Problem 96 - Chemistry The Science in Context

Question

The unit cell of an alloy with a 1: 1 ratio of magnesium and strontium is identical to the unit cell of CsCl. The unit cell edge of MgSr is \(390 \mathrm{pm}\). What is the density of MgSr?

Step-by-Step Solution

Answer

Answer: The density of MgSr alloy is 6.26 g/cm^3.

Step 1: Calculate the volume of the unit cell

Since the unit cell edge length is given as 390 pm, we first need to convert this value to meters: Edge length = 390 pm * (1 m / 10^12 pm) = 3.90 * 10^-10 m The volume (V) of the cubic unit cell can be calculated using the formula: V = a^3, where a is the edge length V = (3.90 * 10^-10 m)^3 = 5.95 * 10^-29 m^3

Step 2: Calculate the mass of MgSr in one unit cell

First, we need to find the molar mass of MgSr: Mg: 24.305 g/mol (Magnesium molar mass) Sr: 87.62 g/mol (Strontium molar mass) Molar mass of MgSr = Mg + Sr = 24.305 + 87.62 = 111.925 g/mol Since there is one Mg and one Sr atom in the unit cell of MgSr (1:1 ratio), we can use the following formula to find the mass of MgSr in one unit cell: Mass = (Molar mass of MgSr / Avogadro's number) * (Number of atoms in one unit cell) Mass = (111.925 g/mol / 6.022 × 10^23 mol^-1) * (2 atoms) Mass = 3.724 * 10^-22 g

Step 3: Calculate the density of MgSr

Now that we have the volume of the unit cell and the mass of MgSr present in it, we can calculate the density: Density = Mass / Volume Density = (3.724 * 10^-22 g) / (5.95 * 10^-29 m^3) Density = 6.26 g/cm^3 The density of MgSr is 6.26 g/cm^3.

Key Concepts

Unit Cell

When we talk about a unit cell, we refer to the basic repeating structure of a crystal. It is the smallest portion of a crystal lattice that, when repeated in all directions, forms the entire crystal. In the exercise, we look at a unit cell similar to that of CsCl, which is cubic in nature.

This means that the edges of the unit cell are all of the same length, and the angles between the edges are all 90 degrees. For MgSr, the edge length is given as 390 pm (picometers), which we typically convert into meters for easier calculations in physics and chemistry.

Understanding the volume of this unit cell is crucial. You can calculate it by cubing the edge length (i.e., multiplying it by itself three times). So, for MgSr with an edge length of 3.90 x 10^-10 meters, the volume will be approximately 5.95 x 10^-29 cubic meters. This precise calculation allows us to determine how much material is in each tiny repeating unit of the crystal.

Molar Mass

Molar mass is an essential concept for understanding how much a substance weighs at the molecular or atomic level. It refers to the mass of one mole of a substance, typically given in grams per mole (g/mol). In the solution, the molar mass of the MgSr is calculated based on the individual molar masses of magnesium (Mg) and strontium (Sr).

To get the molar mass of a compound like MgSr, you simply add the molar masses of the elements involved. Here, we have one atom of magnesium and one of strontium. So, you add 24.305 g/mol (for Mg) and 87.62 g/mol (for Sr), resulting in a molar mass of 111.925 g/mol for MgSr.

This information is utilized to determine the mass of the compound in the unit cell. Given that Avogadro's number (6.022 x 10^23) describes the number of atoms or molecules in one mole, you divide the molar mass by Avogadro's number to find the mass per atom or molecule. For MgSr, we calculate the mass of two atoms in the unit cell since MgSr consists of one magnesium and one strontium each, leading to 3.724 x 10^-22 grams.

Crystal Structure

Crystal structure involves the orderly geometric spatial arrangement of atoms in the crystalline solids. The type of structure affects the properties of the material, including how it interacts with light and the material's density and hardness.

In this case, the problem involves a 1:1 alloy of magnesium and strontium modeled after the cesium chloride (CsCl) structure, which is a simple cubic arrangement. Understanding the crystal structure aids in accurately predicting the number of atoms within a unit cell. For CsCl and similarly for MgSr, the unit cell contains one atom of each element, leading to an effective count of 2 atoms per cell.

This knowledge is fundamental when calculating the material's density and other physical characteristics of a crystal. Knowing the arrangement enables us to understand how atoms within the material are packed and can also explain many of the material's properties, like its density, which is calculated using the mass of atoms in the cell and the volume of the unit cell. For MgSr, this translates into a calculated density of 6.26 g/cm³. This indicates how tightly the atoms are packed in the structure.