Question

The strength of \(0.01 \mathrm{M} \mathrm{Na}_{2} \mathrm{CO}_{3}\) solution in terms of molality is nearly ____ when the density of the solution is \(1.1 \mathrm{~g} / \mathrm{mL}\) (a) \(9 \times 10^{-3}\) (b) \(1.8 \times 10^{-3}\) (c) \(4.5 \times 10^{-3}\) (d) \(1.1 \times 10^{-3}\)

Short answer

The molality is approximately \(9 \times 10^{-3}\), option (a).

Step-by-step solution

Understand the Relationship Between Molarity and Molality

Molarity ( ext{M}) is the number of moles of solute per liter of solution, whereas molality ( ext{m}) is the number of moles of solute per kilogram of solvent. Here, we need to convert from molarity to molality.

Calculate the Volume and Mass of the Solution

Since the density of the solution is given as \(1.1 \text{ g/mL}\), the mass of 1 liter (1000 mL) of this solution is \(1000 \text{ mL} \times 1.1 \text{ g/mL} = 1100 \text{ g}\). This is the mass of the solution.

Determine the Mass of the Solvent

The mass of the solute in the solution can be found from its molarity. \(0.01 \text{ M Na}_2\text{CO}_3\) means there are \(0.01 \text{ moles}\) per liter. The molar mass of \(\text{Na}_2\text{CO}_3\) is \(2(23) + 12 + 3(16) = 106 \text{ g/mol}\). Therefore, the mass of the solute is \(0.01 \text{ moles} \times 106 \text{ g/mol} = 1.06 \text{ g}\). The mass of the solvent (water) in the solution is \(1100 \text{ g} - 1.06 \text{ g} = 1098.94 \text{ g}\) or \(1.09894 \text{ kg}\).

Calculate Molality

Molality is calculated as follows: \[ \text{molality} = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} = \frac{0.01 \text{ moles}}{1.09894 \text{ kg}} \approx 0.0091 \text{ m}\].

Compare Against Options

Comparing \(0.0091 \text{ m}\) against the given options, \((a) \: 9 \times 10^{-3}\) best matches our calculated value.

Key concepts

Molarity and Molality

To solve problems in chemistry relating to solutions, it's important to understand the concept of

• Molarity
• Molality

both of which are measures of concentration of a solution.
Molarity (denoted as \(M\)) is defined as the number of moles of solute present per liter of solution.
This means it's a convenient measure when you're working with reactions that occur in a laboratory setting.
To calculate molarity, the formula is:
\[ \text{Molarity} \ (M) = \frac{\text{moles of solute}}{\text{volume of solution in liters}} \]
Molality (denoted as \(m\)), on the other hand, expresses the concentration as the number of moles of solute per kilogram of solvent.
One of the biggest differences between the two is that molality does not change with temperature, since it isn't dependent on volume which can expand or contract with temperature.
Instead, it is based on mass. To calculate molality, use the formula: \[ \text{Molality} \ (m) = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \] This becomes particularly useful in cases where temperature fluctuations are involved.

Density Calculations

Density is a property of matter that represents how much mass is contained in a given unit of volume.
In chemistry, density can help us convert between mass and volume.
It’s used often in calculations to determine the mass of a solution, especially when the concentration is known.
The formula to find density is: \[ \text{Density} = \frac{\text{mass}}{\text{volume}} \]
This means if you know the density and the volume, you can easily calculate the mass.
In our problem, the density was given as \(1.1 \, \text{g/mL}\).
By multiplying the density by the volume of the solution in milliliters, we were able to find the total mass of the solution.
This is essential when converting between molarity and molality, because it allows us to find out how much of the solution is solvent and how much is solute.

Molar Mass

The molar mass of a substance is an important concept in chemistry.
It is the mass of one mole of a substance, typically expressed in grams per mole (g/mol).
Molar mass is fundamental when performing stoichiometry in chemistry, as it allows conversion between the mass of a sample and the moles of a substance.
To calculate the molar mass of a compound:

• Add up the atomic masses of all atoms in a chemical formula.
• For example, the molar mass of \(\text{Na}_2\text{CO}_3\) is calculated as:
  \[ 2(23 \, \text{g/mol}) + 12 \, \text{g/mol} + 3(16 \, \text{g/mol}) = 106 \, \text{g/mol} \]

This allows us to convert from moles to grams or vice versa in a chemical equation or solution setup.
In the exercise, knowing the molar mass of \(\text{Na}_2\text{CO}_3\) allowed us to find the mass of the solute, which was then used to determine the molality.