Gold is alloyed (mixed) with other metals to increase its...

Gold is alloyed (mixed) with other metals to increase its hardness in making jewelry. (a) Consider a piece of gold jewelry that weighs $9.85 \mathrm{~g}$ and has a volume of $0.675 \mathrm{~cm}^{3}$. The jewelry contains only gold and silver, which have densities of 19.3 and $10.5 \mathrm{~g} / \mathrm{cm}^{3}$, 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. (b) The relative amount of gold in an alloy is commonly expressed in units of carats. Pure gold is 24 carat, 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 carat. State the purity of the gold jewelry in carats.

Gold is alloyed (mixed) with other metals to increase its hardness in making jewelry.
(a) Consider a piece of gold jewelry that weighs $9.85 \mathrm{~g}$ and has a volume of $0.675 \mathrm{~cm}^{3}$. The jewelry contains only gold and silver, which have densities of 19.3 and $10.5 \mathrm{~g} / \mathrm{cm}^{3}$, 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. (b) The relative amount of gold in an alloy is commonly expressed in units of carats. Pure gold is 24 carat, 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 carat. State the purity of the gold jewelry in carats.

Key Concepts

Density

Density is a measure of mass per unit volume and connects the mass and volume of each component in a mixture. It is central to problems involving mixtures because, given the total mass and calculated volume, one can use the known densities of the individual substances to set up relationships and equations to determine the composition.

Volume Additivity in Mixtures

This concept assumes that the total volume of a mixture is equal to the sum of the individual volumes of its components. It simplifies the calculation of the composition by allowing the formulation of a volume equation based on each material's contribution, linking mass and density within the system.

System of Linear Equations in Mixture Problems

When dealing with alloy compositions, setting up a system of linear equations based on the conservation of mass and volume is essential. This approach involves defining variables for the unknown quantities of each component and relating them through equations derived from their densities and the known totals, allowing for the determination of individual masses.

Percentage Composition by Mass

Percentage composition by mass is calculated by dividing the mass of the particular component by the total mass and multiplying by 100. This is a key concept in material science and chemistry for determining the relative amounts of elements or compounds in a mixture or alloy.

Carat Calculation for Gold Purity

The carat rating system is used to express the purity of gold alloys relative to pure gold, which is 24 carats. This involves converting the percentage of gold by mass into carats by comparing it to the 24-carat standard, and it is a widely used convention in jewelry and materials science.

Transcript

00:01 We are asked to consider a piece of jewelry.

00:04 In this particular piece of jewelry, we know that it has a mass of 9 .85 grams.

00:12 We know that it has a volume of 0 .675 cubic centimeters.

00:21 We know that the jewelry contains only gold and silver, and we know that the density of gold is 19 .3 grams per cubic centimeter, and the density of silver is 10%.

00:38 0 .5 grams per cubic centimeter.

00:43 We are asked to calculate the percentage by mass of gold in the jewelry.

00:55 That's part a.

00:58 Okay, let's begin.

01:00 This problem isn't too difficult to do.

01:03 I'm going to just sort of set up, i think i wrote everything here.

01:10 We know that density equals mass divided by volume.

01:16 So we're going to figure out our mass here and then do a little bit of rearranging.

01:22 So let me just put some values into here.

01:27 Our mass that we have is going to equal the density, mass of the object equals the mass of gold plus the mass of silver.

01:51 So the mass of the object is going to be equal to the mass of gold, which will be the density of gold.

02:00 We're just going to write d -a -u times the volume of gold plus the density of silver times the volume of silver.

02:21 And we're going to let the volume of silver equal our given volume 0 .675 cubic centimeters minus the volume of our gold.

02:37 Now we can go ahead and start substituting our values and for this.

02:41 Our mass was 9 .875 grams equals, i'll leave these off, 19 .3 grams per cubic centimeter times i'm going to call this x now just to make it easier plus 10 .5 grams per cubic centimeter times 0 .675 centimeters cubed minus x.

03:24 Okay we can distribute that out now.

03:27 I'm just going to leave this part and i'm going to distribute this and i will get 7 .087.

03:37 5 minus 10 .5 grams per centimeter cubed and grams.

03:49 This will stay the same.

04:03 Now you can see i'm going to subtract this and combine these terms...

Please give Ace some feedback