Problem 92 - Chemistry The Science in Context

Question

Gasoline and water do not mix. Regular grade ( 87 octane) gasoline has a lower density \((0.73 \mathrm{g} / \mathrm{mL})\) than water \((1.00 \mathrm{g} / \mathrm{m})\) \(\mathrm{mL}) .\) A \(100 \mathrm{mL}\) graduated cylinder with an inside diameter of \(3.2 \mathrm{cm}\) contains \(34.0 \mathrm{g}\) of gasoline and \(34.0 \mathrm{g}\) of water. What is the combined height of the two liquid layers in the cylinder? The volume of a cylinder is \(\pi r^{2} b,\) where \(r\) is the radius and \(b\) is the height.

Step-by-Step Solution

Answer

Answer: The combined height of the gasoline and water layers in the cylinder is approximately 10.02 cm.

Step 1: Find the volume of gasoline and water

To find the volume of each liquid layer, we can use the formula: volume = mass / density Find the volume of gasoline and water using their given mass and density: For gasoline: volume_gasoline = (34.0 g) / (0.73 g/mL) = 46.5753425 mL For water: volume_water = (34.0 g) / (1.00 g/mL) = 34.0 mL

Step 2: Calculate the total volume

Now, add the volume of gasoline and water to get the total volume in the cylinder: total_volume = volume_gasoline + volume_water = 46.5753425 mL + 34.0 mL = 80.5753425 mL

Step 3: Find the radius of the cylinder

We are given the diameter of the cylinder, which is 3.2 cm. To find the radius, divide the diameter by 2: radius = diameter / 2 = 3.2 cm / 2 = 1.6 cm

Step 4: Calculate the combined height of liquid layers

Using the total_volume and the radius found in step 3, we can now find the combined height of the liquid layers using the volume formula for a cylinder: total_volume = π * r² * height Rearrange the formula to find the height: height = total_volume / (π * r²) Now, plug in the values: height = 80.5753425 mL / (π * (1.6 cm)²) = 80.5753425 mL / (π * 2.56 cm²) = 80.5753425 mL / (8.042032 cm²) = 10.023044 cm The combined height of the two liquid layers in the cylinder is approximately 10.02 cm.

Key Concepts

Volume of Cylinder

Understanding the volume of a cylinder is key when dealing with liquid layers inside. A cylinder is a 3D shape that has two parallel circular bases connected by a curved surface. To determine the volume of a cylinder, we use the formula:- \[ V = \pi r^2 h \] Where: - \( V \) is the volume - \( r \) is the radius of the base - \( h \) is the height of the cylinderThe radius is half of the diameter of the base, and it is important to ensure that the units are consistent. To find the combined height of two liquids in the graduated cylinder, we use the above formula to equate the sum of their volumes to this expression, solving for the height. This enables us to understand how the height of liquid will be distributed across the container.

Liquid Density

Density is a fundamental property of liquids and is defined as the mass of a substance per unit volume. It is generally expressed in units such as grams per milliliter (g/mL). The formula to find density is:- \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]In the context of our exercise, we look at the density difference between gasoline and water. Gasoline has a density of 0.73 g/mL, and water has a density of 1.00 g/mL. This means that per milliliter, water is denser than gasoline, causing the gasoline to float when they are combined.
Understanding the density helps us figure out the volume each liquid will occupy for a given mass, which is crucial for accurate measurements and calculations in scientific problems.

Mass to Volume Conversion

The conversion from mass to volume is an everyday application of density in practical situations. It allows us to determine the space a given mass of substance will occupy. To perform a mass to volume conversion, you would use the formula:- \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]By rearranging the density equation, this formula allows us to solve for volume. For example, in our exercise, we found that 34 g of gasoline converts to a volume of 46.575 mL when divided by its density of 0.73 g/mL. Similarly, 34 g of water equates to 34 mL using the density of 1.00 g/mL.
This conversion is especially useful in exercises that involve comparing different substances or calculating the volume of complex mixtures, providing a clear understanding of how much space each component takes up.