Question

A small statue that has a mass of \(500.0 \mathrm{~g}\) displaces \(150.5 \mathrm{~mL}\) of water. What is its density in grams per milliliter?

Short answer

The density of the small statue is approximately \(3.32 \, \frac{g}{mL}\).

Step-by-step solution

Write down the given values

We are given: - Mass of the statue: \(m = 500.0 \, g\) - Volume of water displaced: \(V = 150.5 \, mL\)

Write the formula for density

The formula for density is: \(Density = \frac{Mass}{Volume}\) Denoted as: \(ρ = \frac{m}{V}\)

Plug the given values into the formula

Inserting the given values into the formula, we get: \(ρ = \frac{500.0 \, g}{150.5 \, mL}\)

Calculate the density

Now, calculate the density by dividing the mass by the volume: \(ρ = \frac{500.0}{150.5} \, \frac{g}{mL} ≈ 3.32 \, \frac{g}{mL}\)

State the final answer

The density of the small statue is approximately \(3.32 \, \frac{g}{mL}\).

Key concepts

Mass and Volume

Understanding the concepts of mass and volume is essential for many scientific calculations, including those in physics and chemistry. Mass refers to the amount of matter in an object and is usually measured in grams (g) or kilograms (kg). On the other hand, volume indicates the space an object occupies and is often expressed in milliliters (mL) or liters (L).

When working on problems involving these two concepts, it’s important to use the correct units and measuring instruments. Scales are typically used to measure mass, while graduated cylinders or measuring jugs can measure the volume of liquids. For irregularly shaped objects, like the statue in our exercise, we cannot use such direct measurement tools for volume, which is where the displacement method becomes useful.

Displacement Method

The displacement method is a technique used to measure the volume of an irregularly shaped object, one that cannot be easily measured with standard tools like a ruler or measuring cup. This method involves submerging the object in a fluid and measuring how much fluid it displaces.

Here's a simplified explanation of how it's done:

• Fill a graduated cylinder or overflow can with water.
• Submerge the object completely.
• The water level will rise proportional to the volume of the object.
• Measure the difference in water levels to determine the volume of the submerged object.

That's exactly what was done in the given exercise. The statue displaced a certain volume of water, which was then easily measured as 150.5 mL, revealing the volume of the statue. This technique is derived from Archimedes' principle and is particularly useful in our scenario.

Density Formula

Once the mass and volume of an object are known, you can calculate its density, which is the measure of mass per unit volume. The density formula, as used in our original exercise, is a straightforward division:
\[ Density = \frac{Mass}{Volume} \]
In other words, density (ρ) is equal to the mass (m) of an object divided by its volume (V). The units for density could vary, but in this exercise, we're using grams per milliliter (g/mL).

The formula is directly applied to the statue’s known mass and the volume of the water it displaced, showing a density of approximately 3.32 g/mL. This indicates not just how 'heavy' the object is, but how that mass is distributed within the volume it occupies. Density is a fundamental property that can help identify materials, understand buoyancy, and even describe conditions in various environments, such as atmospheric layers or deep-sea pressures.