Question

A seawater sample has a density of $1.03\mathrm{g}/\mathrm{mL}$ and $2.8\mathrm{\%}\mathrm{NaCl}$l by mass. A saturated solution of $\mathrm{NaCl}$ in water is $5.45\mathrm{M}\mathrm{NaCl}.$ How many liters of water would have to be evaporated from $1.00\times {10}^6\mathrm{L}$ of the seawater before $\mathrm{NaCl}$ would begin to crystallize? (A saturated solution contains the maximum amount of dissolved solute possible.)

Short answer

Approximately $9.29\times {10}^4\mathrm{L}$ of water would have to be evaporated from the seawater for NaCl to start crystallizing.

Step-by-step solution

Calculate the total mass of the seawater

The given density of the seawater is $1.03\mathrm{g}/\mathrm{mL}$ or $1030\mathrm{kg}/{\mathrm{m}}^3$, and the volume is $1.00\times {10}^6\mathrm{L}$ or $1000{\mathrm{m}}^3$. Using the formula: Density = Mass / Volume, the total mass of the seawater is $1.03\times {10}^6\mathrm{kg}$.

Calculate the mass of NaCl in seawater

The percentage of NaCl by mass is 2.8%. Therefore, the mass of NaCl in the seawater can be calculated as: $2.8/100\times 1.03\times {10}^6=28840\mathrm{kg}$.

Calculate the moles of NaCl in seawater

We know that the molar mass of NaCl is approximately $58.44\mathrm{g}/\mathrm{mol}$ or $0.05844\mathrm{kg}/\mathrm{mol}$. The number of moles of NaCl in the seawater is thus: $28840/0.05844=4.94\times {10}^5\mathrm{moles}$.

Calculate the amount of water to be evaporated

A saturate solution of NaCl is $5.45\mathrm{M}$, which means there are $5.45\mathrm{moles}$ of NaCl in 1 liter of water. The amount of water in which $4.94\times {10}^5\mathrm{moles}$ of NaCl can be dissolved is $(4.94\times {10}^5)/5.45=9.07\times {10}^4\mathrm{L}$. Since the total volume of the seawater is $1.00\times {10}^6\mathrm{L}$, the amount of water that needs to be evaporated is $1.00\times {10}^6-9.07\times {10}^4=9.29\times {10}^4\mathrm{L}$.

Key concepts

Saturated Solution

In chemistry, a saturated solution is when a solvent contains the maximum amount of solute it can dissolve at a given temperature. If you try to dissolve more solute into a saturated solution, the excess solute will simply remain undissolved. This concept helps in identifying the solubility limit of substances, which varies between different solutes and solvents.

A key point to understand is that a saturated solution is in a state of dynamic equilibrium. This means that the rate of solute dissolving into the solvent equals the rate at which the solute precipitates out. Consequently, the concentration of the dissolved solute remains constant at saturation.

For example, in the problem, seawater is initially not saturated with NaCl. However, when the solution reaches the saturation point (5.45 M), any additional removal of water will result in visible crystallization of NaCl, as solubility limits have been reached.

Molarity

Molarity is a way to express the concentration of a solution, defined as the number of moles of solute per liter of solution. Represented by the symbol "M," it allows chemists to standardize procedures and share detailed instructions about reactions and experiments.

The formula for molarity is given by: $$M=\frac{n}{V}$$where $n$ is the number of moles of solute and $V$ is the volume of the solution in liters.

In the exercise, the saturated solution of NaCl has a molarity of 5.45 M, indicating that each liter of the solution contains 5.45 moles of sodium chloride. This molarity helps us determine how much water must be evaporated for the solution to become saturated and begin forming crystals.

Mass Percentage

Mass percentage expresses the concentration of a component in a mixture as the percentage of the total mass. It is also called percent by mass or mass percent. This is another way to describe a solution's composition, demonstrating the part of the solution that is a specific solute compared to the entire solution's mass.

To calculate mass percentage, use the formula:$$\text{Mass Percentage}=\left(\frac{\text{Mass of solute}}{\text{Total mass of solution}}\right)\times 100\mathrm{\%}$$For example, if a solution contains 28.84 kg of NaCl dissolved in 1030 kg of seawater, the NaCl contributes approximately 2.8% of the total mass, which helps signal how concentrated the solution is.

Understanding mass percentage is crucial when measuring ingredients for lab preparation, quality control, and in processes needing precise ratios for optimal results.