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Chapter 1: Problem 89

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

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

Step by step solution

01

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
02

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.
03

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
04

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.

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