Problem 155 - Introductory Chemistry Atoms First

Question

The density of gold is \(19.3 \mathrm{~g} / \mathrm{mL}\), that of lead is \(11.4 \mathrm{~g} / \mathrm{mL}\), that of iron \(7.8 \mathrm{~g} / \mathrm{mL}\), and that of aluminum \(2.7 \mathrm{~g} / \mathrm{mL}\). A student is given separate samples of three substances \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\), along with a graduated cylinder containing \(50.0 \mathrm{~mL}\) of water. Each sample has a mass of \(200.0 \mathrm{~g}\), and the student finds that the volumes of the samples are A, \(25.64 \mathrm{~mL} ; \mathrm{B}, 10.36 \mathrm{~mL} ;\) and \(\mathrm{C}, 17.54 \mathrm{~mL}\). What is the identity of each substance?

Step-by-Step Solution

Answer

The identities of the substances are: Substance A is Iron, Substance B is Gold, and Substance C is Lead.

Step 1: Calculate the density of substance A

To calculate the density of substance A, we will use the given mass (200.0 g) and volume (25.64 mL) and the formula mentioned above: Density of A = Mass of A / Volume of A = \( \frac{200.0 \mathrm{~g}}{25.64 \mathrm{~mL}} = 7.81 \mathrm{~g/mL} \)

Step 2: Calculate the density of substance B

Similarly, using the mass (200.0 g) and volume (10.36 mL) for substance B, we will calculate its density: Density of B = Mass of B / Volume of B = \( \frac{200.0 \mathrm{~g}}{10.36 \mathrm{~mL}} = 19.30 \mathrm{~g/mL} \)

Step 3: Calculate the density of substance C

Finally, using the mass (200.0 g) and volume (17.54 mL) for substance C, we will calculate its density: Density of C = Mass of C / Volume of C = \( \frac{200.0 \mathrm{~g}}{17.54 \mathrm{~mL}} = 11.40 \mathrm{~g/mL} \)

Step 4: Identify the substances

Now that we have the calculated densities for substances A, B, and C, we can compare them to the given densities of gold, lead, iron, and aluminum: - Density of substance A = 7.81 g/mL ≈ 7.8 g/mL, that of iron. Therefore, substance A is iron. - Density of substance B = 19.30 g/mL ≈ 19.3 g/mL, that of gold. Therefore, substance B is gold. - Density of substance C = 11.40 g/mL ≈ 11.4 g/mL, that of lead. Therefore, substance C is lead. So, the final identification of the substances is as follows: - Substance A: Iron - Substance B: Gold - Substance C: Lead

Key Concepts

Density of Metals

Density is the measure of how much mass is contained in a given volume. The formula for density is \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \), typically expressed in grams per milliliter (g/mL) or kilograms per cubic meter (kg/m³). For metals, density can give us a lot of information about the material since different metals have characteristic densities.

• Gold has a density of 19.3 g/mL, making it one of the heaviest and densest metals.
• Lead, with a density of 11.4 g/mL, is also quite dense but less so than gold.
• Iron has a density of 7.8 g/mL, which is moderate compared to other metals.
• Aluminum is relatively light with a density of 2.7 g/mL.

Knowing the densities of these metals helps identify unknown samples by measuring their mass and volume. By matching a sample's calculated density to these known values, we can determine what metal it might be.

Mass and Volume Relationship

The relationship between mass and volume is crucial in understanding density. Mass is the amount of substance in an object, measured in grams or kilograms, while volume is the amount of space it occupies, measured in milliliters (mL) or liters (L).

When given a mass and you need the volume, you can rearrange the density formula: \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \). Similarly, to calculate mass if you know the density and volume, use: \( \text{Mass} = \text{Density} \times \text{Volume} \).

This basic calculation is used in exercises to determine how a particular material compares to its expected density values. For the exercise, all samples had fixed mass (200.0 g) but their volumes varied:

• Sample A had a volume of 25.64 mL.
• Sample B's volume was 10.36 mL.
• Sample C measured 17.54 mL.

With these values, we can easily find their densities and make comparisons.

Identifying Substances by Density

Identifying substances using density is a practical and straightforward method. Each substance has a unique density, acting as a fingerprint. By measuring a sample's mass and volume, and subsequently calculating its density, you can compare it to known densities of other substances to identify it.

In the given exercise, the calculated densities were:

• Sample A: 7.81 g/mL
• Sample B: 19.30 g/mL
• Sample C: 11.40 g/mL

Comparing these calculations:

• Sample A's density was close to that of iron (7.8 g/mL), so Sample A is identified as iron.
• Sample B's density matched perfectly with gold (19.3 g/mL), thus it is gold.
• Sample C's density aligned with lead (11.4 g/mL), confirming it as lead.

Understanding this method supports its use across scientific and industrial fields. From metallurgy to geology, density can help in sorting materials, quality control, and research purposes.