Problem 149 - Introductory Chemistry Atoms First

Question

The density of an irregularly shaped object is determined by immersing the object in water. If the mass of the object is \(8.34 \mathrm{~g}\) and the water level rises from \(25.00 \mathrm{~mL}\) to \(28.10 \mathrm{~mL}\), what is the density of the object in grams per milliliter?

Step-by-Step Solution

Answer

The density of the irregularly shaped object can be calculated using the formula, Density \(= \frac{Mass}{Volume}\). Given the mass of the object is 8.34 g and the water level rises from 25.00 mL to 28.10 mL, the volume of the object is 3.10 mL. The density is then calculated as \(Density = \frac{8.34}{3.10} ≈ 2.69 \mathrm{~g/mL}\).

Step 1: Identify the given information

We are given the mass of the irregularly shaped object (8.34 g) and the change in the water level (from 25.00 mL to 28.10 mL).

Step 2: Find the volume of the object

When the object is immersed in water, the water level rises from 25.00 mL to 28.10 mL. This means that the volume of the object is equal to the change in water level, which is: Volume = Final water level - Initial water level Volume = 28.10 mL - 25.00 mL Volume = 3.10 mL

Step 3: Calculate the density of the object

Now, we have the mass and volume of the object. We can use the density formula to find the density of the object: Density = Mass / Volume Density = 8.34 g / 3.10 mL Density ≈ 2.69 g/mL The density of the irregularly shaped object is approximately 2.69 grams per milliliter.

Key Concepts

Mass and Volume

Understanding the concepts of mass and volume is essential when we are looking to determine the density of an object. Mass is a measure of the amount of matter in an object and is typically measured in grams (g) or kilograms (kg). Volume, on the other hand, refers to the amount of space an object occupies and is usually measured in milliliters (mL) or liters (L) for liquids and cubic centimeters (cm³) or cubic meters (m³) for solids.

To find the density of an object, you need accurate measurements of both its mass and volume. For regular-shaped objects, volume can be calculated using mathematical formulas based on their dimensions. However, finding the volume of an irregularly shaped object, like in the provided exercise, requires a different approach such as the displacement method.

Displacement Method

The displacement method is a practical way to measure the volume of irregularly shaped objects that might otherwise be difficult to measure directly. This method consists of immersing the object into a fluid and measuring the volume of fluid displaced by the object. The steps are straightforward:

• Fill a graduated cylinder or overflow can with a liquid, typically water, and record the initial level of the liquid.
• Gently lower the object into the liquid and ensure it's fully submerged.
• Record the new level of the liquid after the object's immersion.
• The difference in the liquid level before and after the object is submerged equals the volume of the object.

Using the displacement method allows for precise volume measurements which are crucial for calculating density, especially for objects with complex shapes, as demonstrated in the original exercise.

Density Formula

Once the mass and volume are known, the density formula can be used to find an object’s density. The formula is expressed as:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
This formula indicates that density is the ratio of mass to volume and is often measured in grams per milliliter (g/mL) or kilograms per cubic meter (kg/m³). In the context of the exercise, noticing the units is vital since the mass is provided in grams and the volume in milliliters, making the calculated density direct and straightforward.

The formula highlights that density is an intrinsic property of a material—it remains the same regardless of the amount of substance. Knowing the density of a substance is crucial in many scientific and engineering applications, such as determining whether an object will float or sink in a fluid, specifying materials for structural support, or identifying substances based on their characteristic densities.