Problem 131 - Introductory Chemistry Atoms First

Question

A metal sphere has a radius \(r\) of \(4.00 \mathrm{~cm}\). What is the volume \(V\) of this sphere in cubic centimeters? The formula for the volume of a sphere is \(V=(4 / 3) \pi r^{3}\), where \(\pi=3.14159 .\)

Step-by-Step Solution

Answer

The volume of the sphere with radius \(4.00 \mathrm{~cm}\) is approximately \(268.08 \mathrm{~cm^3}\).

Step 1: Write down the given values and formula

We are given the radius of the sphere, \(r = 4.00 \mathrm{~cm}\). The formula to find the volume of the sphere is \(V=(4 / 3) \pi r^{3}\).

Step 2: Substitute the given radius into the formula

Replace \(r\) in the formula with the given value: \(V = \frac{4}{3}\pi(4.00)^{3}\).

Step 3: Calculate the volume

Perform the calculations: 1. Find the cube of the radius: \((4.00)^{3} = 64\) 2. Multiply the result by \(\frac{4}{3}\): \(\frac{4}{3} \times 64 = 85.33\) 3. Multiply the result by \(\pi\): \(85.33 \times 3.14159 = 268.08\)

Step 4: State the final answer

The volume of the sphere is approximately \(268.08 \mathrm{~cm^3}\).

Key Concepts

Geometry in Chemistry

Understanding the geometry of molecules and particles is pivotal in chemistry. Not all substances assume simple shapes, but spherical models are often used to represent atoms and small molecules. This simplified view becomes exceptionally relevant when considering the packing of atoms in metals or the space occupied by gas molecules.

For instance, metal spheres might represent atoms in a crystalline structure, and knowing the volume of these spheres can help estimate the density of a material. Thus, the geometric concept of volume calculation becomes a fundamental skill for chemists, as it allows the prediction of properties based on the assumed volume of particles.

Mathematical Applications in Science

Mathematics is the language of the universe, and it finds its application in every branch of science. When it comes to understanding the physical world, mathematical formulas such as the volume of a sphere are indispensable. This is because they provide a quantitative understanding of objects and phenomena.

In fields like physics, biology, and environmental science, calculating the volume is critical for quantifying the amount of space an object occupies. From determining the dosage of medication in a spherical capsule to estimating the amount of water that a spherical tank can hold, the applications are endless and underscore the necessity of integrating mathematics seamlessly into scientific pursuits.

Volume Calculation

Volume calculation is not just a mathematical exercise; it has direct implications in real-world scenarios. Understanding how to compute the volume of various shapes enables us to solve problems involving the capacity of containers, the amount of resources required for a project, or even the quantity of fluid that can flow through a system.

It's about converting a theoretical concept into practical knowledge. For students, mastering volume calculations is beneficial for everyday problem-solving and analytical thinking. It also serves as a building block for more complex mathematical and scientific studies.

Sphere Volume Formula

The sphere volume formula, \(V = \frac{4}{3}\pi r^{3}\), is a perfect embodiment of how a simple equation can capture the essence of a three-dimensional object. A sphere is defined as a set of points that are all the same distance, the radius \(r\), from a central point. This symmetry is what makes the formula for a sphere's volume straightforward yet elegant.

To use the formula effectively, attention to detail is key. Ensuring that the radius is correctly measured and cubed, then multiplied by \(\frac{4}{3}\) and \(\text{\(\pi\)}\), allows students to easily calculate the volume of not just abstract spheres, but also tangible spherical objects they might encounter in their studies or daily lives.