Problem 129 - General Chemistry

Question

Platinum metal is used in jewelry; it is also used in automobile catalytic converters. What is the mass of a cube of platinum \(4.40 \mathrm{~cm}\) on an edge? The density of platinum is \(21.4 \mathrm{~g} / \mathrm{cm}^{3}\).

Step-by-Step Solution

Answer

The mass of the platinum cube is approximately 1,823.94 g.

Step 1: Understand the Problem

You are given the edge length of a cube of platinum and its density. You need to find the mass of the cube.

Step 2: Calculate the Volume of the Cube

Use the formula for the volume of a cube, which is given by the cube of its edge length. If the edge of the cube is \(4.40 \text{ cm}\), then the volume \(V\) is calculated as follows:\[V = 4.40^3 \]Calculate the result.

Step 3: Perform the Calculation for Volume

Calculate \(4.40^3\) to find the volume of the cube:\[V = 4.40 \times 4.40 \times 4.40 = 85.184 \text{ cm}^3\]The volume of the cube is \(85.184 \text{ cm}^3\).

Step 4: Use Density to Find Mass

Use the formula for mass, which is given by the density multiplied by the volume. The density \(\rho\) is \(21.4 \text{ g/cm}^3\), and the volume \(V\) is \(85.184 \text{ cm}^3\). The formula is:\[m = V \times \rho \]Substitute the values:\[m = 85.184 \times 21.4\]

Step 5: Perform the Calculation for Mass

Calculate the mass using the given formula:\[m = 85.184 \times 21.4 = 1,823.94 \text{ g}\]Thus, the mass of the platinum cube is approximately \(1,823.94 \text{ g}\).

Key Concepts

Mass of Cube

To find the mass of a cube, you need two primary pieces of information: the volume of the cube and the material's density. Mass is a measure of the amount of matter in an object, often measured in grams or kilograms. For our cube made of platinum, calculating the mass requires using the formula:

\[m = V \times \rho\]
where:

• \(m\) is the mass,
• \(V\) is the volume of the cube, and
• \(\rho\) is the density of the material the cube is made of.

In this exercise, once you have the cube's volume and know the density of platinum (\(21.4 \text{ g/cm}^3\)), you multiply them to find the mass. The calculation helps in understanding how density and volume combine to affect the mass of objects, like a platinum cube.

Volume of Cube

Volume measures the space an object occupies and for a cube, it's calculated by raising the length of one edge to the power of three. That is because a cube has equal sides and its volume is determined by all three dimensions: length, width, and height.

For a cube with an edge length of \(4.40 \text{ cm}\), the formula is:

\[V = \text{edge length}^3\]
Thus, to find the volume you compute:
\[V = 4.40 \times 4.40 \times 4.40 = 85.184 \text{ cm}^3\]
This volume calculation tells us how much space the cube occupies in terms of cubic centimeters, which is essential for determining its mass. Understanding how to calculate volume is a foundational concept in geometry and is frequently applied in various scientific fields.

Platinum Density

Density is essentially the compactness of a material - how much mass is packed into a given volume. It is represented as mass per unit volume, often in units like \(\text{g/cm}^3\). For platinum, a dense metal, this value is \(21.4 \text{ g/cm}^3\).

This means every cubic centimeter of platinum weighs \(21.4\) grams. When you multiply this density by the volume of the platinum cube, you can determine the total mass. Knowing density helps predict whether a material will sink or float in water (since water's density is \(1 \text{ g/cm}^3\)) and is crucial for understanding material properties in both construction and design. Platinum's high density makes it a valuable and sought-after metal for various applications, emphasizing the importance of density in its practical uses.