Problem 71 - Chemistry: An Atoms-Focused Approach

Question

The average density of Earth is \(5.5 \mathrm{g} / \mathrm{cm}^{3} .\) The mass of Venus is \(81.5 \%\) of Earth's mass, and the volume of Venus is \(88 \%\) of Earth's volume. What is the density of Venus?

Step-by-Step Solution

Answer

Given the average density of Earth as \(5.5 \mathrm{g}/\mathrm{cm}^3\), and the ratios of Venus's mass and volume with respect to Earth are \(0.815\) and \(0.88\) respectively, calculate the density of Venus. Solution: The density of Venus is approximately \(5.14 \mathrm{g}/\mathrm{cm}^3\).

Step 1: Write the density formula

To begin with, let's write down the density formula, which states that the density of an object is its mass divided by its volume: $$ \text{density} = \frac{\text{mass}}{\text{volume}} $$

Step 2: Calculate the mass and volume of Earth

Now, let's find the mass and volume of Earth using the given density: $$ \text{density} = \frac{\text{mass}}{\text{volume}} \Rightarrow 5.5 = \frac{\text{mass}_\text{(Earth)}}{\text{volume}_\text{(Earth)}} $$ Since we only need ratios for this calculation, we can leave the mass and volume of Earth as variables at this point.

Step 3: Calculate the mass and volume of Venus

Using the provided ratios, we can calculate the mass and volume of Venus, with respect to Earth: $$ \text{mass}_\text{(Venus)} = 0.815 \times \text{mass}_\text{(Earth)} $$ $$ \text{volume}_\text{(Venus)} = 0.88 \times \text{volume}_\text{(Earth)} $$

Step 4: Calculate the density of Venus

Now, plug in the values for the mass and volume of Venus in the density formula: $$ \text{density}_\text{(Venus)} = \frac{\text{mass}_\text{(Venus)}}{\text{volume}_\text{(Venus)}} = \frac{0.815 \times \text{mass}_\text{(Earth)}}{0.88 \times \text{volume}_\text{(Earth)}} $$

Step 5: Simplify the expression

To simplify the expression, notice that the mass and volume of Earth effectively cancel out, leaving only the constants and the known Earth density: $$ \text{density}_\text{(Venus)} = \frac{0.815 \times \text{mass}_\text{(Earth)}}{0.88 \times \text{volume}_\text{(Earth)}} = 5.5 \cdot \frac{0.815}{0.88} $$

Step 6: Calculate the final answer

Finally, compute the density of Venus by multiplying the constants and the density of Earth: $$ \text{density}_\text{(Venus)} = 5.5 \cdot \frac{0.815}{0.88} \approx 5.14 \mathrm{g}/\mathrm{cm}^3 $$ The density of Venus is approximately \(5.14 \mathrm{g}/\mathrm{cm}^3\).

Key Concepts

Understanding Mass

Mass refers to the amount of matter contained in an object. It is often measured in units like grams or kilograms. Mass does not change regardless of location or the presence of gravity, unlike weight which can change with gravity. For instance, a book has the same mass whether it is on Earth or on the Moon.

In the context of celestial bodies like planets, mass is a crucial factor as it influences the gravitational pull of the planet. For the problem at hand, we know that Venus has a mass that is 81.5% of Earth's mass. This comparison allows us to describe the mass of Venus in terms of Earth's mass without knowing the exact figures for either.

This is useful for calculating ratios and helps us get to the solution without needing exact values. Remember, mass is simply a measure of how much "stuff" is there within an object or a planet, like in this exercise. Understanding mass helps us understand how it will influence other variables like volume and ultimately density.

Exploring Volume

Volume measures the space that an object occupies. It's typically measured in cubic units, such as cubic centimeters (cm\(^3\)) or cubic meters (m\(^3\)). Just like when you fill a box with water, the volume is considered to be the size of that box or the space it takes up.

In our exercise, we are comparing the volume of Venus to that of Earth. Venus's volume is said to be 88% of Earth's volume. This means if we had a container that could hold Earth's volume, Venus would fill 88% of that container. Understanding volume in terms of ratios helps us relate one object to another without needing direct numerical values, which is very handy in astrophysics.

Volume plays a key role in determining density. When calculating the density of Venus, knowing its volume in relation to Earth's helps simplify the math, because we can focus on changes in ratios rather than absolute values.

Linking Ratios to Density

Ratios let us compare quantities directly without needing to know their exact values. In this exercise, both the mass and volume of Venus are expressed as ratios relative to Earth. Understanding ratios is a powerful mathematical tool that simplifies complex problems.

Here, we have Venus's mass as a ratio of Earth's mass (81.5%) and the volume also as a ratio of Earth's volume (88%). These ratios allow us to calculate the density of Venus using the density formula: \[ \text{density} = \frac{\text{mass}}{\text{volume}} \] with relative ease.

When we substitute these ratios into the formula, the mass and volume of Earth cancel out, demonstrating the elegance of working with ratios. This leaves us with a simple multiplication of Earth's density by the mass ratio over the volume ratio, helping us easily determine Venus's density. This approach is efficient not just for planetary science but also for practical calculations where relative quantities are more manageable than absolute values.