# Piece of quartz mixed with gold

1. Jan 15, 2015

### Arcany

1. The problem statement, all variables and given/known data
Piece of quartz containing gold weights 102,5g and average density is 7,98g/cm3. Density of quartz is 2,65g/cm3. How much does gold weight?

2. Relevant equations
ρ=m/V
avg.ρ= m1+m2/V1+V2

3. The attempt at a solution
Using average density I find that volume of this piece is 12,84cm3. ρ=m/V ⇒ V=m/ρ V=102,5/7,98=12,84
Now using formula for calculating average density(avg.ρ= m1+m2/V1+V2) I get that 7,98=m1+m2/12,84. Now 102,5=m1+m2 ⇒m1=102,5-m2 so I place this into the formula.
7,98=(102,5-m2)+m2/12,84 but now -m2 and +m2 cancel each other off and I can't seem to find what to do now. Can you please help?

2. Jan 15, 2015

### Bystander

Write an equation for total volume.

3. Jan 15, 2015

### BiGyElLoWhAt

Well you can redily find the volume, and you are given the weight. You know that the weight of the gold plus the weight of the quartz = total weight and that the volume of the gold plus volume of the quartz = total volume. 2 equations 2 unknowns.

4. Jan 15, 2015

### Arcany

Vtotal=Vquartz+Vgold

5. Jan 15, 2015

### Bystander

And ---- Vquartz = what? And VGold?

6. Jan 15, 2015

### Arcany

Well I know that total volume is 12,84cm3 so 12,84cm3=Vquartz+Vgold so Vquartz=12,84cm3 -Vgold and Vgold=12,84cm3 -Vquartz but I really don't know where to move from here because using one of those in my formula I would run into the same problem as with when I would replace mass.

7. Jan 15, 2015

### BiGyElLoWhAt

Oh, my bad, I thought those two equations would be linearly independent for some reason, and also apparently didn't read your attempt at the solution closely enough.

8. Jan 15, 2015

### Quantum Defect

Are you sure that the problem does not give you the density of gold? Are you allowed to use a handbook to look up the density of gold?

I believe that you do not have enough data with what you have presented. [The data above (total mass, density of quartz, density of sample) can be rewritten to get three equations with four unknowns (two masses and two volumes)]

Another way of thinking about the problem:

Imagine that you have a graph of average density as a function of percent composition of gold. At the left is the density of 100% quartz, at the right is the denisty of 100% gold. You will get a line going from one side to the next (assuming no funny 'mixing' behavior). The problem is that you don't know the right endpoint (i.e. the density of 100% gold), so you don't know what composition corresponds to 7.98 g/cm^3.

For example, suppose that density of gold is 7.98 g/cm^3 (its not, but just suppose). You know then that the sample is 100% gold (if there were any quartz, you would have a lower density). So the mass of gold is 102.5 g.

9. Jan 15, 2015

### haruspex

True, but there is enough information to get a lower bound for the answer, and assuming gold is far more dense than quartz it might not be far off.

10. Jan 16, 2015

### Quantum Defect

I do not think so. What lower bound do you get? What assumptions did you make for the density of gold?

(1) mgold + mquartz = Mtotal

(2) Vgold * rho_gold + Vquartz * rho_quartz = Vtotal * rho_ave

Divide both sides of (2) by Vtotal, and you get an equation for the average density (rho_ave) as a weighted average (by volume fraction) of the densities of the two components. If you plot density versus composition (e.g. volume fraction of gold) you get a straight line. Left side y-intercept (0% Au) = density of quartz; right side y-intercept (100% Au) = density of gold. Without knowing what the right side y-intercept is, you cannot calculate a composition for the quoted average density. You can solve for everything in (2), given the information above, except one of the volumes and rho_gold -- one equation, two unknowns.

11. Jan 16, 2015

### haruspex

We are told
If the gold content is zero those two densities conflict. The lower bound is had by assuming the gold is infinitely dense.

12. Jan 16, 2015

### Quantum Defect

No. If gold is infinitely dense, a trace amount of gold (essentially zero fraction) could easily give you the observed density, no? Look at the equation above in #10. Observed density is a weighted average of the individual densities.

Rho_total = rho_quartz * (1-fraction gold) + rho_gold* (fraction gold)

For a very large density of gold (i.e. very little gold)

(Rho_total-rho_quartz ) = rho_gold*fraction gold (fraction gold << 1)

The fraction gold can be made arbitrarily small, no?

13. Jan 16, 2015

### haruspex

Define fraction - is that by weight or by volume?
If the gold is infinitely dense it occupies no volume. vol quartz = total vol = total mass / avg density.
mass quartz = (total mass / avg density) * density quartz
mass gold = total mass - mass quartz

14. Jan 16, 2015

### Quantum Defect

"Define fraction -" Defined in #10, above, it is fraction by volume.

From # 12: (Rho_total-rho_quartz ) = rho_gold*fraction gold (fraction gold << 1)
Putting in numbers:
(7.98-2.65) g/cm^3 = rho gold * fraction gold (where fraction gold <<1)

5.33 = rho gold * fraction gold

If fraction gold = 0.01, rho_gold= 533 g/cm^3
If fraction gold = 0.001, rho gold = 5,330 g/cm^3
If rho_gold = "infinity" as you state, fraction gold is "0"

There is no lower bound to the fraction of gold, based upon the math.

15. Jan 16, 2015

### haruspex

Yes, there's no lower bound on the volume. But the questions asks for weight (mass), not volume, and there is a lower bound on that.

16. Jan 16, 2015

### Quantum Defect

Aha, I see the error of my ways... sorry for being dense ;)

17. Jan 16, 2015

### haruspex

No problem. You get high marks from me for acknowledging your agreement. Many just go quiet.