Consider a piece of gold jewelry that weighs 9.75 g and has a volume of 0.690 cm^3. The jewelry contains only gold and silver, which have densities of 19.3 g/cm^3 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. 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. Express your answer as an integer.

57 %, 13 karat

Explanation:

We know the total volume of the jewelry is the sum of the volumes of gold and silver, we are also given the densities of gold and silver, and since the volumes are additive, we can setup the following equation=

d = m / v ⇒ v = m/d

Vt = mass gold / d gold + mass silver / d silver

the mass of gold is what we are seeking:

0.690 cm³ = m gold / 19.3 g/cm³ + m silver / 10.5 g/cm³

Now we also know the mass of the jewelry to be 9.75 g, therefore the m of silver can be expressed as 9.75 minus mass of gold, substituting in the above equation:

0.690 cm³ = mass gold / 19.3g/cm³ + (9.75 g - mass gold) / 10.5 g/cm³

The cm³ cancel and we have then:

0.690 g = 0.052 mass gold + 0.929 - 0.095 mass gold

0.690 - 0.929 g = - 0.043 mass gold

- 0.239 = - 0.043 mass gold ⇒ mass gold = 0.239 / 0.043 = 5.56 g

% gold by mass = 5.56 g / 9.75 g = 57 %

For the second part we know that 50 % gold is 12 carat, so to calculate the purity of the gold jewelry in carat we form the proportion:

(12 carat / 50 %) x 57 % = 13 karat