Question

Calculate the molality of each of the following aqueous solutions: (a) \(2.55 M \mathrm{NaCl}\) solution (density of solution \(=1.08 \mathrm{~g} / \mathrm{mL}\) ), (b) 45.2 percent by mass KBr solution.

Short answer

(a) 2.74 mol/kg (b) 6.93 mol/kg

Step-by-step solution

Define Molality

Molality, denoted by \( m \), is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute per kilogram of solvent. The formula is \( m = \frac{n_{solute}}{m_{solvent}} \), where \( n_{solute} \) is the moles of solute, and \( m_{solvent} \) is the mass of the solvent in kilograms.

Calculate Molality for NaCl Solution

Given a \(2.55 \ M \) NaCl solution with a density of \(1.08 \ g/mL\), first find the mass of the solution: \( 1.08 \ g/mL \times 1000 \ mL = 1080 \ g\). The moles of NaCl are \(2.55 \times 1 \ L = 2.55 \ moles\). The molar mass of NaCl is approximately \(58.44 \ g/mol\), so the mass of NaCl is \( 2.55 \times 58.44 = 149.022 \ g\). Subtract the mass of NaCl from the total mass to find the mass of the solvent (water): \(1080 \ g - 149.022 \ g = 930.978 \ g\), which is \(0.930978 \ kg\). The molality is then \( \frac{2.55}{0.930978} = 2.74 \ mol/kg\).

Calculate Molality for KBr Solution

For a 45.2% by mass KBr solution, in a 100 g solution, 45.2 g is KBr and 54.8 g is water. Moles of KBr is given by its mass divided by the molar mass \((119.00 \ g/mol)\), giving \( \frac{45.2}{119.00} = 0.3798 \ moles\). The mass of water is \(54.8 \ g = 0.0548 \ kg\). The molality is \( \frac{0.3798}{0.0548} = 6.93 \ mol/kg\).

Key concepts

Molarity

Molarity is one of the most commonly used measures of solution concentration. It tells us how many moles of a solute are present in one liter of the solution. Molarity is expressed as \( M \) and calculated using the formula:

• \( M = \frac{n_{solute}}{V_{solution}} \)

Here, \( n_{solute} \) is the number of moles of solute and \( V_{solution} \) is the volume of the solution in liters.
This measure helps in understanding how concentrated a solution is and is particularly useful in reactions where precise measurement is crucial.
For example, a \(2.55 \, M\) NaCl solution means there are \(2.55\) moles of NaCl in every liter of the solution.

Density

Density is a physical property that connects the mass of a substance with its volume. It is expressed in units like grams per milliliter (g/mL) or kilograms per liter (kg/L). Calculating the density of a solution combines the mass and volume in the following formula:

• Density = \( \frac{mass}{volume} \)

Understanding density helps us convert between the mass and volume of a solution, which is essential when calculating other concentrations, like molality.
For instance, knowing the density of the NaCl solution allows us to determine the total mass of the solution when given a specific volume, such as 1000 mL of solution weighing 1080 grams due to its density of \(1.08 \, g/mL\).

Percent by Mass

Percent by mass, or mass percent, is a way to express how a solute is dispersed within a solution. It's calculated as:

• \( \text{Percent by Mass} = \left( \frac{mass_{solute}}{mass_{solution}} \right) \times 100 \% \)

This percentage indicates what portion of the solution's total mass is the solute. In the case of a 45.2% KBr solution, 45.2 grams of KBr are present in every 100 grams of the solution.
Understanding percent by mass is crucial when determining how solutions are composed and helps in further calculating specific concentrations like molality.

Solution Concentration

Solution concentration refers to the quantity of solute present in a given quantity of solution. It provides insight into how a solute is distributed within a solvent.
There are various ways to express solution concentration, with each method serving specific purposes based on the nature of the chemical processes involved.

• **Molarity**: Describes moles of solute per liter of solution.
• **Molality**: Focuses on moles of solute per kilogram of solvent.
• **Percent by Mass**: Shows the mass fraction of solute in the solution.

Each measure of concentration can highlight different aspects of a solution's chemical characteristics. Understanding these differences helps in carrying out precise and error-free calculations when preparing solutions for experiments or commercial applications.