Question

You measure the masses and volumes of two cylinders. The mass of cylinder 1 is \(1.35\) times the mass of cylinder 2. The volume of cylinder 1 is \(0.792\) times the volume of cylinder 2 . If the density of cylinder 1 is \(3.85 \mathrm{~g} / \mathrm{cm}^{3}\), what is the density of cylinder 2?

Short answer

The density of cylinder 2 is approximately 2.24 \mathrm{~g/cm}^3.

Step-by-step solution

Identify the Known Quantities

Let the mass of cylinder 2 be represented by \(m_2\), and its volume be \(V_2\). Accordingly, the mass of cylinder 1, \(m_1\), and its volume, \(V_1\), are given by \(m_1 = 1.35m_2\) and \(V_1 = 0.792V_2\). The density of cylinder 1 is given as \(3.85 \mathrm{~g/cm}^3\).

Write the Density Formula for Both Cylinders

Density (\(\rho\)) is mass (\

Express the Mass and Volume of Cylinder 2 in Terms of Cylinder 1

Since we know \(m_1 = 1.35m_2\) and \(V_1 = 0.792V_2\), we can express the mass and volume of cylinder 2 in terms of cylinder 1's mass and volume.

Calculate the Density of Cylinder 2

Using the density formula \(\rho = \frac{m}{V}\) and substituting the respective values, we get \(\rho_2 = \frac{m_2}{V_2} = \frac{m_1}{1.35} * \frac{1}{V_1 * \frac{1}{0.792}}\). To find \(\rho_2\), we can substitute the known density of cylinder 1 into this equation. \(\rho_2 = \frac{3.85}{1.35 * \frac{1}{0.792}}\).

Perform the Calculations to Find Density of Cylinder 2

Now we calculate the density of cylinder 2 by performing the arithmetic operation: \(\rho_2 = \frac{3.85}{1.35 * \frac{1}{0.792}} = 3.85 * \frac{0.792}{1.35}\), which will give us the density of cylinder 2 in \mathrm{~g/cm}^3.

Key concepts

Mass-Volume Relationship

Understanding the mass-volume relationship is crucial when studying the physical properties of materials in chemistry. Mass is a measure of the amount of matter in an object, usually expressed in grams or kilograms. Volume, on the other hand, is the amount of space that an object occupies, typically measured in milliliters or cubic centimeters for solids.To visualize how mass and volume are related, imagine two identical containers filled with different substances. One could be filled with feathers and the other with sand. Even though the volume — the space the containers occupy — is the same, their masses are different due to the density of the substances inside. Density is the link between mass and volume and is a measure of how much mass is contained in a given volume.When solving problems that involve different materials or shapes, like the cylinders in our example, the mass-volume relationship tells us that if two objects have the same volume but different masses, the object with greater mass is denser. Conversely, if they have the same mass but different volumes, the object with the larger volume is less dense. This principle is helpful in determining the density of unknown materials by comparing them to known quantities.

Density Formula

The density formula is a fundamental equation in chemistry that connects mass, volume, and density. It's expressed as \[\begin{equation} \rho = \frac{m}{V} \end{equation}\] where \( \rho \) stands for density, \( m \) for mass, and \( V \) for volume. This concise equation allows us to calculate the density of a substance if we know its mass and volume.For instance, in the exercise, cylinder 1's density is given and can be used alongside its mass-volume relationship to determine the density of cylinder 2. By rearranging the density formula, you can solve for any single variable as long as the other two are known. This versatility makes the density formula an essential tool in a variety of scientific fields beyond just chemistry, including physics, engineering, and even in everyday applications like cooking or nutrition where measurements of ingredients are important.

Problem Solving in Chemistry

In chemistry, problem-solving often involves understanding and applying various concepts to find a solution to a given problem. It usually includes identifying the known and unknown quantities, choosing the appropriate formula, substituting the known values, and algebraically solving for the unknown variable.In our example with the two cylinders, we followed a systematic approach to reach a solution. Firstly, we identified the known quantities: the mass and volume relationships between the two cylinders and the density of cylinder 1. We used the density formula to express these relationships mathematically. Then, we substituted the known values into the equation to solve for the unknown — the density of cylinder 2. Finally, we performed the necessary calculations.This structured approach is powerful for a couple of reasons. Firstly, it breaks the problem down into manageable steps, preventing one from feeling overwhelmed. Secondly, it allows for checking at each step which is essential for finding errors in complex calculations typical in chemistry problems. Remembering to work methodically, to include units for clarity, and to keep track of the values you are calculating ensures accurate and successful problem-solving in chemistry.