[Krynvel]

Hey guys I need some help with a lab question.

A density of a penny is 8.8g/cm³
the composition of the penny is 95% copper, 5% tin and zinc
density of copper is 8.96g/cm³
density of tin is 7.31g/cm³
density of zinc is 7.13g/cm³

The question is how much tin is in the penny?

Here's how I approached this problem,
d(coin)=(d[Cu]*95%)+(d[Zn]*5%-[Sn]%)+(d[Sn]*[Sn]%)
8.8=(8.96*0.95)+(7.13*[Zn]%)+(7.31*[Sn]%)
(0.288-(7.31*[Sn]%))/7.13=[Zn]% <- Plug this back into the original equation

I ended up here and I got stuck.
0.288=(0.288-7.31*[Sn]%)+(7.31*[Sn]%)

I tried using limits on the original equation: if [Zn]% approached 0, what % composition would Tin approach and it gave me an answer that I couldn't accept which was less than 1%.

A guiding hand to point me to the fault of my logic would be greatly appreciated.

[chenbeier]

Not Impossible,  You dont know tin eiher zinc. Your calculation is wrong. 5% = Sn + Zn

[Borek]

d(coin)=(d[Cu]*95%)+(d[Zn]*5%-[Sn]%)+(d[Sn]*[Sn]%)

Right direction, but wrong execution. Assume x to be the fraction of tin and y to be the fraction of zinc and rewrite the equation in terms of these unknowns.

What do you know about sum of x and y?

[Krynvel]

I know the sum of x and y to add to 5% of the total density which is 0.44 g/cm³
or 5%-%[Sn]=%[Zn]
Plug that into the originial equation...

So what I got was:
(0.05)(8.8g/cm³)=(7.13g/cm³*(0.05-%[Sn]))+7.31%[Sn]
Simplify:
0.44g/cm³=0.3565g/cm³+0.18%[Sn]
Simplify:
0.0835=0.18%[Sn]
Simplify:
0.46[of 5% of total density]=%[Sn]
Reworded:
%[Sn]=2.3%

Now my question...
Why couldn't I use d_coin=(d_copper*%copper)+(d_zinc*%zinc)+(d_tin*%tin)

Whenever I used that equation I got
0.288g/cm³for the 5% of total density by subbing in copper density and %copper and subtracting that value from the density of the coin.

[Borek]

I know the sum of x and y to add to 5% of the total density which is 0.44 g/cm³
or 5%-%[Sn]=%[Zn]
Plug that into the originial equation...

So what I got was:
(0.05)(8.8g/cm³)=(7.13g/cm³*(0.05-%[Sn]))+7.31%[Sn]

No, this is not correct, I feel like you are trying to make a shortcut and it fires away. Please write the whole equation, including information you have on all three metals.

[Enthalpy]

The method is basically wrong. Whoever asked the question is mistaken.

In alloys, volumes don't add. Bell bronze and Invar have a bigger volume than their constituents. NiTi changes its volume completely between austenitic and martensitic forms. TiAl has less volume than its constituents.

But there is an interesting correlation between the volume change when alloying and Young's modulus (the stiffness if you wish).

The same happens with liquids, for instance a mixture of ethanol and water occupies less volume than the sum of the constituents.

[Borek]

The method is basically wrong. Whoever asked the question is mistaken.

Yes, but let's tackle the problem in only possible way first, then we will discuss why it is only an approximation.

As of now we have to correct math behind the simplified model.

[AWK]

The discussion on this topic has died down.
After a week, it's time to explain what the error is. Calculating this problem is possible assuming the additivity of the volume. As Enthalpy rightly pointed out, there may be surprises, but for the total content of Zn + Sn = 5%, surprises are unlikely to be expected.
For both 5% of binary alloys of Cu with Sn or Zn, manufacturers give a density of 8.85 (assuming volume additivity, the calculated values ​​are 8.860 and 8.846 respectively - i.e. the error is less than 2%).
There are no data for three-component alloys with such a low content of Zn and Sn, but with a high probability it can be expected that their density will be within these limits (8.846 to 8.860). And only for the density within such limits the problem is solvable. For Sn content 4 to 1% and Zn up to 5% I got densities 8.857, 8.855, 8.852 and 8.849, respectively.

[Enthalpy]

Out of curiosity (and not, obviously and of course, because of any tendency for polemics whatsoever), I compared the density of brass with that of copper and zinc.

kg/dm³
8.96  Cu
7.14  Zn

Measured vs computed by adding volumes. Alloys given in mass composition.
8.86  8.85  Cu95Zn5
8.80  8.74  Cu90Zn10
8.75  8.63  Cu85Zn15
8.67  8.53  Cu80Zn20
8.53  8.32  Cu70Zn30
I stop here because brass with more Zn can contain a body-centred phase.

The difference is much bigger than I expected, possibly because pure Zn isn't a face-centred cubic like these brasses are.

Added volumes would let tell "this is Cu90Zn10" from the observed density of Cu85Zn15. Ouch.