When making a new boolit alloy, having doubts about whether or not the volumes of the metals will still add up the same after mixing together as before mixing them is a quite legitimate concern. That is because lots of things out there in the real world don’t add up the same afterward as before mixing, good examples of that inequality are ethanol and water, or sodium chloride (good old table salt) and water, which do not. In fact with salt and water, as long as the salt is given time to become totally dissolved into the water, the volume of water does not change noticeably even when a considerable amount of salt is dissolved into it, making it greatly different from metals. Adding a whole saltshaker of salt to a nearly full glass of water without any ice in it at the table in a restaurant is an old magician’s trick that has earned those folks who know about such things a bit of pocket change or a few free dinners by collecting on bets! Don’t try it at home, though, it usually gets the owner of the saltshaker accused of rigging the bet by having “trick salt” in the saltshaker.
Metals behave totally different, though, and with few exceptions, when alloyed the volumes of metals WILL ADD UP nearly the same afterward as before. The only times when the before and after metal volumes do not add up pretty close to the same are a few very definite cases when the crystal systems of the two constituent metals are different, such as one being in the hexagonal system and the other being in the face-centered or body-centered cubic systems. Fortunately for tin and most other commonly used alloy constituents that are in the hexagonal system, their volumes add up very nearly equally with the metals of the two cubic system.
The formula for determining the specific gravity of an alloy from its constituent specific gravities is quite straightforward, and those of us who are familiar with electronics formulas will immediately recognize it as being derived from the same mathematical principle as the formula for determining resistance R and inductance L in parallel circuits, and capacitance C in series circuits. Besides metallurgy and electronics, the basic principle is also used in other areas of physics and chemistry for determining the specific heats of mixtures of materials. It has a lot of uses! The formula is as follows:
The reciprocal of the specific gravity of the new alloy is equal to the reciprocal of the specific gravity of the first metal times its decimal part of the weight of the alloy + the reciprocal of the specific gravity of the second metal times its decimal part of weight of the alloy + the reciprocal of the specific gravity of the third metal times its decimal part of the weight of the alloy + the reciprocal of the specific gravity of the fourth metal times its decimal part of the weight of the alloy + the reciprocal of the specific gravity of the fifth metal times its decimal part of the weight of the alloy + etc, carried out as many times as there are different metals in the resulting new alloy.
If there are only two metals in the new alloy then the formula is:
The reciprocal of the specific gravity of the new alloy is equal to the reciprocal of the specific gravity of the first metal times its decimal part of the weight of the alloy + the reciprocal of the specific gravity of the second metal times its decimal part of the weight of the alloy.
For an example I’ll use the values for a common lead-based Babbitt or bearing alloy, that age-hardens, quenches, and also works okay for hard cast projectiles.
Because a lot of our younger members are still in school I will try to make the math as easy as possible so everybody can have a shot at being able to use this or at least get an idea of what is going on.
The alloy test sample weighs 1672 grains and was made from 1521.52 grains lead (chemist’s symbol Pb) with 150.48 grains antimony (chemist’s symbol Sb). Taking 1521.52 grains divided by 1672 grains gives me 0.910 decimal parts lead in 1.00 decimal part of the alloy. Taking 150.48 grains divided by 1672 grains gives 0.090 decimal parts antimony per 1.00 decimal part of the alloy.
The specific gravity of lead is 11.34 and its reciprocal is 1/11.34 or 0.0881834 in decimal form.
The specific gravity of antimony is 6.68 and its reciprocal is 1/6.68 or 0.1497 in decimal form.
The reciprocal of a number is found by simply taking 1 and dividing it by that particular number, for example the reciprocal of 5 is 1/5, or one fifth, and it is 0.2 in decimal form or decimal numbers. Another example is the reciprocal of 6 is 1/6, or one sixth, and it is 0.16666666666 etc in decimal form or decimal numbers, but quite often just rounded off to 0.167 to keep it simple.
Now we use the following formula:
Reciprocal of the specific gravity = (reciprocal of the specific gravity of lead times decimal parts of lead in the alloy) + (reciprocal of the specific gravity of antimony times decimal parts of antimony in the alloy)
When the math instructions are inside of a pair of () it means that all the adding, subtracting, multiplying, dividing, and exponential operations inside them have to be done first before anything else outside of them can be added or subtracted to or from those numbers because if not done that way the answer will be totally wrong.
Using actual math terms:
1/spgrv = (1/spgrv Pb x dec parts Pb in alloy)+(1/spgrv Sb x dec parts Sb in alloy)
Plugging in the actual numbers gives us:
1/spgrv = (1/11.34 x 0.910) + (1/6.68 x 0.090)
Now turning the “math crank” it grinds out
1/spgrv = (0.0881834 x 0.910) + (0.1497 x 0.090)
turning the “math crank” again it grinds out
1/spgrv = 0.080247 + 0.013473
turning the “math crank” one more time it grinds out
1/spgrv = 0.09372
At this point we still don’t have the answer yet because we have the reciprocal of the number we are trying to get to. To get the reciprocal just take 1 and divide the above number into it.
So we have 1/ 0.09372, which turning the “math crank” one more time makes it grind out the value of 10.6700848. Rounding this off to a significant number of places beyond the decimal point so it means something real gives us a specific gravity of 10.67 for this alloy. This jibes awful darned close to the specific gravity found by weighing the test sample under water to find the weight of its volume of water and then dividing the dry weight by the weight of its water volume. The calculations for that were in the earlier thread, but the test results were found to be 10.65 and that is 99.8% of the theoretical specific gravity of 10.67 that was just calculated.
With just two metals in an alloy it CAN be used for determining the percentages of those metals when the specific gravities of the alloy as well as both metals in it are known, but not their amounts. That is done by equating all values in the equation to one particular “like term” and then solving for the unknown. I don’t have time right now to rearrange the equation to solve for that, but if anyone who is a math whiz wants to do that and post it, go right ahead since it would be quite useful.
As already pointed out by garandsrus with the table he so kindly posted for us, this formula will not work using it backwards to discover the percentages of a metal in an alloy when there are THREE OR MORE metals in it. That is because by varying the amounts of three metals in an alloy it can still give one specific gravity just by varying the ratios of the ingredients over a rather wide range! But working forward like I did up above it is dead-on accurate for finding the RESULTING specific gravity when the amounts and specific gravities of all the constituents are known.