Question

The strength of \(0.01 \mathrm{M} \mathrm{Na}_{2} \mathrm{CO}_{3}\) solution in terms of molality is nearly \(\quad\) 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 of the solution is approximately \(9 \times 10^{-3}\), corresponding to option (a).

Step-by-step solution

Understand the Relationship between Molarity and Molality

Molarity (\(M\)) is the number of moles of solute per liter of solution. Molality (\(m\)), on the other hand, is the number of moles of solute per kilogram of solvent. We need to convert the given molarity into molality.

Find Mass of the Solution

Given the molarity of the solution is \(0.01 \, \text{M}\). This implies 0.01 moles of \(\text{Na}_2\text{CO}_3\) in 1 liter of solution.Assuming 1 liter = 1000 mL and density = \(1.1 \, \text{g/mL}\), the mass of 1 liter of solution is \(1000 \, \text{mL} \times 1.1 \, \text{g/mL} = 1100 \, \text{g}\).

Calculate the Mass of Solute

To find the mass of solute (\(\text{Na}_2\text{CO}_3\)), use its molar mass:\(\text{Molar mass of \(\text{Na}_2\text{CO}_3\)} = 2(23)\text{Na} + 12\text{C} + 3(16)\text{O} = 106 \, \text{g/mol}\).Thus, the mass of 0.01 moles of sodium carbonate is \(0.01 \, \text{mol} \times 106 \, \text{g/mol} = 1.06 \, \text{g}\).

Find the Mass of the Solvent

Subtract the mass of the solute from the total mass of the solution to find the mass of the solvent:\(\text{Mass of solvent} = 1100 \, \text{g} - 1.06 \, \text{g} = 1098.94 \, \text{g}\).Since we require mass in kilograms for molality, convert it: \(1098.94 \, \text{g} = 1.09894 \, \text{kg}\).

Calculate Molality

Molality (\(m\)) is defined as the moles of solute divided by the kilograms of solvent.So, \[m = \frac{0.01 \, \text{moles}}{1.09894 \, \text{kg}} \approx 9.1 \times 10^{-3} \, \text{mol/kg}.\]This roughly corresponds to option (a) \(9 \times 10^{-3}\).

Key concepts

Molarity vs Molality

Molarity and molality are both measures used to express the concentration of solutes in a solution, but they differ in what they take into account. Molarity (\(M\)) is defined as the number of moles of solute per liter of solution. It takes the volume of the entire solution into account, which includes both solute and solvent. This is why molarity can change with temperature, as volume typically changes with temperature fluctuations.
Molality (\(m\)), conversely, is expressed as the number of moles of solute per kilogram of solvent. Because molality relies solely on the mass of the solvent, it remains unaffected by temperature changes. This makes molality a more reliable measurement in scenarios where temperature change is expected and significant. Thus, understanding the differences between these two is crucial for determining the most appropriate context to use each measure. When precision is needed regardless of temperature shifts, molality is often preferred.

• Molarity is volume-based: moles per liter of solution.
• Molality is mass-based: moles per kilogram of solvent.
• Molality is temperature-independent, unlike molarity.

Density and Solution Conversion

Density plays a crucial role in converting between molarity and molality. Density is defined as mass per unit volume, which in this context, is often used to find the mass of a given volume of the solution. Given that the molarity provides us with the number of moles per liter, a known density allows us to calculate the mass of that liter, thereby opening the path to find molality through mass transformation.
For example, if the density of a solution is 1.1 g/mL , then 1 liter (or 1000 mL) of this solution would have a mass of 1100 g . This mass includes both the solvent and solute. To convert molarity to molality, we subtract the mass of the solute (calculated using moles and molar mass) from the total mass to find the mass of the solvent. Finally, converting this mass into kilograms sets up for calculating molality by using the moles of solute and kilograms of solvent.
So, always keep the importance of density in mind whenever you are required to switch between molarity and molality.

Molar Mass Calculation

Calculating the molar mass is a fundamental step when converting between molarity and molality, as it allows us to determine the mass of the solute from the moles given by the molarity. The molar mass of a compound can be found by adding together the atomic masses of all the atoms present in the compound, which are readily available on the periodic table.
For instance, to calculate the molar mass of sodium carbonate (\(\text{Na}_2\text{CO}_3\)), you would add the atomic masses of two sodium atoms, one carbon atom, and three oxygen atoms. This results in: \[2(23) + 12 + 3(16) = 106 \, \text{g/mol}\]. Knowing this, you can find the mass of a certain number of moles. If a solution has \(0.01\) moles of \(\text{Na}_2\text{CO}_3\), the mass of the solute is calculated as: \(0.01 \, \text{mol} \times 106 \, \text{g/mol} = 1.06 \, \text{g}\).
This helps in finding the mass of the solute required for the specific conversion calculations, highlighting why understanding molar mass is key in such chemical assessments.