Problem 63 - General Chemistry

Question

One sphere has a radius of \(4.52 \mathrm{~cm} ;\) another has a radius of \(4.72 \mathrm{~cm}\). What is the difference in volume (in cubic centimeters) between the two spheres? Give the answer to the correct number of significant figures. The volume of a sphere is \((4 / 3) \pi r^{3},\) where \(\pi=3.1416\) and \(r\) is the radius.

Step-by-Step Solution

Answer

The volume difference between spheres is 54.1 cm³.

Step 1: Calculate the Volume of the First Sphere

To calculate the volume of a sphere with a radius, use the formula: \[ V = \frac{4}{3} \pi r^3 \]For the first sphere with a radius of 4.52 cm, we substitute into the formula: \[ V_1 = \frac{4}{3} \times 3.1416 \times (4.52)^3 \]Calculating this, we find: \[ V_1 \approx 4.52^3 \times 3.1416 \times \frac{4}{3} \approx 386.44 \text{ cm}^3 \]

Step 2: Calculate the Volume of the Second Sphere

Now, use the same formula for the sphere with a radius of 4.72 cm:\[ V_2 = \frac{4}{3} \times 3.1416 \times (4.72)^3 \]Computing, we find: \[ V_2 \approx 4.72^3 \times 3.1416 \times \frac{4}{3} \approx 440.50 \text{ cm}^3 \]

Step 3: Find the Difference in Volumes

Subtract the volume of the first sphere from the volume of the second sphere:\[ \Delta V = V_2 - V_1 = 440.50 \text{ cm}^3 - 386.44 \text{ cm}^3 \]This gives us: \[ \Delta V \approx 54.06 \text{ cm}^3 \]

Step 4: Adjust for Significant Figures

The radius values provided have three significant figures. Therefore, our answer should have three significant figures. The calculated difference in volume is 54.06 cm³, so rounding appropriately, we get: 54.1 cm³.

Key Concepts

Significant Figures

Significant figures are essential in ensuring that calculations reflect the precision of the measurements provided. When you see a number like 4.52 cm, it means the measurement was precise to three significant digits. This impacts how you approach calculations and report final answers.
Significant figures therefore help in maintaining consistency and accuracy. This is especially crucial in scientific measurements, where a slight change in precision can affect the results.

• Count all non-zero digits as significant.
• Zeroes between non-zero digits are significant.
• Leading zeroes are not significant.
• Trailing zeroes in decimals are significant.

In the original exercise, since the radii provided (4.52 cm and 4.72 cm) consist of three significant figures, all related calculations like the sphere volumes and their differences should be reported to three significant figures as well.

Sphere Radius

The radius of a sphere is the distance from its center to any point on its surface. It's a crucial measurement because it directly impacts the volume of the sphere. When you know the radius, you can use it to calculate other properties of the sphere, such as its volume and surface area.
The formula for volume, \(V = \frac{4}{3}\pi r^3\), shows the radius raised to the third power, indicating its significant influence. A small increase in radius can lead to a much larger increase in volume. Thus, precise measurement of the radius is essential for accurate volume calculation.

• The radius is often given in units such as centimeters or meters, depending on the context.
• Measurement accuracy is critical. A millimeter discrepancy in radius can lead to considerable volume differences.

In the exercise, the radii were 4.52 cm and 4.72 cm respectively, showing the necessity of considering these values with their complete precision for calculating accurate volumes.

Volume Calculation

Volume calculation for a sphere involves applying the specific formula \[ V = \frac{4}{3} \, \pi \, r^3 \]. This formula calculates the amount of space a sphere occupies, based on its radius. By substituting the radius value into the formula, you can find the sphere's volume in cubic units.
To use the formula:

• Measure the radius accurately.
• Substitute the radius value into the formula.
• Ensure the use of the correct value for \(\pi\), typically 3.1416 for most calculations.
• Calculate to the correct number of significant figures, as needed for precision.

In the given problem, calculating the volume for each sphere separately with their respective radii (4.52 cm and 4.72 cm), we find their individual volumes. The difference is then computed by subtracting the smaller volume from the larger one, followed by rounding off the result to three significant figures. This method ensures accuracy and adheres to the significant figure rules.