Question

A copper cube has a mass of \(87.2 \mathrm{~g}\). Find the edge length of the cube. (The density of copper is \(8.96 \mathrm{~g} / \mathrm{cm}^{3}\), and the volume of a cube is equal to the edge length cubed.)

Short answer

The edge length of the copper cube is approximately 4.45 cm.

Step-by-step solution

Identify the Formula for Volume of the Cube

For a cube with edge length 'a', the volume 'V' is given by the formula \(V = a^3\).

Calculate the Volume of the Cube

Using the mass (m) of the cube and the density (D) of copper, calculate the volume of the cube using the formula \(V = \frac{m}{D}\). Substitute the given values and compute the volume.

Calculate the Edge Length

Take the cube root of the volume to find the edge length 'a' of the cube. The cube root of a number 'x' is the number which, when multiplied by itself three times, gives 'x'. This can be represented as \(a = \sqrt[3]{V}\). Compute the cube root of the cube's volume to find the edge length.

Key concepts

Mass and Volume Relationship

Understanding the relationship between mass and volume is essential for solving many types of problems in science, particularly in chemistry and physics. Mass refers to the amount of matter in an object and is measured in grams (g) or kilograms (kg). Volume, on the other hand, is the space that an object occupies and is usually measured in cubic meters (m³) or cubic centimeters (cm³).

In the context of density problems, mass and volume are directly related through the equation for density: \[ Density = \frac{Mass}{Volume} \]
This equation shows that if you have two of the three variables (mass, volume, or density), you can solve for the third. When students encounter a problem where they must find an unknown volume from known mass and density, they should rearrange the density formula to solve for volume, which is \[ Volume = \frac{Mass}{Density} \]
In our exercise example, we were given the mass of a copper cube and needed to find its volume using the density of copper. This step is crucial before we can proceed to determine the cube's dimensions.

Chemistry Calculations

Chemistry calculations involve a variety of computations that are used to understand the physical and chemical properties of substances. Central to many of these calculations is the concept of molarity, reaction stoichiometry, and as seen in our exercise, density. Density is a property that describes how much mass is contained within a certain volume of a material.

It's critical for students to become proficient with chemistry calculations as they are not only important for laboratory work but also for understanding theoretical concepts. For example, calculating the density of a substance requires precise measurement and conversion of units. In the case of the copper cube problem, we use the density to indirectly measure the volume of the cube by manipulating the density formula mentioned in the previous section.

Moreover, such chemistry calculations are not limited to simple substances but extend to complex mixtures, solutions, and compounds. Learning to perform these calculations with accuracy and understanding the logic behind them can empower students to tackle more advanced problems in chemistry.

Volume of a Cube

The volume of a cube is a fundamental concept in geometry that students typically encounter early in their studies. Calculating the volume is quite straightforward—the volume (V) of a cube is found by cubing the length of one of its edges (a): \[ V = a^3 \]
In practical terms, if we know the edge length of a cube, we can determine how much space it occupies. Conversely, if we know the volume, we can determine the length of an edge by taking the cube root of the volume.

In our exercise, after finding the volume of the copper cube using the mass and density, the final step is to determine the cube's edge length. This is done by taking the cube root of the volume: \[ a = \sqrt[3]{V} \]
This step concludes the process, as it provides the cube's edge measurement which was the goal in the exercise. Grasping the concept of the volume of a cube is essential as it helps students visualize and solve real-world problems involving spatial reasoning and dimensions.