Gold is alloyed (mixed) with other metals to increase its hardness in making jewelry.

A) Consider a piece of gold jewelry that weighs 9.40 g and has a volume of 0.675 cm3. The jewelry contains only gold and silver, which have densities of 19.3 g/cm3 and 10.5 g/cm3, respectively. If the total volume of the jewelry is the sum of the volumes of the gold and silver that it contains, calculate the percentage of gold (by mass) in the jewelry.

B) The relative amount of gold in an alloy is commonly expressed in units of karats. Pure gold is 24-karat, and the percentage of gold in an alloy is given as a percentage of this value. For example, an alloy that is 50% gold is 12-karat. State the purity of the gold jewelry in karat.

A) 54.04%

B) 13-karat

Explanation:

A) From the problem we have

1) Mg + Ms = 9.40 g

2) Vg + Vs = 0.675 cm³

Where M stands for mass, V stands for volume, and g and s stand for gold and silver respectively.

We can rewrite the first equation using the density values:

3) Vg * 19.3 g/cm³ + Vs * 10.5 g/cm³ = 9.40

So now we have a system of two equations (2 and 3) with two unknowns:

We express Vg in terms of Vs:

Vg + Vs = 0.675 cm³ Vg = 0.675 - Vs

We replace the value of Vg in equation 3:

Vg * 19.3 + Vs * 10.5 = 9.40 (0.675-Vs) * 19.3 + Vs * 10.5 = 9.40 13.0275 - 19.3Vs + 10.5Vs = 9.40 - 8.8 Vs + 13.0275 = 9.40 Vs = 0.412 cm³

Now we calculate Vg:

Vg + Vs = 0.675 cm³ Vg + 0.412 cm³ = 0.675 cm³ Vg = 0.263 cm³

We calculate Mg from Vg:

0.263 cm³ * 19.3 g/cm³ = 5.08 g

We calculate the mass percentage of gold:

5.08 / 9.40 * 100% = 54.04%

B)

We multiply 24 by the percentage fraction:

24 * 54.04/100 = 12.97-karat ≅ 13-karat