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Chapter 1: Problem 45

Precious metals and gems are measured in troy weights in the English system: $$ \begin{aligned} 24 \text { grains } &=1 \text { pennyweight (exact) } \\ 20 \text { pennyweight } &=1 \text { troy ounce (exact) } \\ 12 \text { troy ounces } &=1 \text { troy pound (exact) } \\ 1 \text { grain } &=0.0648 \mathrm{~g} \\ 1 \text { carat } &=0.200 \mathrm{~g} \end{aligned} $$ a. The most common English unit of mass is the pound avoirdupois. What is 1 troy pound in kilograms and in pounds? b. What is the mass of a troy ounce of gold in grams and in carats? c. The density of gold is \(19.3 \mathrm{~g} / \mathrm{cm}^{3}\). What is the volume of a troy pound of gold?

Short Answer

Expert verified
a. 1 troy pound is approximately 0.373 kg and 0.823 pounds avoirdupois. b. The mass of a troy ounce of gold is approximately 31.104 grams and 155.52 carats. c. The volume of a troy pound of gold is approximately 19.34 cubic centimeters.

Step by step solution

01

Convert troy pound to grams

First, we need to convert 1 troy pound to grams using the given conversion rates. We know that 1 troy pound is equal to 12 troy ounces, and 1 troy ounce is equal to 20 pennyweight, and 1 pennyweight is equal to 24 grains. We will use these conversions to find out how many grams is 1 troy pound. 1 troy pound = 12 troy ounces = 12 * 20 pennyweight = 240 pennyweight Now, we'll convert 240 pennyweight to grams. 240 pennyweight = 240 * 24 grains = 5760 grains Finally, we'll convert 5760 grains to grams. 1 grain = 0.0648 g 5760 grains = 5760 * 0.0648 g ≈ 373.248 g
02

Convert grams to kilograms and pounds avoirdupois

Now we have the equivalent mass of 1 troy pound in grams, we'll convert it to kilograms and pounds avoirdupois. 1 kg = 1000 g 373.248 g ≈ 0.373 kg 1 pound avoirdupois = 453.592 g 373.248 g ≈ 373.248/453.592 ≈ 0.823 pounds avoirdupois So, 1 troy pound is approximately 0.373 kg and 0.823 pounds avoirdupois. #b. Calculate the mass of a troy ounce of gold in grams and carats#
03

Convert troy ounce to grams

We know 1 troy ounce is equal to 20 pennyweight, and 1 pennyweight is equal to 24 grains. Using these conversions, we'll find out how many grams are in 1 troy ounce. 1 troy ounce = 20 pennyweight = 20 * 24 grains = 480 grains Now, we'll convert 480 grains to grams. 480 grains = 480 * 0.0648 g ≈ 31.104 g
04

Convert grams to carats

Now we have the equivalent mass of 1 troy ounce in grams, we'll convert it to carats. 1 carat = 0.200 g 31.104 g = 31.104/0.200 ≈ 155.52 carats So, the mass of a troy ounce of gold is approximately 31.104 grams and 155.52 carats. #c. Calculate the volume of a troy pound of gold#
05

Recall the mass of a troy pound in grams

From part (a), we calculated that the mass of 1 troy pound is approximately 373.248 g.
06

Use the density formula to find the volume

We know the density of gold is 19.3 g/cm³. Using the formula Density = Mass/Volume, we can find the volume of a troy pound of gold. Volume = Mass/Density Volume = 373.248 g / 19.3 g/cm³ ≈ 19.34 cm³ So, the volume of a troy pound of gold is approximately 19.34 cubic centimeters.

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