Problem 57 - Chemistry: The Central Science in SI Units, Global Edition

Question

Brass is a substitutional alloy consisting of a solution of copper and zinc. A particular sample of yellow brass consisting of \(65.0 \%\) Cu and \(35.0 \%\) Zn by mass has a density of \(8470 \mathrm{~kg} / \mathrm{m}^{3}\). (a) What is the molality of \(\mathrm{Zn}\) in the solid solution? (b) What is the molarity of \(Z n\) in the solution?

Step-by-Step Solution

Answer

The molality of Zn in the solid solution is \(8.246 \mathrm{~mol/kg}\), and the molarity of Zn in the solution is \(6.986 \times 10^7 \mathrm{~mol/m}^3\).

Step 1: To find the mass per unit volume of each element i.e. Cu and Zn, we will multiply the density of the overall alloy with the percentage mass of each element. Given mass percentages: Volume of sample: \(1 \mathrm{~m}^3\) So, mass of Cu: \(0.65 \times 8470 = 5500.5 \mathrm{~kg}\) And, mass of Zn: \(0.35 \times 8470= 2964.5 \mathrm{~kg}\) ##Step 2: Calculate the number of moles per unit volume for Zn##:

To find the number of moles per unit volume for Zn, we will divide the mass of Zn by its molar mass. Molar mass of Zn: \(65.38 \mathrm{~g/mol}\) Number of moles of Zn: \(\frac{2964.5 \times 10^3 \mathrm{~g}}{65.38 \mathrm{~g/mol}} = 45364.3 \mathrm{~mol}\) ##Step 3: Calculate the molality of the solution##:

Step 2: To calculate the molality of the solution, we need to divide the number of moles of Zn by the mass of the solvent in kg (Copper in this case). Molality = \(\frac{45364.3 \mathrm{~mol}}{5500.5 \mathrm{~kg}} = 8.246 \mathrm{~mol/kg}\) So, the molality of Zn in the solid solution is \(8.246 \mathrm{~mol/kg}\). ##Step 4: Calculate the molarity of the solution##:

To calculate the molarity of the solution, we need to divide the number of moles of Zn by the total volume of the solution. Since we know the density of the alloy and the total mass of the alloy, we can compute the total volume of the alloy, where: Total mass of the alloy: \(5500.5 \mathrm{~kg} \times (\frac{1000 \mathrm{~g}}{1 \mathrm{~kg}})= 5.5 \times 10^6 \mathrm{~g}\) Density of the alloy: \(8470 \mathrm{~kg/m}^3 = 8.47 \times 10^6 \mathrm{~g/m}^3\) Volume of alloy: \(\frac{5.5 \times 10^6 \mathrm{~g}}{8.47 \times 10^6 \mathrm{~g/m}^3}= 0.000649 \mathrm{~m}^3\) Molarity = \(\frac{45364.3 \mathrm{~mol}}{0.000649 \mathrm{~m}^3} = 6.986 \times 10^7 \mathrm{~mol/m}^3\) So, the molarity of Zn in the solution is \(6.986 \times 10^7 \mathrm{~mol/m}^3\).

Key Concepts

Substitutional Alloys

Substitutional alloys are fascinating mixtures of metals where the atoms of the different metals replace each other in the crystal lattice. Essentially, they form when metals of similar atomic size and crystal structure mix together. In these alloys, the constituent elements dissolve in each other and create a uniform structure. A classic example is brass, which is an alloy made from copper (Cu) and zinc (Zn).

• The structure of substitutional alloys allows for the properties of the alloy—such as hardness, ductility, and corrosion resistance—to be tailored by altering the composition of the metals involved.
• Brass, for example, is valued for its acoustic properties and resistance to corrosion, making it an ideal material for musical instruments and durable fittings.

When substitution occurs, the primary metal retains its lattice, while the secondary metal substitutes some lattice points without significantly altering the original lattice. Think of it like swapping similar-sized Lego pieces; everything still fits together neatly.

Density Calculation

To understand any material's density, we need to look at its mass per unit volume. Density is a key property that distinguishes different materials and is measured in kg/m³. In the context of an alloy like brass, the density of the alloy is determined by the combined densities of the metals that make it up.

• Calculating density requires multiplying the mass percentages of the individual metals by the total density of the alloy. For instance, if given a sample of brass with a density of 8470 kg/m³, calculating the mass of copper and zinc separately helps us understand their individual contributions.
• This is done by multiplying the density by their respective mass percentages, like so: Mass of Cu = 0.65 x 8470 kg/m³ = 5500.5 kg, and Mass of Zn = 0.35 x 8470 kg/m³ = 2964.5 kg.

The overall concept of density helps in understanding how tightly the atoms are packed in the alloy and how it will behave in various applications.

Molarity Calculation

Molarity, often symbolized as M, is a concentration unit that describes the number of moles of a solute per liter of solution. It becomes especially useful when dealing with chemical reactions in liquid solutions. While molality considers mass, molarity focuses on volume, making it sensitive to temperature changes.

• To find the molarity of Zn in an alloy sample, we calculate the number of moles of Zn and divide it by the total volume of the alloy. In our example, having calculated already that Zn has 45364.3 mol, and knowing the volume of the solution (0.000649 m³), we use the formula: \[ \text{Molarity} = \frac{\text{number of moles}}{\text{volume (in liters)}} \]
• This gives us a molarity of approximately 6.986 x 10⁷ mol/m³, which tells us how concentrated Zn is within that specific sample.

This concept is essential for chemists who need to ensure that substances react in precise proportions, facilitating accurate experimental and industrial processes.