Question

A 25.0 -cm-long cylindrical glass tube, sealed at one end, is filled with ethanol. The mass of ethanol needed to fill the tube is found to be 45.23 g. The density of ethanol is 0.789 \(\mathrm{g} / \mathrm{mL}\) . Calculate the inner diameter of the tube in centimeters.

Short answer

The inner diameter of the cylindrical glass tube is approximately 1.71 cm.

Step-by-step solution

Determine the volume of ethanol in the cylindrical tube

First, we need to determine the volume of ethanol in the cylindrical tube. We can do this using the formula: Volume = Mass / Density The mass of ethanol is given as 45.23 g, and the density as 0.789 g/mL. Plugging these values into the formula, we get: Volume = 45.23 g / (0.789 g/mL) Calculating this gives: Volume ≈ 57.316 mL

Conversion of volume unit

As we are looking for the diameter in centimeters, we need to convert the volume from milliliters (mL) to cubic centimeters (cm³) to keep the units consistent. Luckily, there is a 1:1 relationship between mL and cm³: 1 mL = 1 cm³ So, the volume of ethanol in the cylindrical tube is 57.316 cm³.

Calculate the cross-sectional area of the tube

Now we will use the volume formula for a cylinder to find the cross-sectional area of the tube. The formula is: Volume = Length × Cross-sectional area We know the volume and length of the tube, so we can solve for the cross-sectional area. Rearranging the formula, we have: Cross-sectional area = Volume / Length Plugging in the values, we get: Cross-sectional area = 57.316 cm³ / 25.0 cm Calculating this gives: Cross-sectional area ≈ 2.29264 cm²

Calculate the inner diameter of the tube

The cross-sectional area is a circle since the tube is cylindrical. The formula for the area of a circle is: Area = π × (Diameter / 2)² We can now solve for the diameter. Rearranging the formula, we have: Diameter = 2 × √(Area / π) Plugging in the cross-sectional area value, we get: Diameter = 2 × √(2.29264 cm² / π) Calculating this gives: Diameter ≈ 1.708596 cm To summarize, the inner diameter of the cylindrical glass tube is approximately 1.71 cm.

Key concepts

Volume of a Cylinder

A cylinder is a three-dimensional shape with circular ends and straight sides, like a can. Understanding how to calculate its volume is key, especially in tasks like finding out how much liquid it can hold. The volume of a cylinder is found using the formula:
\[ \text{Volume} = \text{Length} \times \text{Cross-sectional area} \]This formula highlights that the volume is influenced by two main factors:

• The height or length of the cylinder
• The area of its circular base

To connect with our example, imagine the glass tube as a cylinder. To find its volume knowing it holds ethanol, you'll use the Volume = Mass/Density formula.
Given that ethanol's mass is 45.23 g and its density is 0.789 g/mL, calculate:\[ \text{Volume} = \frac{45.23 \text{ g}}{0.789 \text{ g/mL}} \]This results in about 57.316 mL or cm³, illustrating how the volume allows us to plan for capacity in various applications.

Ethanol Properties

Ethanol, a common type of alcohol with many uses, has properties that are crucial for experiments and industrial purposes. Knowing ethanol’s density is beneficial for calculations like finding volumes or amounts needed.
Ethanol's density is 0.789 g/mL, implying:

• It's lighter than water which has a density of 1 g/mL.
• It interacts differently with materials compared to other solvents.

When working with ethanol, considerations include its density and its physical behaviors, like evaporation or mixing qualities. These are especially relevant when working with volumes and masses, such as when filling a glass tube.
For example, by knowing ethanol’s density, you can determine how much space a certain mass of it will occupy. This highlights why understanding ethanol’s properties is crucial for calculations in scientific and practical applications.

Cylinder Dimensions

When dealing with cylindrical shapes, like glass tubes, knowing how to determine its dimensions accurately is essential. The critical dimensions are:

• Height or Length: This is the straight distance from one end of the cylinder to the other. In our scenario, this is given as 25.0 cm.
• Diameter: The distance across the circle that makes up the ends of the cylinder. It’s also directly related to calculating the area of the circle.

To find the diameter when we know the volume and length, we first compute the cross-sectional area using the rearranged volume formula:
\[ \text{Cross-sectional area} = \frac{\text{Volume}}{\text{Length}} \]Finally, relating it to a circle’s area \( \pi \times (\text{Diameter} / 2)^2 \), solving for the Diameter results in:\[ \text{Diameter} = 2 \times \sqrt{\frac{\text{Area}}{\pi}} \]In the given scenario, using these calculations leads us to an inner diameter of approximately 1.71 cm, demonstrating how geometry helps in practical measurement tasks.