Question

If you dissolve \(2.00 \mathrm{g}\) of \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\) in \(750 \mathrm{g}\) of water, what is the molality of \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2} ?\) What is the total molality of ions in solution? (Assume total dissociation of the ionic solid.

Short answer

The molality of \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\) is 0.0163 mol/kg; total molality of ions is 0.0489 mol/kg.

Step-by-step solution

Determine Molar Mass of Solute

Calculate the molar mass of \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\). Calcium (\(\mathrm{Ca}\)) has a mass of 40.08 g/mol, nitrogen (\(\mathrm{N}\)) has a mass of 14.01 g/mol, and oxygen (\(\mathrm{O}\)) has a mass of 16.00 g/mol. Calculate as follows:\[\text{Molar mass} = 40.08 + 2(14.01 + 3 \times 16.00) = 164.10\, \text{g/mol}\]

Convert Grams of Solute to Moles

Use the molar mass to convert the mass of \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\) into moles.\[\text{Moles} = \frac{2.00\, \text{g}}{164.10\, \text{g/mol}} = 0.0122\, \text{mol}\]

Convert Solvent Mass to Kilograms

Convert the mass of water from grams to kilograms for the molality calculation.\[750\, \text{g} = 0.750\, \text{kg}\]

Calculate Molality of the Solute

Molality (\(m\)) is defined as moles of solute per kilogram of solvent.\[m = \frac{0.0122\, \text{mol}}{0.750\, \text{kg}} = 0.0163\, \text{mol/kg}\]

Determine Total Ions Produced

Determine the total number of ions formed after dissociation. \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\) dissociates into 3 ions: 1 \(\mathrm{Ca}^{2+}\) ion and 2 \(\mathrm{NO}_{3}^{-}\) ions.

Calculate Total Molality of Ions

Multiply the molality of the solute by the number of ions to find the total molality of ions.\[\text{Total molality of ions} = 0.0163\, \text{mol/kg} \times 3 = 0.0489\, \text{mol/kg}\]

Key concepts

Molar Mass

Understanding molar mass is crucial when solving chemistry problems. Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It serves as a conversion factor to bridge the number of atoms in an element or molecules in a compound to its mass and vice versa.

For calcium nitrate, \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\), we calculated the molar mass using the atomic masses of its constituent elements. The molar mass calculation involves:

• Calcium (Ca): 40.08 g/mol
• Nitrogen (N): 14.01 g/mol
• Oxygen (O): 16.00 g/mol

The formula for calcium nitrate is composed of one calcium atom, two nitrogen atoms, and six oxygen atoms, leading to the molar mass of 164.10 g/mol. This number was derived by summing the atomic masses as follows: \(\mathrm{Molar\ mass} = 40.08 + 2(14.01 + 3 \times 16.00) = 164.10\, \text{g/mol}\).

This calculation is fundamental for converting mass to moles, allowing us to employ the molar quantity in further problem-solving steps.

Dissociation of Ionic Compounds

Ionic compounds like calcium nitrate dissociate into individual ions when dissolved in water. This dissociation is key for determining ion concentration in solutions.

Calcium nitrate, \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\), is an ionic compound that splits into its constituent ions fully in water. The dissociation can be represented as:

• One calcium ion \(\left( \mathrm{Ca}^{2+} \right)\)
• Two nitrate ions \(\left( \mathrm{NO}_{3}^{-} \right)\)

Upon dissolving, each formula unit of calcium nitrate produces a total of three ions. Understanding this concept of dissociation is important when calculating ion concentration and molality as it triples the number of ions compared to the number of moles of the initial compound.

This insight helps us determine not only the splitting behavior but also prepares us for determining the total ion content in molal units.

Calcium Nitrate

Calcium nitrate is an ionic salt with practical importance in chemistry and agriculture. Its chemical formula is \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\), which highlights its composition of one calcium ion bonded to two nitrate ions.

When dissolved in water, calcium nitrate dissociates completely, releasing its ions into the solution. This behavior is typical for many ionic salts because of the strong interactions with water molecules, allowing them to disperse their ions across the liquid.

Calcium nitrate is often employed in fertilizers due to its solubility and role as a calcium and nitrogen nutrient source. It’s essential for students to not just memorize its formula but also appreciate its behavior in solution, helping to better understand the nature of ionic compounds as a whole.

Ion Concentration

Ion concentration is vital for understanding solution chemistry. It tells us how many ions are present in a given quantity of solvent and can change the chemical and physical properties of a solution.

When calcium nitrate dissociates in water, it forms three distinct ions from one formula unit. The calculation of ion concentration involves multiplying the molality of the original solute by the number of ions produced.
In our calculation, the molality of calcium nitrate was found to be 0.0163 mol/kg. Since dissociation yields three ions, the total molality of ions in the solution is given by \(\text{Total molality of ions} = 0.0163\, \text{mol/kg} \times 3 = 0.0489\, \text{mol/kg}\).

Understanding ion concentration helps in various practical applications, such as predicting electrolyte behavior in batteries or biological systems where ion balance is crucial.