Piece of quartz mixed with gold

[Arcany]

Homework Statement

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?

Homework Equations

ρ=m/V
avg.ρ= m₁+m₂/V₁+V₂

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?

 

[Bystander]

Write an equation for total volume.

 

[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.

 

[Arcany]

Write an equation for total volume.

V_total=V_quartz+V_gold

 

[Bystander]

And ---- V_quartz = what? And V_Gold?

 

[Arcany]

And ---- V_quartz = what? And V_Gold?

Well I know that total volume is 12,84cm³ so 12,84cm³=V_quartz+V_gold so V_quartz=12,84cm³ -V_gold and V_gold=12,84cm³ -V_quartz 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.

 

[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.

 

[Quantum Defect]

Homework Statement

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?

Homework Equations

ρ=m/V
avg.ρ= m₁+m₂/V₁+V₂

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?

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.

 

[haruspex]

you do not have enough data with what you have presented.

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.

 

[Quantum Defect]

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.

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.

 

[haruspex]

I do not think so. What lower bound do you get?

We are told

Piece of quartz containing gold ... average density is 7,98g/cm3. Density of quartz is 2,65g/cm3.

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

 

[Quantum Defect]

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.

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?

 

[haruspex]

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?

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

 

[Quantum Defect]

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

"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.

 

[haruspex]

"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.

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.

 

[Quantum Defect]

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.

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

 

[haruspex]

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

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