The British Sovereign coin is an alloy of gold and copper having a total mass of 7.988 g and is 22-karat gold.
(a) Find the mass of gold in kg using that # of karats = 24 x (mass of gold)/(total mass).
(b) Calculate the volumes of gold and copper used to manufacture the coin.
(c) Calculate the density of the coin.
a. calculate using the formula
b. knowing the masses of copper and gold, and densities, find the volume
c. density=totalmass/totalvolume
(a) To find the mass of gold in kg, we need to use the formula:
# of karats = 24 x (mass of gold)/(total mass)
Given that the British Sovereign coin is 22-karat gold and has a total mass of 7.988 g:
22 = 24 x (mass of gold)/(7.988 g)
To solve for the mass of gold, we rearrange the equation as follows:
mass of gold = (22/24) x 7.988 g
mass of gold = 0.9167 x 7.988 g
mass of gold ≈ 7.32 g
Now, to convert the mass of gold from grams to kilograms, divide it by 1000:
mass of gold in kg = 7.32 g / 1000
mass of gold in kg ≈ 0.00732 kg
Therefore, the mass of gold in the British Sovereign coin is approximately 0.00732 kg.
(b) To calculate the volumes of gold and copper used to manufacture the coin, we need to know the densities of gold and copper and apply the respective mass formulas:
Density of gold = mass of gold / volume of gold
Density of copper = mass of copper / volume of copper
We can rearrange these equations to solve for the volume of gold and copper:
Volume of gold = mass of gold / density of gold
Volume of copper = mass of copper / density of copper
However, we first need to find the mass of copper. Since we know the total mass of the coin is 7.988 g and the mass of gold is 7.32 g, we can calculate the mass of copper:
mass of copper = total mass - mass of gold
mass of copper = 7.988 g - 7.32 g
mass of copper ≈ 0.668 g
Given that the density of gold is approximately 19,320 kg/m^3 and the density of copper is approximately 8,960 kg/m^3:
Volume of gold = 7.32 g / 19,320 kg/m^3
Volume of copper = 0.668 g / 8,960 kg/m^3
To calculate the volumes, we need to convert the mass from grams to kilograms. Dividing by 1000 gives:
Volume of gold ≈ 3.79 x 10^-7 m^3
Volume of copper ≈ 7.46 x 10^-8 m^3
Therefore, the volume of gold used to manufacture the coin is approximately 3.79 x 10^-7 m^3, and the volume of copper used is approximately 7.46 x 10^-8 m^3.
(c) To calculate the density of the coin, we need to know the total volume and the total mass of the coin:
Density of the coin = total mass / total volume
Given that the total mass is 7.988 g and the total volume is the sum of the volumes of gold and copper calculated in part (b):
Total volume = Volume of gold + Volume of copper
Total volume ≈ 3.79 x 10^-7 m^3 + 7.46 x 10^-8 m^3
Total volume ≈ 5.54 x 10^-7 m^3
Now we can calculate the density of the coin:
Density of the coin = 7.988 g / 5.54 x 10^-7 m^3
To convert the mass from grams to kilograms, we divide by 1000:
Density of the coin ≈ 7.988 g / (5.54 x 10^-7 m^3) / 1000
Density of the coin ≈ 1.44 x 10^4 kg/m^3
Therefore, the density of the British Sovereign coin is approximately 1.44 x 10^4 kg/m^3.