Problem 44 - Chemistry The Science in Context

Question

Aluminum \((d=2.70 \mathrm{g} / \mathrm{mL})\) and strontium \((d=2.64 \mathrm{g} / \mathrm{mL})\) have nearly the same density. If we manufacture two cubes, each containing 1 mole of one element or the other, which cube will be smaller? What are the dimensions of this cube?

Step-by-Step Solution

Answer

Answer: The aluminum cube will be smaller with dimensions approximately 2.15 cm x 2.15 cm x 2.15 cm.

Step 1: Calculate the mass of 1 mole for aluminum and strontium cubes

To do this, we will use the formula n = m/M where n is the number of moles, m is the mass, and M is the molar mass. Since we want to find the mass, we can rearrange the formula as m = n*M. The molar mass of aluminum is approximately 27 g/mol, and for strontium, it is approximately 88 g/mol. For aluminum: m_Al = n * M_Al m_Al = 1 mol * 27 g/mol = 27 g For strontium: m_Sr = n * M_Sr m_Sr = 1 mol * 88 g/mol = 88 g

Step 2: Calculate the volume of the aluminum and strontium cubes

Now, we will use the density formula to calculate the volume of each cube. The formula is density = mass/volume, which can be rearranged to volume = mass/density. For aluminum: V_Al = m_Al / d_Al V_Al = 27 g / (2.70 g/mL) = 10 mL For strontium: V_Sr = m_Sr / d_Sr V_Sr = 88 g / (2.64 g/mL) = 33.33 mL

Step 3: Compare the volumes and determine which cube is smaller

Comparing the volumes, we see that the aluminum cube is smaller as it has a volume of 10 mL, whereas the strontium cube has a volume of 33.33 mL.

Step 4: Calculate the dimensions of the smaller cube (aluminum)

Since the volume of a cube is V = a^3, where a is the side length, we can find the side length by taking the cube root of the volume. a_Al = V_Al^(1/3) a_Al = (10 mL)^(1/3) ≈ 2.15 cm The dimensions of the smaller aluminum cube are approximately 2.15 cm x 2.15 cm x 2.15 cm.

Key Concepts

Molar Mass

Molar mass is a key concept in chemistry, which refers to the mass of one mole of a substance. It is expressed in grams per mole (g/mol). This value allows us to relate the mass of a sample to the number of molecules or atoms it contains. For instance, the molar mass of aluminum is about 27 g/mol, while for strontium, it is around 88 g/mol. This difference in molar mass is due to the number of protons and neutrons in the atomic nuclei of these elements. Understanding molar mass is crucial when dealing with calculations of moles and finding the mass of a particular substance, providing a foundation for many chemical equations and reactions.

Density Formula

The density of a substance is an important physical property that is defined as mass per unit volume. It is often expressed in grams per milliliter (g/mL) or kilograms per cubic meter (kg/m³). The density formula is given by:\[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]This equation can be rearranged to solve for mass or volume, depending on the information given. In the context of cubes, knowing the density allows us to calculate their volumes when the mass is known. For example, in this exercise, the aluminum cube with a mass of 27 grams and a density of 2.70 g/mL results in a calculated volume of 10 mL.Likewise, knowing the density of strontium helps determine the volume of its cube when the mass is 88 grams, yielding a larger volume of 33.33 mL.

Cube Dimensions

Cube dimensions are determined by understanding the concept of the cube's volume. A cube is a three-dimensional shape where all sides have equal lengths. The volume \(V\) of a cube is calculated using the formula:\[V = a^3\]where \(a\) is the side length of the cube. To find the dimensions of a cube if the volume is known, simply take the cube root of the volume.For instance, a cube having a volume of 10 mL will have side lengths calculated as:\[a = \sqrt[3]{10} ≈ 2.15 \text{ cm}\]This cube root calculation provides us with the length of each side of the cube, allowing us to visualize and understand the physical size of the object we are studying.

Element Properties

Elements have unique properties that affect their physical and chemical behavior. These properties include molar mass, density, and atomic structure. For example, both aluminum and strontium have similar densities, meaning they have similar mass-to-volume ratios, yet their molar masses and atomic structures differ. The atomic number and arrangement of electrons differentiate elements. Aluminum is a lighter metal with a lower molar mass and is typically more reactive. Strontium is a heavier alkaline-earth metal. Understanding these properties helps us explain why the aluminum cube is smaller despite its similar density to strontium. It's pivotal in predictively applying these properties during practical calculations involving the creation of substances at particular scales.