Question

If you want a solution that is \(0.100 \mathrm{m}\) in ions, what mass of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) must you dissolve in \(125 \mathrm{g}\) of water? (Assume total dissociation of the ionic solid.)

Short answer

Dissolve 0.5918 g of \(\mathrm{Na}_2\mathrm{SO}_4\) in 125 g of water.

Step-by-step solution

Understand the molality formula

Molality (abla mabla) is defined as the number of moles of solute per kilogram of solvent. The formula is: \[ \text{Molality (m)} = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \] We need a molality of 0.100m and the mass of the solvent (water) is given as 125 g, which is 0.125 kg.

Calculate the moles of ions

Since \(\mathrm{Na_{2}SO_{4}}\) dissociates into 3 ions \((2\mathrm{Na^+} + \mathrm{SO_4^{2-}})\), a 0.100m solution requires 0.100 moles of ions per kg of solvent. Hence, for 0.125 kg of water (solvent), we need: \[ 0.100 \text{ mol/kg} \times 0.125 \text{ kg} = 0.0125 \text{ moles of ions} \]

Determine the moles of \(\mathrm{Na}_2\mathrm{SO}_4\) required

Since \(\mathrm{Na}_2\mathrm{SO}_4\) dissociates into 3 ions, the number of moles of \(\mathrm{Na}_2\mathrm{SO}_4\) needed is: \[ \frac{0.0125 \text{ moles of ions}}{3} = 0.004167 \text{ moles of } \mathrm{Na}_2\mathrm{SO}_4 \]

Find the molar mass of \(\mathrm{Na}_2\mathrm{SO}_4\)

The molar mass of \(\mathrm{Na}_2\mathrm{SO}_4\) is calculated as follows: \(2 \times 22.99 \text{ (Na)} + 32.07 \text{ (S)} + 4 \times 16.00 \text{ (O)} = 142.05 \text{ g/mol}\).

Calculate the mass of \(\mathrm{Na}_2\mathrm{SO}_4\) needed

To find the required mass, multiply the moles needed by the molar mass: \[ 0.004167 \text{ moles} \times 142.05 \text{ g/mol} = 0.5918 \text{ g} \] Therefore, 0.5918 g of \(\mathrm{Na}_2\mathrm{SO}_4\) is needed.

Key concepts

Moles of ions

When dealing with solutions, understanding the concept of moles of ions is crucial. A mole is a unit that measures the amount of substance, useful for counting particles like atoms or molecules.
In this exercise, sodium sulfate (\(\text{Na}_2\text{SO}_4\)) is used as the solute, which will completely dissociate in solution. When \(\text{Na}_2\text{SO}_4\) dissolves in water, it separates into its constituent ions:

• 2 moles of sodium ions (\(\text{Na}^+\))
• 1 mole of sulfate ions (\(\text{SO}_4^{2-}\))

Every mole of \(\text{Na}_2\text{SO}_4\) therefore generates 3 moles of ions. In a solution with a molality of 0.100m in ions, this means that for every kilogram of solvent, we need a total of 0.100 moles of ions. This is a key point: the concentration of ions, rather than the compound itself, determines the behavior of the solution.

Sodium sulfate dissociation

Understanding sodium sulfate dissociation is vital for calculating the amount needed to achieve a desired ion concentration. When dissolved, sodium sulfate (\(\text{Na}_2\text{SO}_4\)) splits into three ions:

• Two sodium ions (\(\text{Na}^+\))
• One sulfate ion (\(\text{SO}_4^{2-}\))

This process is referred to as dissociation. Each \(\text{Na}_2\text{SO}_4\) molecule generates 3 ions, which is crucial in determining how many ions are in solution.
If you aim for a 0.100m solution in terms of ions with 0.125 kg of water, it is essential to consider this dissociation. The total moles of ions must match the desired molality when taking into account the breakdown into 3 parts. For these calculations, we use the fact that 0.0125 moles of ions in 0.125 kg of water would meet the 0.100m ion concentration required.

Calculation of molar mass

The molar mass is the mass of one mole of a chemical substance. For sodium sulfate (\(\text{Na}_2\text{SO}_4\)), calculating the molar mass involves adding up the atomic masses of all the atoms in the formula.
Here’s how you calculate it:

• Sodium (\(\text{Na}\)) has an atomic mass of about 22.99 g/mol, and there are two sodium atoms, so: \(2 \times 22.99 = 45.98 \text{ g/mol}\).
• Sulfur (\(\text{S}\)) has a mass of 32.07 g/mol.
• Oxygen (\(\text{O}\)) has a mass of 16.00 g/mol, and there are four oxygen atoms, so: \(4 \times 16.00 = 64.00 \text{ g/mol}\).

Adding these together, you get the molar mass of \(\text{Na}_2\text{SO}_4\):
\(45.98 + 32.07 + 64.00 = 142.05 \text{ g/mol}\).
This comprehensive understanding allows us to find out how much of the compound we need to achieve the desired solution molality; in our case, to dissolve 0.004167 moles of \(\text{Na}_2\text{SO}_4\) to get 0.5918 grams.