Question

The British gold sovereign coin is an alloy of gold and copper having a total mass of 7.988 g, and is 22-karat gold 24 x (mass of gold)/(total mass) (a) Find the mass of gold in the sovereign in kilograms using the fact that the number of karats kc (b) Calculate the volumes of gold and copper, respectively, used to manufacture the coin. m3 volume of gold m3 volume of copper (c) Calculate the density of the British sovereign coin. kg/m3

Answer 1

Answers:

(a) $0.0073kg$

(b) Volume gold: $3.79(10)^{-7}m^{3}$, Volume cupper: $7.6(10)^{-8}m^{3}$

(c) $17633.554kg/m^{3}$

Explanation:

(a) Mass of gold

We are told the total mass $M$ of the coin, which is an alloy  of gold and copper is:

$M=m_{gold}+m_{copper}=7.988g=0.007988kg$   (1)

Where  $m_{gold}$ is the mass of gold and $m_{copper}$ is the mass of copper.

In addition we know it is a 22-karat gold and the relation between the number of karats $K$ and mass is:

$K=24\frac{m_{gold}}{M}$   (2)

Finding ${m_{gold}$:

$m_{gold}=\frac{22}{24}M$   (3)

$m_{gold}=\frac{22}{24}(0.007988kg)$   (4)

$m_{gold}=0.0073kg$   (5)  This is the mass of gold in the coin

(b) Volume of gold and cupper

The density $\rho$ of an object is given by:

$\rho=\frac{mass}{volume}$

If we want to find the volume, this expression changes to: $volume=\frac{mass}{\rho}$

For gold, its volume $V_{gold}$ will be a relation between its mass $m_{gold}$  (found in (5)) and its density $\rho_{gold}=19.30g/cm^{3}=19300kg/m^{3}$:

$V_{gold}=\frac{m_{gold}}{\rho_{gold}}$   (6)

$V_{gold}=\frac{0.0073kg}{19300kg/m^{3}}$   (7)

$V_{gold}=3.79(10)^{-7}m^{3}$   (8)  Volume of gold in the coin

For copper, its volume $V_{copper}$ will be a relation between its mass $m_{copper}$  and its density $\rho_{copper}=8.96g/cm^{3}=8960kg/m^{3}$:

$V_{copper}=\frac{m_{copper}}{\rho_{copper}}$   (9)

The mass of copper can be found by isolating $m_{copper}$ from (1):

$M=m_{gold}+m_{copper}$  

$m_{copper}=M-m_{gold}$  (10)

Knowing the mass of gold found in (5):

$m_{copper}=0.007988kg-0.0073kg=0.000688kg$  (11)

Now we can find the volume of copper:

$V_{copper}=\frac{0.000688kg}{8960kg/m^{3}}$   (12)

$V_{copper}=7.6(10)^{-8}m^{3}$   (13)  Volume of copper in the coin

(c) Density of the sovereign coin

Remembering density is a relation between mass and volume, in the case of the coin the density $\rho_{coin$ will be a relation between its total mass $M$ and its total volume $V$:

$\rho_{coin}=\frac{M}{V}$ (14)

Knowing the total volume of the coin is:

$V=V_{gold}+V_{copper}=3.79(10)^{-7}m^{3}+7.6(10)^{-8}m^{3}=4.53(10)^{-7}m^{3}$ (15)

$\rho_{coin}=\frac{0.007988kg}{4.53(10)^{-7}m^{3}}$ (16)

Finally:

$\rho_{coin}=17633.554kg/m^{3}}$ (17)  This is the total density of the British sovereign coin