Problem 99 - Chemistry and Chemical Reactivity

Question

A piece of copper has a mass of \(0.546 \mathrm{g} .\) Show how to set up an expression to find the volume of this piece of copper in units of liters. (Copper density \(=8.96 \mathrm{g} / \mathrm{cm}^{3} .\) )

Step-by-Step Solution

Answer

The volume of the copper piece is approximately 0.0000609375 liters.

Step 1: Understanding the Problem

We are given the mass of a piece of copper and its density, and we're asked to calculate the volume in liters. The formula for volume based on mass and density is \( V = \frac{m}{\rho} \), where \( m \) is mass and \( \rho \) is density.

Step 2: Calculate the Volume in Cubic Centimeters

First, we calculate the volume in cubic centimeters using the formula \( V = \frac{m}{\rho} \). Plugging in the values, \( V = \frac{0.546 \text{ g}}{8.96 \text{ g/cm}^3} \). Calculating this gives \( V \approx 0.0609375 \text{ cm}^3 \).

Step 3: Convert Cubic Centimeters to Liters

Since 1 liter equals 1000 cubic centimeters, we can convert the volume from cubic centimeters to liters. Thus, \( V \) in liters is given by \( V_{liters} = \frac{0.0609375 \text{ cm}^3}{1000} \approx 0.0000609375 \text{ liters} \).

Key Concepts

Density

Density is a fundamental concept in physics and chemistry. It refers to the amount of mass contained in a given volume. Understanding density helps us relate mass and volume of a substance. The formula for density is:\[\rho = \frac{m}{V}\]where:

• \(\rho\) is the density, usually expressed in grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³).
• \(m\) is the mass of the object.
• \(V\) is the volume of the object.

In the case of copper, a common example used in calculations, its density is given as 8.96 g/cm³. This means that each cubic centimeter of copper weighs 8.96 grams. Grasping this concept allows us to predict how much space a particular quantity of mass will occupy, which is crucial in numerous scientific and engineering applications.

Mass to Volume Conversion

To find the volume of a substance from its mass, knowing its density is essential. Using the formula for density, rearranged to \[V = \frac{m}{\rho}\]we can easily convert mass to volume. This is particularly useful for substances like metals where knowing just the mass might not be sufficient when analyzing their physical distribution in space.

• First, determine the mass of the object. For example, a copper piece might weigh 0.546 grams.
• Next, use the density value, for copper, it's 8.96 g/cm³.
• Finally, plug these values into the formula to get the volume in cubic centimeters \(V = \frac{0.546 \,\text{g}}{8.96 \,\text{g/cm}^3} \approx 0.0609 \,\text{cm}^3\).

Thus, when you know the mass and density, determining the volume becomes a simple task. This conversion process is widely applicable in various fields such as material science, engineering, and everyday problem-solving involving matter.

Unit Conversion

Unit conversion is a necessary skill in math and science. It allows us to change measurements to units that are more convenient or standardized. Converting volume from cubic centimeters to liters is a common conversion, especially useful in chemistry and physics.One liter is equivalent to 1000 cubic centimeters. Therefore, to convert a volume in cubic centimeters to liters, one can use the formula:\[V_{liters} = \frac{V_{cm^3}}{1000}\]For instance, if you have calculated the volume of copper as 0.0609375 cm³, the conversion to liters would be:\[V_{liters} = \frac{0.0609375}{1000} \approx 0.0000609375 \,\text{liters}\]This conversion ensures that you can easily communicate and compare measurements, while sticking to the SI units that are globally accepted. Unit conversions like this are fundamental in science to maintain consistency and clarity in quantitative analysis.