Problem 91 - Chemistry and Chemical Reactivity

Question

An ancient gold coin is \(2.2 \mathrm{cm}\) in diameter and \(3.0 \mathrm{mm}\) thick. It is a cylinder for which volume = (\pi) (radius) \(^{2}\) (thickness). If the density of gold is 19.3 \(\mathrm{g} / \mathrm{cm}^{3},\) what is the mass of the coin in grams?

Step-by-Step Solution

Answer

The mass of the coin is 22.0 g.

Step 1: Convert Measurements

First, convert the thickness from millimeters to centimeters. Since there are 10 millimeters in a centimeter, the thickness of the coin is \(3.0 \text{ mm} = 0.3 \text{ cm}\). The diameter is already given in centimeters.

Step 2: Calculate the Radius

The radius is half of the diameter. Given that the diameter is \(2.2 \text{ cm}\), the radius is \(\frac{2.2}{2} = 1.1 \text{ cm}\).

Step 3: Calculate the Volume

Use the formula for the volume of a cylinder: \(V = \pi r^2 h\). Plug in the radius \(r = 1.1 \text{ cm}\) and the thickness \(h = 0.3 \text{ cm}\): \[ V = \pi (1.1)^2 (0.3) = \pi (1.21) (0.3) = 0.363 \pi \text{ cm}^3 \]

Step 4: Calculate the Volume Approximation

Use \(\pi \approx 3.14\) to approximate the volume: \[ V \approx 0.363 \times 3.14 = 1.14042 \text{ cm}^3 \]

Step 5: Calculate the Mass

Use the density formula \(\text{mass} = \text{volume} \times \text{density}\). With density \(19.3 \text{ g/cm}^3\): \[ \text{mass} = 1.14042 \times 19.3 = 22.036 \text{ g} \]

Step 6: Round to Proper Significant Figures

Considering the significant figures in the given values, the mass should be rounded to three significant figures: The mass is \(22.0 \text{ g}\).

Key Concepts

Volume of a Cylinder

To find the mass of our ancient gold coin, we first need to calculate its volume since this is a critical step for further calculations. The formula for the volume of a cylinder is:
\[ V = \pi r^2 h \]
where:

• \( V \) is the volume of the cylinder,
• \( r \) is the radius of the base,
• \( h \) is the height or thickness of the cylinder.

In our case, the ancient coin can be considered similar to a small cylinder. Given that its diameter is 2.2 centimeters, you calculate the radius by dividing the diameter by two, resulting in 1.1 centimeters.
The thickness, converted into centimeters, is 0.3 cm.
By plugging these values into the formula, we calculate the volume as:
\[ V = \pi (1.1)^2 (0.3) = 0.363\pi \text{ cm}^3 \]
Approximating \( \pi \) to 3.14, we find the volume to be approximately 1.14042 cm³.

Density

Understanding density is crucial when relating mass and volume. Density is defined as the mass per unit volume of a substance and is expressed by the formula:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
For the gold coin, the density is given as 19.3 grams per cubic centimeter (g/cm³), a typical value for gold.
After determining the volume of the coin to be approximately 1.14042 cm³, we use the relationship of density to calculate mass:
\[ \text{Mass} = \text{Density} \times \text{Volume} \]
Plugging in our numbers, we have:
\[ \text{Mass} = 19.3 \cdot 1.14042 = 22.036 \text{ g} \]
This step shows how the density of a material informs us about how much a certain volume of it will weigh.

Significant Figures

Significant figures are important in scientific measurements and calculations, as they indicate the precision of a measurement. When calculating the mass of the coin, we start with measurements that have significant figures.
For instance, the diameter and thickness of the coin both have two significant figures. Consequently, all calculations derived from these measurements need an appropriate number of significant figures to maintain consistency.
When we initially calculate the mass as 22.036 grams, we notice it has more significant figures than the input data.
To be accurate and reflect the data's precision, we round the mass to the correct number of significant figures, which is three in this instance, producing a final mass value of 22.0 grams.
Rounding according to significant figures ensures that our results are reliable concerning the initial measurement's precision.

Unit Conversion

Before performing calculations, it's often necessary to convert units to ensure consistency and accuracy. Unit conversion becomes crucial when measurements are not presented in the same system or when they differ in scale.
In the case of the gold coin problem, the diameter is already in centimeters, but the thickness is given in millimeters. Since standard calculations particularly in volume use consistent units, we must convert millimeters to centimeters; knowing there are 10 millimeters in a centimeter helps us here.
This conversion is simple:

• Convert 3.0 mm to cm by dividing 3.0 by 10: \( 3.0 \text{ mm} = 0.3 \text{ cm} \).

Having the same unit for all measurements allows us to plug them into equations without errors related to unit mismatches.
Accurate and proper unit conversion is a vital skill in scientific measurements and calculations.