Problem 44 - Introductory Chemistry Atoms First

Question

The measured density of lead is \(11.4 \mathrm{~g} / \mathrm{mL}\). What volume in milliliters will \(1.50\) pounds of lead occupy? \([1\) pound \(=453.6 \mathrm{~g}]\)

Step-by-Step Solution

Answer

1.50 pounds of lead will occupy 59.7 milliliters of volume.

Step 1: Convert mass from pounds to grams

To convert the mass of lead from pounds to grams, use the conversion factor given in the question: 1 pound = 453.6 grams. Multiply the mass in pounds by the conversion factor: \(1.50 \textrm{ pounds} \times \frac{453.6 \textrm{ grams}}{1 \textrm{ pound}}\).

Step 2: Calculate the mass of lead in grams

After converting the mass of lead to grams, the equation looks like the following: \(1.50 \textrm{ pounds} \times \frac{453.6 \textrm{ grams}}{1 \textrm{ pound}} = 680.4 \textrm{ grams}\). So, the mass of lead is 680.4 grams.

Step 3: Use the density formula to calculate the volume of lead

The density formula is defined as Density = Mass / Volume, or \(D = \frac{M}{V}\). We can rearrange this formula to solve for the volume: Volume = Mass / Density, or \(V = \frac{M}{D}\).

Step 4: Substitute the values and calculate the volume

Now, substitute the values of mass and density into the volume formula: \(V = \frac{680.4 \textrm{ grams}}{11.4 \textrm{ g/mL}}\).

Step 5: Calculate the volume of lead in milliliters

After substituting the values, the equation looks like the following: \(V = \frac{680.4 \textrm{ grams}}{11.4 \textrm{ g/mL}} = 59.7 \textrm{ mL}\). So, 1.50 pounds of lead will occupy 59.7 milliliters of volume.

Key Concepts

Mass Conversion

When dealing with different units of mass, conversion is an essential step to ensure calculations are accurate and in the desired unit. Mass conversion involves changing from one unit of mass to another, such as pounds to grams or kilograms to ounces. This is important when using formulas that require a specific unit, ensuring consistency and accuracy.
To convert mass, use a conversion factor, which is a ratio that represents how many of one unit equals another unit. For example, when converting pounds to grams, the conversion factor is:

• 1 pound = 453.6 grams

To apply the conversion factor, multiply the mass in the original unit by the conversion factor. For instance, converting 1.50 pounds to grams involves multiplying 1.50 by 453.6, resulting in 680.4 grams. This ensures the mass is in the proper unit for further calculations.

Density Formula

The density formula is crucial when you need to link mass and volume. Density is defined as the mass of a substance per unit volume, expressed in units like grams per milliliter (g/mL) or kilograms per cubic meter (kg/m³).
The formula to calculate density is:

• \[D = \frac{M}{V}\]

Where:

• \(D\) is the density.
• \(M\) is the mass of the substance.
• \(V\) is the volume occupied by the substance.

If you know the mass and density of a substance, you can rearrange the formula to find the volume:

• \[V = \frac{M}{D}\]

This equation is instrumental in quantifying how much space a given mass of a material will occupy.

Volume Calculation

Calculating volume is a fundamental step in many physics and chemistry problems, especially when using the density formula. Once you have the mass and density, finding the volume becomes a straightforward process.
To find the volume from mass and density, use the rearranged density formula:

• \[V = \frac{M}{D}\]

Where:

• \(V\) is the volume.
• \(M\) is the mass.
• \(D\) is the density.

For example, if you have 680.4 grams of lead and its density is 11.4 g/mL, substitute these values into the formula to find that the volume is 59.7 mL. This result allows you to understand the space occupied by a certain mass of material.

Unit Conversion

Unit conversion is an essential skill in science and mathematics. It ensures that all measurements and calculations are consistent and comparable. Converting units often requires a conversion factor, a ratio that equates one unit to another, which enables seamless transitions between metric and imperial systems.
For example, converting the density of a material from g/mL to kg/m³ may require several steps and conversion factors. Consider the density of lead at 11.4 g/mL:

• To convert to kg/m³, consider:
  • 1 g = 0.001 kg
  • 1 mL = 0.000001 m³
  Substituting these into the density value will yield the equivalent in kg/m³.

Always ensure to clearly document and follow the correct chain of conversion factors for accuracy. This process helps maintain uniformity in scientific studies and practical applications.