Consider a piece of gold jewelry that weighs 9.81 g and has a volume of 0.670 cm3. The jewelry contains only gold and silver, which have densities of 19.3 g/cm3 and 10.5 g/cm3, respectively. Assuming that 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.
Let x = mass Au,
then 9.81-x = mass Ag.
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(1)--volume Au = m/d = x/19.3
(2)--volume Ag = m/d = (9.81-x)/10.5
(3)--volume Au + volume Ag = 0.670
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Substitute (1) and (2) into (3), solve for x and 9.81-x, then
mass % Au = (mass Au/9.81)*100 = ?
mass % Ag = (mass Ag/9.81)*100 = ?
Post your work if you get stuck.
How do you find x? I understand how to set up the problem, but I don't know how to get x.
To find the percentage of gold (by mass) in the jewelry, we need to calculate the mass of gold and silver present in the jewelry, and then determine the ratio of gold mass to the total jewelry mass.
Step 1: Calculate the mass of gold in the jewelry.
The density of gold is 19.3 g/cm3, and the volume of the jewelry is 0.670 cm3.
Mass of gold = Volume of gold × Density of gold
Mass of gold = 0.670 cm3 × 19.3 g/cm3
Mass of gold = 12.931 g
Step 2: Calculate the mass of silver in the jewelry.
To find the mass of silver, we subtract the mass of gold from the total mass of the jewelry.
Mass of silver = Total mass of jewelry - Mass of gold
Mass of silver = 9.81 g - 12.931 g (We have a negative value here, indicating that our assumption is incorrect. The total weight is less than the weight of gold. Therefore, the percentage of gold is greater than 100%.)
Since it is not possible to have more than 100% of a substance, we can conclude that our assumption about the composition of the jewelry is incorrect.
To calculate the percentage of gold (by mass) in the jewelry, you need to use the formula:
Percentage of gold (by mass) = (Mass of gold / Total mass of the jewelry) * 100
To find the mass of gold, you can use the formula:
Mass of gold = Volume of gold * Density of gold
In this case, you know the volume of the jewelry and the densities of gold and silver, so you can calculate the mass of gold. Let's break down the steps:
Step 1: Calculate the mass of gold:
Given:
Density of gold = 19.3 g/cm3
Volume of gold = Volume of jewelry = 0.670 cm3
Using the formula:
Mass of gold = Volume of gold * Density of gold
Mass of gold = 0.670 cm3 * 19.3 g/cm3
Mass of gold = 12.931 g
Step 2: Calculate the mass of silver:
Given:
Density of silver = 10.5 g/cm3
Volume of silver = Total volume of the jewelry - Volume of gold = 0.670 cm3 - 0.670 cm3 (since the jewelry only contains gold and silver)
Volume of silver = 0 cm3 (The volume of silver is zero)
Since the volume of silver is zero, the mass of silver is also zero.
Step 3: Calculate the total mass of the jewelry:
The total mass of the jewelry is the sum of the mass of gold and the mass of silver.
Total mass of the jewelry = Mass of gold + Mass of silver
Total mass of the jewelry = 12.931 g + 0 g (since the mass of silver is zero)
Total mass of the jewelry = 12.931 g
Step 4: Calculate the percentage of gold (by mass) in the jewelry:
Using the formula:
Percentage of gold (by mass) = (Mass of gold / Total mass of the jewelry) * 100
Percentage of gold (by mass) = (12.931 g / 12.931 g) * 100
Percentage of gold (by mass) = 100%
Therefore, the jewelry is 100% gold (by mass).