Chapter 13: Problem 57

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?

Short Answer

Expert verified

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 by step solution

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##:

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 \(6.986 \times 10^7 \mathrm{~mol/m}^3\).