Question

Calculate each of the following quantities: (a) Mass (g) of solute needed to make \(475 \mathrm{~mL}\) of \(5.62 \times 10^{-2} \mathrm{M}\) potassium sulfate (b) Molarity of a solution that contains \(7.25 \mathrm{mg}\) of calcium chloride in each milliliter (c) Number of \(\mathrm{Mg}^{2+}\) ions in each milliliter of \(0.184 \mathrm{M}\) magnesium bromide

Short answer

(a) 4.65 g of potassium sulfate. (b) 6.53 x 10^-2 M. (c) 1.11 x 10^20 Mg^2+ ions.

Step-by-step solution

- Calculate the Mass of Solute Needed (Part a)

First, calculate the number of moles of potassium sulfate needed. Use the formula for molarity: \[M = \frac{n}{V}\] where M is molarity, n is the number of moles, and V is the volume in liters. Rearrange to find the number of moles: \[n = M \times V\] For this problem: \[M = 5.62 \times 10^{-2} \text{ M} \] \[V = 475 \text{ mL} = 0.475 \text{ L}\] Thus: \[n = 5.62 \times 10^{-2} \text{ M} \times 0.475 \text{ L} = 2.67 \times 10^{-2} \text{ moles}\] Next, calculate the mass using the molar mass of potassium sulfate (\(K_2SO_4\)): \[K: 39.10 \text{ g/mol} \quad S: 32.07 \text{ g/mol} \quad O: 16.00 \text{ g/mol}\] Molar mass: \[2(39.10) + 32.07 + 4(16.00) = 174.27 \text{ g/mol}\] Finally, use the mass formula: \[\text{mass} = n \times \text{molar mass}\] \[\text{mass} = 2.67 \times 10^{-2} \text{ mol} \times 174.27 \text{ g/mol} = 4.65 \text{ g}\]

- Calculate Molarity of Solution (Part b)

To find the molarity of the calcium chloride solution, first convert the mass of calcium chloride to grams: \[7.25 \text{ mg} = 7.25 \times 10^{-3} \text{ g}\] Then, use the formula for molarity: \[M = \frac{n}{V}\] where n is the number of moles and V is volume in liters. First, calculate the number of moles of calcium chloride (\(CaCl_2\)) using its molar mass: \[Ca: 40.08 \text{ g/mol} \quad Cl: 35.45 \text{ g/mol}\] Molar mass: \[40.08 + 2 \times 35.45 = 110.98 \text{ g/mol}\] Number of moles: \[n = \frac{7.25 \times 10^{-3} \text{ g}}{110.98 \text{ g/mol}} = 6.53 \times 10^{-5} \text{ moles}\] Volume: \[1 \text{ mL} = 1 \times 10^{-3} \text{ L}\] Thus, molarity: \[M = \frac{6.53 \times 10^{-5} \text{ moles}}{1 \times 10^{-3} \text{ L}} = 6.53 \times 10^{-2} \text{ M}\]

- Number of Mg ions (Part c)

For calculating the number of \(\mathrm{Mg}^{2+} \) ions, use Avogadro's number (\(6.022 \times 10^{23} \) ions/mol). The given molarity of magnesium bromide \((\mathrm{MgBr_2})\) is 0.184 M. The number of moles of \( \mathrm{Mg}^{2+} \) ions in 1 mL of solution is: \[0.184 \text{ moles/L} \times 1 \text{ mL} \times \frac{1 \text{ L}}{1000 \text{ mL}} = 1.84 \times 10^{-4} \text{ moles}\] Next, calculate the number of ions: \[\text{Number of ions} = 1.84 \times 10^{-4} \text{ moles} \times 6.022 \times 10^{23} \text{ ions/mol} \times 1 = 1.11 \times 10^{20} \text{ ions}\]

Key concepts

Molar Mass

Molar mass is a fundamental concept in chemistry, representing the mass of one mole of a given substance. To find it, you sum the atomic masses of all the atoms in a molecule. For instance, the molar mass of potassium sulfate (\(K_2SO_4\)) can be found by adding the atomic masses of potassium (\(K\)), sulfur (\(S\)), and oxygen (\(O\)) as follows:
- Potassium (2 atoms): \(2 \times 39.10 \text{ g/mol}\)
- Sulfur (1 atom): \(32.07 \text{ g/mol}\)
- Oxygen (4 atoms): \(4 \times 16.00 \text{ g/mol}\)
Adding them up, the molar mass of \(K_2SO_4\) is \(174.27 \text{ g/mol}\). Understanding molar mass is crucial for converting between mass and moles in various chemical calculations.

Number of Moles

The number of moles is a measure of the quantity of a substance. It connects the mass of a substance to the number of particles it contains. The formula to find the number of moles (\(n\)) is:
\(n = \frac{\text{mass}}{\text{molar mass}}\)
For example, if you have \(7.25 \text{ mg} \) of calcium chloride (\(CaCl_2\)), you first convert it to grams: \(7.25 \text{ mg} = 7.25 \times 10^{-3} \text{ g}\). Then, using the molar mass of \(CaCl_2\) (\(110.98 \text{ g/mol}\)), you calculate the number of moles:
\(n = \frac{7.25 \times 10^{-3} \text{ g}}{110.98 \text{ g/mol}} = 6.53 \times 10^{-5} \text{ moles}\). Knowing the number of moles helps in determining molarity and other properties of the solution.

Avogadro's Number

Avogadro's number (\(6.022 \times 10^{23}\)) is the number of atoms, ions, or molecules in one mole of a substance. It's a constant that allows chemists to count particles by weighing them. For instance, to find the number of \(Mg^{2+} \) ions in 1 mL of \(0.184 \text{ M} \) magnesium bromide (\(MgBr_2\)), first find the number of moles of \(Mg^{2+}\) ions:
\(0.184 \text{ moles/L} \times 1 \text{ mL} \times \frac{1 \text{ L}}{1000 \text{ mL}} = 1.84 \times 10^{-4} \text{ moles}\)
Then, multiply by Avogadro's number to get the actual number of ions:
\(\text{Number of ions} = 1.84 \times 10^{-4} \text{ moles} \times 6.022 \times 10^{23} \text{ ions/mol} = 1.11 \times 10^{20} \text{ ions}\).
This concept is crucial for understanding the scale of reactions and properties of substances at the molecular level.

Solution Concentration

Solution concentration, commonly expressed as molarity (\(M\)), indicates the amount of solute dissolved in a given volume of solvent. It’s calculated using the formula:
\(M = \frac{n}{V}\)
where \(M\) is molarity, \(n\) is the number of moles, and \(V\) is the volume in liters. For example, to find the molarity of a calcium chloride solution containing \(7.25 \text{ mg/mL}\), convert the mass to grams: \(7.25 \text{ mg} = 7.25 \times 10^{-3} \text{ g}\). Then, calculate the moles of solute using its molar mass: \(n = 6.53 \times 10^{-5} \text{ moles}\). Since the volume is \(1 \text{ mL} = 1 \times 10^{-3} \text{ L}\), the molarity is:
\(M = \frac{6.53 \times 10^{-5} \text{ moles}}{1 \times 10^{-3} \text{ L}} = 6.53 \times 10^{-2} \text{ M}\).
Understanding solution concentration helps in preparing solutions with precise properties for chemical reactions and experiments.