total mass 9.85 grms and total volume 0.65 cm ^3 . The jewlry contain only gold and silver which has densities 19.3 g/cm^3 amd 10.5 g/cm^3.calculate the % of mass of gold in jewellery
To calculate the percentage of gold in the jewelry, we need to know the masses of gold and silver in the jewelry.
Let's assume that x grams of gold and (9.85 - x) grams of silver are present in the jewelry.
The density of gold is 19.3 g/cm^3, so the volume of gold (Vg) can be calculated using the formula:
Vg = mg / ρg
Similarly, the volume of silver (Vs) can be calculated using the formula:
Vs = ms / ρs
Where mg and ms are the masses of gold and silver respectively, and ρg and ρs are their respective densities.
Given that the total volume of the jewelry is 0.65 cm^3, we can write the equation:
Vg + Vs = 0.65 cm^3
Substituting the volume formulas, we have:
mg / ρg + ms / ρs = 0.65
Since we have two unknowns (mg and ms), we need another equation to solve for them. We can use the total mass of the jewelry to form the second equation:
mg + ms = 9.85
Now we have a system of two equations that we can solve simultaneously.
First, rearrange the equation mg / ρg + ms / ρs = 0.65 to solve for mg:
mg / ρg = 0.65 - (ms / ρs)
mg = (0.65 - (ms / ρs)) * ρg
Substitute this value of mg in the second equation:
(0.65 - (ms / ρs)) * ρg + ms = 9.85
Now solve this equation for ms:
(0.65ρg - (ms / ρs)ρg) + ms = 9.85
0.65ρg + (1 - (ρg / ρs))ms = 9.85
Multiply through by ρs:
0.65ρgρs + (ρs - ρg)ms = 9.85ρs
Simplifying further:
0.65ρgρs - ρgms + ρsms = 9.85ρs
Rearrange to solve for ms:
ms(ρs - ρg) + 0.65ρgρs = 9.85ρs
ms = (9.85ρs - 0.65ρgρs) / (ρs - ρg)
Substituting the given values:
ms = (9.85 * 10.5 - 0.65 * 19.3 * 10.5) / (10.5 - 19.3)
Evaluate ms to find the mass of silver.
Now, the mass of gold can be calculated as:
mg = 9.85 - ms
Finally, the percentage of gold in the jewelry can be calculated using the formula:
% gold = (mg / (mg + ms)) * 100
Substitute the values of mg and ms to find the percentage of gold in the jewelry.