Question

A solution is prepared by diluting \(225 \mathrm{~mL}\) of \(0.1885 \mathrm{M}\) aluminum sulfate solution with water to a final volume of \(1.450 \mathrm{~L}\). Calculate (a) the number of moles of aluminum sulfate before dilution. (b) the molarities of the aluminum sulfate, aluminum ions, and sulfate ions in the diluted solution.

Short answer

Answer: The molarities in the diluted solution are as follows: (a) Aluminum sulfate: 0.02926 M (b) Aluminum ions: 0.05852 M (c) Sulfate ions: 0.08778 M

Step-by-step solution

Calculate the number of moles of aluminum sulfate before dilution

To calculate the number of moles of aluminum sulfate before dilution, use the formula: Moles = Molarity × Volume The volume given is in mL, so we need to convert it to L: \(225 \mathrm{~mL} = 225 \div 1000 = 0.225 \mathrm{~L}\) Now we can use the initial molarity of the solution: Moles of aluminum sulfate = \(0.1885 \mathrm{M} \times 0.225 \mathrm{~L} = 0.04242 \mathrm{~mol}\) (a) The number of moles of aluminum sulfate before dilution is \(0.04242 \mathrm{~mol}\).

Calculate the molarities of aluminum sulfate, aluminum ions, and sulfate ions in the diluted solution

Now that we have the number of moles of aluminum sulfate and the final volume of the solution, we can calculate the molarities in the diluted solution. New molarity of aluminum sulfate = Moles of aluminum sulfate / Final volume New molarity of aluminum sulfate = \(0.04242 \mathrm{~mol} \div 1.450 \mathrm{~L} = 0.02926 \mathrm{M}\) (b1) The molarity of aluminum sulfate in the diluted solution is \(0.02926 \mathrm{~M}\). For each mole of aluminum sulfate (\(\mathrm{Al_2(SO_4)_3}\)) that dissociates, two moles of aluminum ions (\(\mathrm{Al^{3+}}\)) and three moles of sulfate ions (\(\mathrm{SO_4^{2-}}\)) are produced. Thus, the molarities of aluminum ions and sulfate ions will be 2 times and 3 times the molarity of aluminum sulfate, respectively. Molarity of aluminum ions = \(2 \times 0.02926 \mathrm{M} = 0.05852 \mathrm{M}\) Molarity of sulfate ions = \(3 \times 0.02926 \mathrm{M} = 0.08778 \mathrm{M}\) (b2) The molarity of aluminum ions in the diluted solution is \(0.05852 \mathrm{~M}\). (b3) The molarity of sulfate ions in the diluted solution is \(0.08778 \mathrm{~M}\).

Key concepts

Molarity

Molarity is a measurement of concentration in chemistry, specifically it is a way to express the concentration of a solute in a solution. In simpler terms, it tells us how much of a substance (solute) is dissolved in a certain volume of liquid (solvent). The formula to calculate molarity is given by the ratio of moles of solute to the volume of the solution in liters. Mathematically, it is described by the equation:
code>Molarity (M) = \( \frac{\text{moles of solute}}{\text{volume of solution in liters}} \)
In the context of our textbook problem, we initially calculate the number of moles of aluminum sulfate using the formula outlined in the solution, and after dilution, we determine the new molarity by dividing the moles of solute by the new, larger volume of solution. This concept is vital, as understanding molarity allows one to predict how a solution will react in different contexts, such as in chemical reactions or in biological systems.

Moles of Solute

The concept of moles of solute is central to the study of chemistry. A mole is a unit that represents a specific number of particles, atoms, ions, or molecules. It allows chemists to count particles by weighing, as it relates the mass of a substance to the amount of substance. The mole is based on Avogadro's number, which is approximately \(6.022 \times 10^{23}\) entities per mole.

When computing the moles of solute in a given volume, you use the formula:
code>Moles = Molarity \(\times\) Volume
In our example, the initial volume of the aluminum sulfate solution is converted into liters before the calculation, ensuring that the unit for volume is consistent with the molarity unit, which is moles per liter. The resulting number of moles is a measure of the quantity of the aluminum sulfate present before its dilution, and it remains unchanged regardless of the volume of the solution, provided no additional solute is added.

Molar Concentration

Molar concentration, often just called concentration, is another term for molarity. It is widely used in chemistry to describe the amount of a substance in a given volume of solution. In the practice of chemistry, knowing the molar concentration of a solute is essential for tasks like preparing solutions of desired strengths, performing titrations, or calculating yields of reactions.

As we've seen with the aluminum sulfate example, to calculate the molar concentration after dilution, we take the number of moles of solute and divide it by the new volume (post-dilution). This concept is not only key in solving textbook problems but also in real-life applications where precise measurements are crucial, such as in pharmaceuticals, environmental testing, and in research laboratories. To expand our example, it's important to note that when a compound like \(\mathrm{Al_2(SO_4)_3}\) dissociates in water, it affects the concentration of the resulting ions, which are all relevant for understanding chemical equilibria and reaction dynamics.