Question

A graduated cylinder is filled to the 40.00 -mL mark with a mineral oil. The masses of the cylinder before and after the addition of the mineral oil are \(124.966 \mathrm{~g}\) and \(159.446 \mathrm{~g}\), respectively. In a separate experiment, a metal ball bearing of mass \(18.713 \mathrm{~g}\) is placed in the cylinder and the cylinder is again filled to the 40.00 -mL mark with the mineral oil. The combined mass of the ball bearing and mineral oil is \(50.952 \mathrm{~g}\). Calculate the density and radius of the ball bearing. [The volume of a sphere of radius \(r\) is \(\left.(4 / 3) \pi r^{3} .\right]\)

Short answer

The density of the ball bearing is \(0.862 \mathrm{~g/cm}^3\) and its radius is \(1.50 \mathrm{~cm}\)

Step-by-step solution

Determine Mass and Volume of Mineral Oil

First, find the mass of the mineral oil by subtracting the mass of the empty cylinder from the mass of the cylinder with the mineral oil. This gives us \(159.446 \mathrm{~g}\) - \(124.966 \mathrm{~g}\) = \(34.480 \mathrm{~g}\). Next, we need to convert the volume of the oil from mL to cm³, as the SI unit of volume is cubic meter (m³). However, it's more convenient to use cm³ (where \(1 \mathrm{~mL}\) = \(1 \mathrm{~cm}^3\)) for this problem. Therefore, the volume of the mineral oil is \(40.00 \mathrm{~mL}\) = \(40.00 \mathrm{~cm}^3\)

Calculate the Density of Mineral Oil

Density is calculated by dividing mass by volume. Using the mass and volume from Step 1, we find that the density of the mineral oil is \(34.480 \mathrm{~g}\) / \(40.00 \mathrm{~cm}^3\) = \(0.862 \mathrm{~g/cm}^3\)

Find the Volume of Oil Displaced by Ball Bearing

When the ball bearing is added to the cylinder filled with oil, the oil level stays the same which means the ball bearing displaces an amount of oil equal to its own volume. To find this volume, we subtract the combined mass of the ball bearing and oil from the mass of the oil we found in Step 1. This gives us \(50.952 \mathrm{~g}\) - \(34.480 \mathrm{~g}\) = \(16.472 \mathrm{~g}\). Then, we divide this mass by the density of the oil to find the volume of oil displaced: \(16.472 \mathrm{~g}\) / \(0.862 \mathrm{~g/cm}^3\) = \(19.10 \mathrm{~cm}^3\)

Calculate the Radius of Ball Bearing

The volume of a sphere is given by the formula \(V = 4/3 * \pi * r^3\), where V is the volume and r is the radius. Since we have the volume of the sphere, we can rearrange this formula to solve for the radius: \(r = ((3*V)/(4*\pi))^{1/3}\). Substituting the volume of the ball bearing we found in Step 3, we find its radius: \(r = ((3 * 19.10 \mathrm{~cm}^3) /(4 * \pi))^{1/3}\) = \(1.50 \mathrm{~cm}\)

Key concepts

Understanding the Graduated Cylinder

A graduated cylinder is a common laboratory tool used for measuring the volume of a liquid. It has markings along its side that indicate the volume it contains. These markings allow precise measurements. In the exercise, the cylinder is first filled to the 40.00 mL mark with mineral oil. This allows us to accurately assess the volume of the oil.

Graduated cylinders are favored for their accuracy and ease of use. When using them, it's important to observe the meniscus. The meniscus is the curve at the surface of the liquid, and measurements are typically taken at the lowest point of the meniscus at eye level. This ensures that readings are not skewed by surface tension.

• Always read the liquid level at eye level.
• Ensure the cylinder is on a flat surface for accurate measurement.
• Use the same unit (mL or cm³) consistently, as 1 mL = 1 cm³ in this context.

Properties of Mineral Oil

Mineral oil, used in this exercise, is a colorless and odorless oil derived from petroleum. It serves many purposes, including as a lubricant and a hydraulic fluid. Understanding its density is vital for various applications.

The density of a substance is defined as its mass per unit volume. In the exercise, we found the mass of the mineral oil to be 34.480 g, with a volume of 40.00 mL. This gives it a density of 0.862 g/cm³. Density is an important property because it helps us understand how substances will interact with each other.

• Makes sure to always distinguish between mass (measured in grams) and volume (measured in mL or cm³).
• Remember that density = mass/volume.
• The density formula helps determine whether substances will float or sink in each other.

Exploring Volume Displacement

Volume displacement is a method used to determine the volume of an irregularly-shaped object, such as a metal ball bearing. In this context, when you drop an object into a liquid, it pushes the liquid out of the way, displacing it. The amount of liquid displaced can be measured to find the volume of the object.

In the exercise, the ball bearing is added to the cylinder filled with mineral oil, keeping the liquid level constant. This implies that the oil displaced is equal to the volume of the ball bearing. By finding the change in mass—subtracting the combined mass of oil and bearing from the initial mass—we find the displaced oil's mass. Dividing this by the density gives us the volume displaced.

• Identify the volume displacement with: **Volume displaced = Change in mass / Density**.
• Ensure the object is fully submerged for accurate displacement measurements.
• This method is ideal for finding the volume of objects that can't be easily measured directly.

Calculating Sphere Volume and Radius

The volume of a sphere is critical in calculating its radius, especially for objects like ball bearings. The formula for the volume of a sphere is \(V = \frac{4}{3} \pi r^3\), where \(V\) is the volume and \(r\) is the radius. In this exercise, the volume determined by displacement was 19.10 cm³.

By rearranging the formula, we solve for the radius: \(r = \left(\frac{3V}{4\pi}\right)^{\frac{1}{3}}\). Substituting our calculated volume into this formula gives us the radius of the ball bearing. Calculating these accurately is important in fields like manufacturing, where precision is key.

• Use the volume displacement method to find exact measurements.
• Solve for radius by rearranging the sphere formula.
• Carefully substitute values to ensure correct results.