Question

You are given three cubes, \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\); one is magnesium, one is aluminum, and the third is silver. All three cubes have the same mass, but cube A has a volume of \(25.9 \mathrm{~mL}\), cube B has a volume of \(16.7 \mathrm{~mL}\), and cube \(C\) has a volume of \(4.29 \mathrm{~mL}\). Identify cubes \(A, B\), and \(C\).

Short answer

Cube A is magnesium, Cube B is aluminum, Cube C is silver.

Step-by-step solution

- Determine the density of each cube

The mass is the same for each cube, but different volumes result in different densities. Use the formula for density: \[ \text{Density} = \frac{\text{mass}}{\text{volume}} \]

- Assign known densities to the metals

Look up the densities of magnesium, aluminum, and silver. The respective densities are:\[ \text{Density of magnesium} = 1.74 \frac{\text{g}}{\text{cm}^3} \]\[ \text{Density of aluminum} = 2.70 \frac{\text{g}}{\text{cm}^3} \]\[ \text{Density of silver} = 10.49 \frac{\text{g}}{\text{cm}^3} \]

- Calculate the densities of the cubes

Assume each cube has a mass of 1 gram for simplicity. Calculate the densities:\[ \text{Density of cube A} = \frac{1}{25.9} \approx 0.0386 \frac{\text{g}}{\text{mL}} \]\[ \text{Density of cube B} = \frac{1}{16.7} \approx 0.0599 \frac{\text{g}}{\text{mL}} \]\[ \text{Density of cube C} = \frac{1}{4.29} \approx 0.233 \frac{\text{g}}{\text{mL}} \]

- Match the calculated densities with metal densities

Compare the calculated densities to the known densities. The calculated densities do not match directly because of the mass assumption, so reconsider using the fact that volumes inversely relate to the densities:\[ \text{Density of cube with largest volume (A)} \approx \text{Density of magnesium} \]\[ \text{Density of cube with middle volume (B)} \approx \text{Density of aluminum} \]\[ \text{Density of cube with smallest volume (C)} \approx \text{Density of silver} \]

- Identify each cube

Using the volume to density relationship:Cube A (largest volume) matches with Magnesium,Cube B (middle volume) matches with Aluminum,Cube C (smallest volume) matches with Silver.

Key concepts

density formula

Density is a fundamental concept in chemistry. It describes how much mass is packed into a given volume. The density formula, represented as \(\text{Density} = \frac{\text{mass}}{\text{volume}}\), allows us to calculate the density of a substance if we know its mass and volume. This formula shows that density is directly proportional to mass and inversely proportional to volume. If you increase the mass while keeping the volume constant, density increases. Conversely, if you increase the volume with the same mass, density decreases.

metal densities

Different metals have unique densities due to the arrangement and type of atoms they contain. For instance, magnesium, aluminum, and silver have specific densities which are quite distinct from each other. Magnesium's density is about 1.74 \(\frac{\text{g}}{\text{cm}^3}\), making it less dense than aluminum, which has a density of approximately 2.70 \(\frac{\text{g}}{\text{cm}^3}\). Silver is much denser, with a density of 10.49 \(\frac{\text{g}}{\text{cm}^3}\). Knowing these values helps in identifying metals based on their density. This is particularly useful when dealing with objects of the same mass but different volumes.

volume and mass relationship

The relationship between volume and mass is crucial in understanding density. Since density is the ratio of mass to volume, any change in either will affect the density. For example, in our exercise, each cube has the same mass. However, because their volumes differ, their densities change accordingly. A larger volume with the same mass results in a lower density, while a smaller volume results in higher density. This principle is foundational in solving many chemistry problems involving identification of substances.

identifying substances based on density

Density is a powerful tool for identifying substances. By calculating the density of an unknown material and comparing it to known values, you can often determine what the material is. In our exercise, we used volumes and the known densities of magnesium, aluminum, and silver to identify which material each cube was made of. Because cube A had the largest volume, it had the smallest density, closely matching magnesium. Cube B had a medium volume, resulting in a density that matched aluminum. Finally, cube C, with the smallest volume, had the highest density, aligning with silver.

chemistry problem-solving

Chemistry problem-solving often involves applying formulas and known values to find unknowns. An organized, step-by-step approach is essential. For the given exercise, we started by using the density formula, \(\text{Density} = \frac{\text{mass}}{\text{volume}}\). Next, we looked up the densities of common metals. We then calculated the densities of the cubes assuming a mass to compare with the metal densities. By correlating calculated densities with known densities, we identified each cube's material. This systematic approach simplifies complex problems and leads to correct conclusions.