 1, x = mass fraction of metal 1
same for metals 2, 3
The formula continues for more than 3 metals.
I have developed an Excel spreadsheet that calculates density automatically. If anyone wants a copy, send me an email at firstname.lastname@example.org
P.S.: Interesting thought: About 165,000 tonnes of gold have been mined throughout history. If all of this gold was gathered into a cube it would only be about 21 meters on a side. At $1,550/oz, its value would be $8.5 trillion, quite a bit less than the almost $15 trillion debt of the US government. Yikes!