Problem 67 - Chemistry Matter and Change

Question

The density of aluminum is 2.7 \(\mathrm{g} / \mathrm{mL} .\) What is the volume of 8.1 \(\mathrm{g}\) ?

Step-by-Step Solution

Answer

The volume of 8.1 g of aluminum is 3.0 mL.

Step 1: Identify Given Values

The density of aluminum is given as \(2.7 \, \text{g/mL}\) and the mass is \(8.1 \, \text{g}\). We need to find the volume.

Step 2: Write the Formula for Density

The formula for density \(D\) is \(D = \frac{m}{V}\), where \(m\) is the mass and \(V\) is the volume.

Step 3: Rearrange the Formula to Solve for Volume

To find the volume, solve for \(V\) by rearranging the formula: \(V = \frac{m}{D}\).

Step 4: Substitute Known Values into the Formula

Substitute \(m = 8.1 \, \text{g}\) and \(D = 2.7 \, \text{g/mL}\) into the formula for volume: \(V = \frac{8.1}{2.7}\).

Step 5: Calculate the Volume

Perform the division: \(V = \frac{8.1}{2.7} = 3.0\). Therefore, the volume of \(8.1 \, \text{g}\) of aluminum is \(3.0 \, \text{mL}\).

Key Concepts

Mass

Mass is a fundamental concept in physics and chemistry that refers to the amount of matter within an object. It is often measured in grams (g) or kilograms (kg).

Since mass measures the quantity of material in an object, it is independent of external conditions like gravity. This means that an object's mass remains the same whether it's on Earth, the moon, or floating in space.

• Mass is different from weight, which is a force exerted due to gravity acting on a mass.
• In our exercise, the mass is given as 8.1 grams, which helps determine other properties like volume when paired with density.

Understanding mass is crucial for solving problems involving density and volume. It allows us to calculate how much material is present in a certain volume when the density is known.

Volume

Volume is the amount of space that an object or substance occupies. It's generally measured in units like liters (L), milliliters (mL), or cubic meters (m³). In our problem, volume is the quantity we need to find, given the mass and density.

To better understand this:

• Volume describes how much space an object takes up. Think of it like filling a box with water to see how much it can hold.
• It can vary for the same mass if the object is solid, liquid, or gas. This is because each state of matter occupies space differently.

In the context of density calculations, volume tells us about the space taken by a certain amount of mass at a given density.

Density Formula

Understanding the density formula is key to solving problems involving mass and volume. Density (\(\rho\)) is defined as the mass (m) of an object divided by its volume (V): \[ \rho = \frac{m}{V} \]
This formula relates these three properties in a meaningful way.

• For a given substance, density is usually a constant. It's a characteristic property, like the density of aluminum, which is \(2.7 \, \text{g/mL}\) in our exercise.
• Understanding this formula allows us to extract unknown values, such as finding volume when given mass and density.

By rearranging this formula based on what you need, you can solve a variety of problems in everyday and scientific situations.

Calculations

Calculations involving density require precise substituting and rearranging of formulas. To find unknown values like volume, follow these steps:

• Start by writing the density formula: \( D = \frac{m}{V} \)
• If you need to find the volume (V), rearrange the formula to \( V = \frac{m}{D} \).
• Substitute the given values into this formula. For example, with a mass of \(8.1 \, \text{g}\) and a density of \(2.7 \, \text{g/mL}\), the equation becomes \(V = \frac{8.1}{2.7}\).
• Perform the calculation to find the volume. Here, it results in \(3.0 \, \text{mL}\).

Density-related calculations are foundational in many fields. Whether it's in chemistry labs or engineering projects, these calculations help us understand material properties and design systems accordingly.