Problem 74 - Chemistry: An Atoms-Focused Approach

Question

The sun is a sphere with an estimated mass of \(2 \times 10^{30} \mathrm{kg} .\) If the radius of the sun is \(7.0 \times 10^{5} \mathrm{km}\) what is the average density of the sun in units of grams per cubic centimeter? The volume of a sphere is \(\frac{4}{3} \pi r^{3}\)

Step-by-Step Solution

Answer

Answer: The average density of the sun is approximately \(1.43 \mathrm{g/cm}^3\).

Step 1: Convert mass and radius units

We will first convert the mass of the sun from kilograms to grams, since our answer should be in grams per cubic centimeter. To do so, we can use the conversion factor \(1 \mathrm{kg} = 1000 \mathrm{g}\): \[2 \times 10^{30} \mathrm{kg} \cdot \frac{1000 \mathrm{g}}{1 \mathrm{kg}} = 2 \times 10^{33} \mathrm{g}\] We also need to convert the radius given in kilometers to centimeters using the conversion factors \(1 \mathrm{km} = 1000 \mathrm{m}\) and \(1 \mathrm{m} = 100 \mathrm{cm}\): \[7.0 \times 10^5 \mathrm{km} \cdot \frac{1000 \mathrm{m}}{1 \mathrm{km}} \cdot \frac{100 \mathrm{cm}}{1 \mathrm{m}} = 7.0 \times 10^{10} \mathrm{cm}\]

Step 2: Calculate the volume of the sun

Now that we have the radius in centimeters, we can find the volume using the given formula for the volume of a sphere \(\frac{4}{3}\pi r^3\): \[\frac{4}{3} \pi (7.0 \times 10^{10} \mathrm{cm})^{3} = 1.4 \times 10^{33} \mathrm{cm}^3\]

Step 3: Calculate the average density of the sun

Finally, using the formula for density, which is mass divided by volume, we can calculate the average density in grams per cubic centimeter: \[\frac{2 \times 10^{33} \mathrm{g}}{1.4 \times 10^{33} \mathrm{cm}^3} = \frac{2}{1.4} \mathrm{g/cm}^3 \approx 1.43 \mathrm{g/cm}^3\] The average density of the sun is approximately \(1.43 \mathrm{g/cm}^3\).

Key Concepts

Unit Conversion

Unit conversion is an essential step in ensuring that measurements are in the correct form for calculations. In this exercise, we begin by converting the mass of the sun from kilograms to grams. This conversion is straightforward because the kilogram is part of the metric system, and we know that:

• 1 kilogram (kg) is equal to 1000 grams (g).

Thus, the sun's mass is given as \(2 \times 10^{30} \text{kg}\). To convert it to grams, multiply by 1000: \[2 \times 10^{30} \text{kg} \times \frac{1000 \text{g}}{1 \text{kg}} = 2 \times 10^{33} \text{g}\].
Next, we convert the radius of the sun from kilometers to centimeters. The sequence of conversions is as follows:

• 1 kilometer (km) = 1000 meters (m)
• 1 meter (m) = 100 centimeters (cm)

For the radius \(7.0 \times 10^{5} \text{km}\), use both conversions:\[7.0 \times 10^5 \text{km} \times \frac{1000 \text{m}}{1 \text{km}} \times \frac{100 \text{cm}}{1 \text{m}} = 7.0 \times 10^{10} \text{cm}\].
This ensures that all units are consistent and in the required form for further calculation.

Sphere Volume Formula

The volume of a sphere is a key part of calculating the average density of spherical objects like the sun. The formula for the volume \(V\) of a sphere is given by:\[ V = \frac{4}{3} \pi r^3 \]where \(r\) is the radius of the sphere and \(\pi\) is a mathematical constant approximately equal to 3.14159. In this exercise, we have already converted the radius of the sun to centimeters as \(7.0 \times 10^{10} \text{cm}\).
To find the volume, plug the radius value into the formula:\[ V = \frac{4}{3} \pi (7.0 \times 10^{10} \text{cm})^3 \]
Carrying out the calculations gives us the volume:\[ V = 1.4 \times 10^{33} \text{cm}^3 \]
This calculation shows that knowing the radius of the sun and using the sphere volume formula, you can determine the volume, a critical component in calculating density.

Density Formula

The density of an object is a measure of how much mass it has in a given volume. The formula to calculate density \(\rho\) is given by:\[ \rho = \frac{\text{mass}}{\text{volume}} \]This is expressed in grams per cubic centimeter (g/cm\(^3\)) for our exercise. With the mass of the sun converted to grams (\(2 \times 10^{33} \text{g}\)) and its volume calculated in cubic centimeters (\(1.4 \times 10^{33} \text{cm}^3\)), we can find the average density.
Substitute the values into the density formula:\[ \rho = \frac{2 \times 10^{33} \text{g}}{1.4 \times 10^{33} \text{cm}^3} \]
Calculating this gives:\[ \rho \approx 1.43 \text{g/cm}^3 \]
The result shows an average density of the sun as approximately 1.43 grams per cubic centimeter. This highlights how density is a function of both mass and volume, and why correct unit conversion is vital for accuracy.