[KEP]

Hello everyone,

I need some help regarding the correct formula for determining the density of an alloy.

For this example I'll use cartridge brass (C260) which is 70 % copper and 30 % zinc, by weight.

The density of copper is 8.96 g/cm3 and that of zinc is 7.14 g/cm3 (cm3 or cc).

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The first formula, a popular one so it seems but which is incorrect, assumes volume fractions:

(70 cc * 8.96 g/cc + 30 cc * 7.14 g/cc) / 100 cc = (627.2 g + 214.2 g) / 100 cc = 841.4 g / 100 cc = 8.414 g/cc

I say it's incorrect because alloys are not made like that, but by using mass fractions.

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The second formula, which I had relied upon until now, assumes mass fractions:

100 g / (70 g / 8.96 g/cc + 30 g / 7.14 g/cc) = 100 g / (7.8125 cc + 4.2017 cc) = 100 g / 12.0142 cc = 8.323 g/cc

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And a third formula which uses the molar mass for each of the constituents.

The standard atomic weight (and corresponding molar mass) for copper is 63.546 and for zinc is 65.38.

70 g / 63.546 g/mol = 1.1016 mol

30 g / 65.38 g/mol = 0.4589 mol

1.106 mol + 0.4589 mol = 1.5605 mol for the whole alloy

Now for the mole fractions:

1.1016 mol / 1.5605 mol = 0.706

0.4589 mol / 1.5605 mol = 0.294

Then I use the density:

0.706 * 8.96 g/cc + 0.294 * 7.14 g/cc = 6.326 g/cc + 2.099 g/cc = 8.425 g/cc

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I need the formula for accurately calculating the mass of some metal parts that I design using a CAD program.
Until now I had relied upon the second formula but after some research a few days ago I came upon the third one which gave me some doubts.

The thing is that across the internet various manufacturers of different alloys give the density in the data sheet, and that value is way off of what I get using either formula, even the first incorrect one.

http://www.farmerscopper.com/cartridge-brass-260-c260-c26000.html
https://www.ajoster.com/sites/default/files/downloads/C-26000_B36_CARTRIDGE-BRASS_AJ-OSTER.pdf
https://www.diehl.com/cms/files/Diehl_Metall_Strip_MB30_V3_M-SM.pdf

https://www.wisetool.com/density.htm

No matter what I use, I don't get the results that are given on sites like these. And I could go on to list more examples than the one given above.

What am I missing?

[mjc123]

The mass fraction formula would be the correct one, assuming no volume of mixing, i.e. the volume of 100g alloy is the same as the sum of the volumes of 70g pure copper and 30 g pure zinc. But this is often, if not usually, untrue, and is evidently untrue in this case. Then there is no theoretical formula that will give you the density; you have to rely on empirical data.

[Borek]

In other words: volumes are not additive. When you mix V_A of substance A with V_B of substance B, final volume is almost never exactly V_A+V_B.

[AWK]

On ajoster.com you can find six densities of different brass alloys. Try the calibration curve method.

[Enthalpy]

Some alloys have a volume very different from the added volumes of the constituents. Brass isn't an extreme case. These alloys have often other interesting properties. Young's modulus seems to be strongly influenced by the packing density.

Bronze bell, Cu with 20% Sn and no Pb, has more volume than its constituents, and its E-modulus is quite low. That makes it radiate less sound power and prolongs the decay, when radiation is the main loss.

Invar, Fe with 36% Ni, has excess volume, and its E-modulus is quite low. This one too has a long sound decay.

Nearly equimolar TiAl is denser than expected and its E~165GPa widely exceeds both alloying elements (105GPa and 70GPa). The thermal expansion too (10.8ppm/K) is lower than any mixture law would predict (8.7ppm/K and 23.5ppm/K for the constituents).

[KEP]

Thanks to everyone for their responses.

I understand now.

I was so caught up on the whole "volume fractions vs mass fractions formula" that I failed to see the blatantly obvious.
And that is: volumes are not additive.

But even if they were, in the case of metal alloys such as in my example above, there is still the matter of shrinkage.
Metals expand when they melt (hence their lower density) and then shrink back when cooled and solidified.
Even if volumes would be additive in the case of brass they would only be when liquid.
Because an alloy doesn't necessarily have to shrink the same amount as the weighted average of its constituents' shrinkage amount.

One alloy might shrink less and therefore the volume would be greater than expected and the density less than expected and another might do the opposite.
And on top of shrinkage there is the matter of interstitials and intermetallics which can also affect the final volume.
Pretty hard to derive a prediction formula considering such variables.

So yeah, from now on I'll have to rely on the values given in data sheets for various alloys.

[Enthalpy]

Just as side notes, some materials expand when freezing. Not only water, also some metals.
Volumes of liquid mixtures aren't additive neither. I have no example for metals in my head, but water and ethanol is a known case.

[Borek]

Change of the volume (shrinkage or expansion) at solidification doesn't change anything here. One would just use density of liquids for estimating density of a liquid mixture and density of solids to estimate density of a solid mixture. Final result would be wrong in both cases due to the same principle.