Problem 57 - Chemistry The Central Science

Question

Brass is a substitutional alloy consisting of a solution of copper and zinc. A particular sample of red brass consisting of \(80.0 \% \mathrm{Cu}\) and \(20.0 \% \mathrm{Zn}\) by mass has a density of \(8750 \mathrm{~kg} / \mathrm{m}^{3} .\) (a) What is the molality of \(\mathrm{Zn}\) in the solid solution? (b) What is the molarity of \(\mathrm{Zn}\) in the solution?

Step-by-Step Solution

Answer

The molality of Zn in the solid solution is \(3.825 \: \text{mol/kg}\), and the molarity of Zn in the solution is \(26.76 \: \text{mol/L}\).

Step 1: Convert percentages to masses

Assume we have 100 grams of brass. Since the sample is 80.0% Cu and 20.0% Zn by mass, we can calculate the masses of each element as follows: Mass of Cu = 80.0% × 100 grams = 80 grams Mass of Zn = 20.0% × 100 grams = 20 grams

Step 2: Determine moles of Zn and Cu

Now, determine the moles of Zn and Cu in the brass by dividing their masses by their respective molar masses: Molar mass of Cu = 63.55 g/mol Molar mass of Zn = 65.38 g/mol Moles of Cu = (80 g)/(63.55 g/mol) = 1.259 mol Moles of Zn = (20 g)/(65.38 g/mol) = 0.306 mol

Step 3: Find the mass and volume of the solution

Since we considered 100 grams of brass, the mass of the solution is 100 grams. Now, find the volume of the solution using the given density: Density = (Mass of solution) / (Volume of solution) 8750 kg/m^3 = (100 g) / (Volume of solution) Convert the density from kg/m^3 to g/cm^3: 8750 kg/m^3 × (1 kg/1000 g) × (100^3 cm^3/1 m^3) = 8.75 g/cm^3 Now, solve for the volume of the solution: Volume of solution = (100 g) / (8.75 g/cm^3) = 11.43 cm^3 (convert to L): 11.43 cm^3 × (1 L / 1000 cm^3) = 0.01143 L

Step 4: Calculate molality and molarity of Zn

Now, we can calculate the molality and molarity of Zn in the solution: Molality(m) = (Moles of Zn) / (Mass of Cu in kg) = (0.306 mol) / (0.080 kg) = 3.825 mol/kg Molarity(M) = (Moles of Zn) / (Volume of solution in L) = (0.306 mol) / (0.01143 L) = 26.76 mol/L The molality of Zn in the solid solution is 3.825 mol/kg, and the molarity of Zn in the solution is 26.76 mol/L.

Key Concepts

Molality Calculation

Molality is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute per kilogram of solvent. In our example involving brass, we consider the zinc as the solute, and copper acts as the solvent. Molality is often represented by the symbol \(m\).
Calculation of molality is beneficial in scenarios where temperature fluctuates, since molality is not affected by temperature changes. This is because the mass of the solvent remains constant regardless of temperature shifts, as opposed to volume.
In our exercise, we calculated the molality of zinc in brass by first determining the moles of zinc, which is 0.306 mol. This is based on using zinc’s mass (20 g) and its molar mass (65.38 g/mol).
Then, taking copper’s mass (80 g) and converting it into kilograms (0.080 kg), we applied the formula:

• Molality (\(m\)) = Moles of solute (Zn) / Mass of solvent (Cu) in kg
• \(m = \frac{0.306 \, \text{mol}}{0.080 \, \text{kg}} = 3.825 \, \text{mol/kg}\)

Thus, the molality of zinc in this substitutional solid solution is 3.825 mol/kg.

Molarity Calculation

Molarity is a common measure of concentration, defined as the number of moles of solute per liter of solution. It is denoted by \(M\). In this context, we are calculating the molarity of zinc within the brass sample.
To calculate molarity, the volume of the solution is required in liters. This is where the density of brass comes into play, helping us determine the solution's volume from its mass. Given the density of brass as 8750 kg/m³, it was converted to grams per cubic centimeter (\(g/cm^3\)) for compatibility with our calculations:

• 8750 kg/m³ \(\times\) (1 kg/1000 g) \(\times\) (100³ cm³/1 m³) = 8.75 g/cm³

From here, we use the equation:

• Volume = Mass / Density

The mass of brass (100 g) divided by density (8.75 g/cm³) gives us a volume of 11.43 cm³, or 0.01143 liters when converted.
Using this volume, we calculated zinc's molarity with:

• Molarity (\(M\)) = Moles of solute (Zn) / Volume of solution in liters
• \(M = \frac{0.306 \, \text{mol}}{0.01143 \, \text{L}} = 26.76 \, \text{mol/L}\)

Therefore, the molarity of zinc in the brass is 26.76 mol/L.

Density and Volume Relationship

Density is a physical property defined as mass per unit volume. Understanding the relationship between mass, volume, and density is crucial, especially in the context of calculating concentrations like molarity and molality.
In this exercise, knowing the density of brass allowed us to compute the volume of the brass sample. The formula used was:

• Density = Mass / Volume

Rearranging this formula helps in finding the volume if the mass and density are known:

• Volume = Mass / Density

This aspect is crucial for conversion purposes, especially when you need to translate a mass measurement into a volume to determine molarity. In our case, the density was given as 8750 kg/m³, which needed conversion to compatible units like g/cm³ to accurately find the volume of the 100 g brass sample.
Remember, these conversions are important for maintaining unit consistency in calculations. Just like converting density units was essential here, keeping track of unit consistency ensures correct and meaningful results in chemical calculations.