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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

01

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##:
02

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攕uch as hardness, ductility, and corrosion resistance鈥攖o 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/m鲁. 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/m鲁, 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/m鲁 = 5500.5 kg, and Mass of Zn = 0.35 x 8470 kg/m鲁 = 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 m鲁), 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 10鈦� mol/m鲁, 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.

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Most popular questions from this chapter

Which of the following in each pair is likely to be more soluble in hexane, \(\mathrm{C}_{6} \mathrm{H}_{14}:\) (a) \(\mathrm{CCl}_{4}\) or \(\mathrm{CaCl}_{2}\), (b) benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) or glycerol, \(\mathrm{CH}_{2}(\mathrm{OH}) \mathrm{CH}(\mathrm{OH}) \mathrm{CH}_{2} \mathrm{OH},\) (c) octanoic acid, \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{COOH},\) or acetic acid, \(\mathrm{CH}_{3} \mathrm{COOH}\) ? Explain your answer in each case.

(a) A sample of hydrogen gas is generated in a closed container by reacting \(1.750 \mathrm{~g}\) of zinc metal with \(50.0 \mathrm{~mL}\) of \(1.00 \mathrm{M}\) hydrochloric acid. Write the balanced equation for the reaction, and calculate the number of moles of hydrogen formed, assuming that the reaction is complete. (b) The volume over the solution in the container is 150 mL. Calculate the partial pressure of the hydrogen gas in this volume at \(25^{\circ} \mathrm{C}\), ignoring any solubility of the gas in the solution. (c) The Henry's law constant for hydrogen in water at \(25^{\circ} \mathrm{C}\) is \(7.7 \times 10^{-6} \mathrm{~mol} / \mathrm{m}^{3}-\mathrm{Pa}\). Estimate the number of moles of hydrogen gas that remain dissolved in the solution. What fraction of the gas molecules in the system is dissolved in the solution? Was it reasonable to ignore any dissolved hydrogen in part (b)?

The density of acetonitrile \(\left(\mathrm{CH}_{3} \mathrm{CN}\right)\) is \(0.786 \mathrm{~g} / \mathrm{mL}\) and the density of methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) is \(0.791 \mathrm{~g} / \mathrm{mL}\). A solution is made by dissolving \(25.0 \mathrm{~mL}\) of \(\mathrm{CH}_{3} \mathrm{OH}\) in \(100 \mathrm{~mL}\) of \(\mathrm{CH}_{3} \mathrm{CN}\) (a) What is the mole fraction of methanol in the solution? (b) What is the molality of the solution? (c) Assuming that the volumes are additive, what is the molarity of \(\mathrm{CH}_{3} \mathrm{OH}\) in the solution?

The maximum allowable concentration of lead in drinking water is 9.0 ppb. (a) Calculate the molarity of lead in a 9.0ppb solution. (b) How many grams of lead are in a swimming pool containing 9.0 ppb lead in \(60 \mathrm{~m}^{3}\) of water?

The concentration of gold in seawater has been reported to be between 5 ppt (parts per trillion) and 50 ppt. Assuming that seawater contains 13 ppt of gold, calculate the number of grams of gold contained in \(1.0 \times 10^{3}\) gal of seawater.
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