Problem 68 - Chemistry Matter and Change

Question

An object with a mass of 7.5 g raises the level of water in a graduated cylinder from 25.1 mL to 30.1 mL. What is the density of the object?

Step-by-Step Solution

Answer

The density of the object is \(1.5 \ g/cm^3\).

Step 1: Identify the given information

We are given the following information: - Mass of the object, m = 7.5 g - Initial water level in the graduated cylinder, V₁ = 25.1 mL - Final water level in the graduated cylinder after the object is submerged, V₂ = 30.1 mL Now, let's find the volume of the object.

Step 2: Calculate the volume of the object using the water displacement method

To find the volume of the object, we will use the water displacement method. The volume of the object is equal to the difference between the final water level and the initial water level. Volume of the object, V = V₂ - V₁ Plug in the values: V = 30.1 mL - 25.1 mL V = 5.0 mL

Step 3: Convert the volume unit

To calculate the density, we need to make sure the volume is in cubic centimeters (cm³). Since 1 mL is equivalent to 1 cm³, we can directly use the volume as: V = 5.0 cm³

Step 4: Calculate the density of the object

We can now use the mass and volume to calculate the density of the object using the density formula, which is: Density (ρ) = mass (m) / volume (V) Plug in the values: ρ = m / V ρ = 7.5 g / 5.0 cm³

Step 5: Solve for the density

Divide the mass by the volume to get the density: ρ = 7.5 g / 5.0 cm³ ρ = 1.5 g/cm³ Now, we have found the density of the object. The object has a density of 1.5 g/cm³.

Key Concepts

Water Displacement Method

Understanding the water displacement method is crucial for accurately determining the volume of an irregularly shaped object. In your daily life, this method might remind you of how taking a bath causes the water level to rise. Similarly, when an object is submerged in water, it displaces an amount of water equal to its own volume. The key here is observing the change in water level in a graduated cylinder.

The steps are straightforward: start by noting the initial level of water (V₁). When the object is submerged, the water level rises to a new mark (V₂). The volume of the object (V) is simply the difference between these two readings, mathematically expressed as
V = V₂ - V₁.
Such direct volume measurement via water displacement is especially handy in a classroom setting or any situation requiring quick and precise volume determinations for objects that don't have simple geometric shapes.

Volume Measurement

Volume is a fundamental concept in science, representing the three-dimensional space occupied by a substance or an object. Precise volume measurement is essential for a multitude of applications, including cooking, manufacturing, and especially in scientific experiments where it is critical to have accurate data. There are multiple methods to measure volume depending on the state and shape of the material.

For liquids, common laboratory equipment like graduated cylinders, pipettes, or burettes serve as the perfect tools. For solids with regular shapes, formulas based on geometry are used. For irregular solids, as mentioned, the water displacement method is often the go-to. It's important to remember that volume units can vary (liters, milliliters, cubic centimeters, etc.), and conversions may be necessary for certain calculations.

Mass and Volume

Mass and volume are two fundamental physical quantities in science, often measured in grams and milliliters or cubic centimeters, respectively. The mass of an object refers to the amount of matter it contains and remains constant, irrespective of its location. On the other hand, volume is the space that object occupies, and it can vary dramatically, especially when dealing with gases that are highly responsive to temperature and pressure changes.

In experiments and calculations, the consistency in the measurement units used for mass and volume is vital. Even though the metric system is standard, it's useful to be vigilant about the units involved in the calculation to avoid errors—mixing up milliliters with liters, or grams with kilograms, can lead to incorrect conclusions.

Density Formula

Density is a measure of how much mass is contained in a given volume and is exemplified by the density formula:
\[\begin{equation}\rho = \frac{m}{V}\end{equation}\]
where \(\rho\) is the density, \(m\) is the mass, and \(V\) is the volume. This seemingly simple equation is powerful because it allows us to compare the 'compactness' of different materials, crucial for applications ranging from shipbuilding (where buoyancy is key) to cooking (think about the density of dough affecting bread texture).

For the homework exercise, by plugging in the object's mass and volume that we already determined, the density is calculated as:\[\begin{equation}\rho = \frac{7.5\,g}{5.0\,cm^3} = 1.5\,g/cm^3\end{equation}\]
The principle behind this is simple yet profound, affecting not just academic exercises but also real-world engineering, environmental studies, and even our daily decisions, such as choosing the right materials for household needs.