Problem 36 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) What is the mass of a silver cube whose edges measure 2.00 \(\mathrm{cm}\) each at \(25^{\circ} \mathrm{C} ?\) The density of silver is \(10.49 \mathrm{~g} / \mathrm{cm}^{3}\) at \(25^{\circ} \mathrm{C}\). (b) The density of aluminum is \(2.70 \mathrm{~g} / \mathrm{cm}^{3}\) at \(25^{\circ} \mathrm{C}\). What is the weight of the aluminum foil with an area of \(0.5 \mathrm{~m}^{2}\) and a thickness of \(0.5 \mathrm{~mm} ?\) (c) The density of hexane is \(0.655 \mathrm{~g} / \mathrm{mL}\) at \(25^{\circ} \mathrm{C} .\) Calculate the mass of \(1.5 \mathrm{~L}\) of hexane at this temperature.

Step-by-Step Solution

Answer

(a) The mass of the silver cube is 83.92 g. (b) The weight of the aluminum foil is 6.615 N. (c) The mass of the 1.5 L of hexane is 982.5 g.

Step 1: (a) Finding the mass of the silver cube

We're given the edge length of the silver cube, and we know that the volume of a cube can be calculated using the formula: Volume = edge_length³. First, calculate the volume of the silver cube: Edge length = 2.00 cm Volume of the cube = (2.00 cm)³ = 8.00 cm³ Now, we can use the given density to find the mass. Density is the ratio of mass to volume, so we have: Density = Mass / Volume Solving for mass, we get: Mass = Density * Volume Now, plug in the density of silver (10.49 g/cm³) and the volume (8.00 cm³) and calculate the mass: Mass = (10.49 g/cm³) * (8.00 cm³) = 83.92 g

Step 2: (b) Finding the weight of the aluminum foil

First, we need to convert the area and thickness of the aluminum foil into cm: Area = 0.5 m² = 5000 cm² Thickness = 0.5 mm = 0.05 cm Now, find the volume of the aluminum foil by multiplying its area by its thickness: Volume of the foil = Area * Thickness = 5000 cm² * 0.05 cm = 250 cm³ Now, we can use the given density to find the mass of the aluminum foil. Using the same equation as before, plug in the density of aluminum (2.70 g/cm³) and the volume (250 cm³) and calculate the mass: Mass = (2.70 g/cm³) * (250 cm³) = 675 g Now, we convert the mass to weight. Recall that weight is the force exerted on an object due to gravity, and it can be calculated using the equation: Weight = Mass * gravity, where gravity is approximately 9.8 m/s². First, convert the mass from grams to kilograms: Mass = 675 g = 0.675 kg Now, calculate the weight: Weight = (0.675 kg) * (9.8 m/s²) = 6.615 N

Step 3: (c) Finding the mass of hexane

Given the volume of hexane, 1.5 L, first convert it to mL: Volume = 1.5 L = 1500 mL Now, using the given density, we'll find the mass of hexane. Again, using the Density = Mass / Volume equation and solving for the mass: Mass = Density * Volume Plug in the density of hexane (0.655 g/mL) and the volume (1500 mL) and calculate the mass: Mass = (0.655 g/mL) * (1500 mL) = 982.5 g

Key Concepts

Volume Calculation

Understanding how to calculate volume is a fundamental concept in density calculations. When dealing with three-dimensional objects, volume becomes a crucial factor for determining mass based on density. In this context, we focus on calculating the volume of a cube, as it forms one of the simplest geometric shapes. Given a cube, you can find its volume by cubing the length of one of its edges. For instance, if a cube has an edge length of 2.00 cm, its volume is calculated as:\[ \text{Volume} = \text{edgetime\ length}\ ^3 = (2.00 \, \text{cm})^3 = 8.00 \, \text{cm}^3 \]This calculation is crucial as it provides the necessary volume required to determine the mass of the object when its density is known.

Mass Determination

Once you have the volume of an object, determining its mass using its density becomes straightforward. The relationship between mass, density, and volume is given by the formula:\[ \text{Mass} = \text{Density} \times \text{Volume} \]For example, with a cube of silver with a volume of 8.00 cm³ and a density of 10.49 g/cm³, the mass can be calculated as follows:\[ \text{Mass} = 10.49 \, \text{g/cm}^3 \times 8.00 \, \text{cm}^3 = 83.92 \, \text{g} \]This formula is widely used in physics and chemistry to determine the mass of an object when given its density and volume, providing essential information for further calculations and applications.

Cube Dimensions

Cubical dimensions are easy to compute, making them a popular choice for introductory volume and density problems. A cube is defined by having all sides of equal length. This symmetry allows for straightforward volume calculations, as the formula simply becomes the edge length raised to the power of three.

• The formula for a cube is: \[ \text{Volume} = a^3 \]
• Where \( a \) represents the length of the edge of the cube.

For example, if a silver cube's edges measure 2.00 cm, calculating its volume becomes simply solving \( (2.00 \, \text{cm})^3 \), resulting in a volume that can then be used to find its mass given the density.

Unit Conversion

Unit conversion is a crucial skill in scientific calculations, ensuring all measurements are in compatible units. In this exercise, converting different units is essential, such as meters to centimeters and millimeters to centimeters for working with volumes in cubic centimeters.When converting:- Convert area from square meters to square centimeters by multiplying by 10,000 \[ 0.5 \, \text{m}^2 \to 5000 \, \text{cm}^2 \] - Convert thickness from millimeters to centimeters by dividing by 10 \[ 0.5 \, \text{mm} \to 0.05 \, \text{cm} \] These conversions are necessary for aligning the units with those of density measurements often expressed in grams per cubic centimeter (g/cm³), ensuring accurate and meaningful calculations.