Problem 64 - General Chemistry

Question

A solid circular cylinder of iron with a radius of \(1.500 \mathrm{~cm}\) has a ruler etched along its length. What is the volume of iron contained between the marks labeled \(3.20 \mathrm{~cm}\) and \(3.50 \mathrm{~cm} ?\) The volume of a circular cylinder is \(\pi r^{2} l,\) where \(\pi=3.1416, r\) is the radius, and \(l\) is the length.

Step-by-Step Solution

Answer

The volume of iron between the marks is approximately 2.121 cm³.

Step 1: Identifying the Parameters

The problem gives us the radius of the cylinder as \( r = 1.500\, \text{cm} \) and provides two length measurements: \( 3.20 \, \text{cm} \) and \( 3.50 \, \text{cm} \). We need to find the volume of the section between these two marks.

Step 2: Calculate the Length of the Cylinder Section

Find the length \( l \) of the cylinder section between the given marks. Subtract the smaller length from the larger one: \( l = 3.50\, \text{cm} - 3.20\, \text{cm} = 0.30\, \text{cm} \).

Step 3: Formula for Volume of a Cylinder

Recall the formula for the volume of a cylinder: \( V = \pi r^2 l \). We will use this formula to calculate the volume of our specific section of the cylinder.

Step 4: Substitute the Parameters into the Formula

Substitute \( r = 1.500 \, \text{cm} \), \( l = 0.30 \, \text{cm} \), and \( \pi = 3.1416 \) into the formula:\[ V = 3.1416 \times (1.500)^2 \times 0.30 \].

Step 5: Calculate the Volume

Perform the calculations:1. Compute \( 1.500^2 = 2.250 \).2. Multiply by \( \pi \): \( 3.1416 \times 2.250 = 7.0686 \).3. Multiply by the length \( l \): \( 7.0686 \times 0.30 = 2.12058 \text{ cm}^3 \).

Step 6: Conclusion

The volume of iron contained between the marks labeled \( 3.20 \, \text{cm} \) and \( 3.50 \, \text{cm} \) is approximately \( 2.121 \, \text{cm}^3 \), after rounding to three decimal places.

Key Concepts

Cylinder Volume Calculation

Calculating the volume of a cylinder is important in various fields, such as engineering and chemistry. The basic formula to calculate the volume of a circular cylinder is given by: \[ V = \pi r^2 l \]where:

• \( V \) is the volume of the cylinder.
• \( \pi \approx 3.1416 \) is the mathematical constant.
• \( r \) represents the radius of the cylinder's base.
• \( l \) is the length or height of the cylinder.

The formula basically calculates the area of the circle formed by the base and then multiplies it by the height of the cylinder to find the volume.
For instance, consider the problem of finding the volume of a section of an iron cylinder with radius 1.500 cm and length between 3.20 cm and 3.50 cm. First, determine the height of this section by subtracting the smaller length from the larger one: 0.30 cm. Then, substitute the values into the formula as shown in earlier steps to find the volume of the cylinder section.

Geometry in Chemistry

Geometry plays a crucial role in chemistry, particularly when working with various types of containers and physical forms of substances. The geometric understanding of shapes, such as cylinders, can help chemists determine how much space a substance occupies or how much is required to fill a particular container.
In the context of a chemical laboratory, cylinders are often used for precise measurements. Graduated cylinders, for example, use this geometry to measure liquids. The formula for cylinder volume not only helps in determining the volume of solids like the iron cylinder but also enables chemists to calculate the capacity of cylindrical containers. This is vital in many chemical processes like reactions, where accurate volumes are necessary.
Understanding the geometric principles of cylinder shapes helps chemists in designing experiments and ensures the accuracy of their measurements, whether they are dissolving solids like metals or calculating the amount needed for a reaction.

Measurement Techniques in Chemistry

Accurate measurement is fundamental in chemistry, where even small inaccuracies can lead to significant errors in experimental results. Different techniques are used for measuring volumes, masses, and other quantities in chemistry, and choosing the right one depends on the nature of the material being measured and the required precision.
When measuring liquids, chemists often use cylindrical containers such as cylinders and burettes for their precision. For measuring solid volumes or calculating the volumes of containers like our solid iron cylinder example, knowledge of geometric formulas is essential.
For measuring small distances or differences, like the length between two marks on a cylinder, using a vernier caliper or a ruler with fine divisions can be very effective. These tools ensure the accuracy needed in experimental procedures. The ability to accurately measure different physical properties is crucial for reproducing experiments and achieving reliable results. Understanding volume measurement of cylinders via geometry, as seen in the example, showcases the importance of these techniques in practical chemistry applications.