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 \( \text{Ca(NO}_3\text{)}_2 \) is 0.0163 mol/kg, and the total molality of ions is 0.0489 mol/kg.

Step-by-step solution

Calculate Moles of Solute

First, find the molar mass of \( \text{Ca(NO}_3\text{)}_2 \). The atomic masses are approximately: Ca = 40.08, N = 14.01, O = 16.00. Therefore, the molar mass is:\[ 40.08 + 2(14.01) + 6(16.00) = 164.10 \ \text{g/mol} \]Now calculate the moles of \( \text{Ca(NO}_3\text{)}_2 \):\[ \text{moles} = \frac{2.00 \ \text{g}}{164.10 \ \text{g/mol}} \approx 0.0122 \ \text{mol} \]

Calculate Molality of Solute

Molality (\( m \)) is defined as moles of solute per kilogram of solvent. The weight of water is 750 g, which is 0.750 kg:\[ m = \frac{0.0122 \ \text{mol}}{0.750 \ \text{kg}} \approx 0.0163 \ \text{mol/kg} \]

Determine the Number of Ions

\( \text{Ca(NO}_3\text{)}_2 \) dissociates into one \( \text{Ca}^{2+} \) ion and two \( \text{NO}_3^{-} \) ions:\[ \text{Ca(NO}_3\text{)}_2 \rightarrow \text{Ca}^{2+} + 2\text{NO}_3^{-} \]Thus, one formula unit yields 3 ions.

Calculate Total Molality of Ions

Since each molecule of \( \text{Ca(NO}_3\text{)}_2 \) results in 3 ions, the total molality of ions is 3 times the molality of \( \text{Ca(NO}_3\text{)}_2 \):\[ 3 \times 0.0163 \ \text{mol/kg} = 0.0489 \ \text{mol/kg} \]

Key concepts

Dissociation

Dissociation is the process by which a compound splits into two or more constituent ions or molecules in a solution. When ionic compounds like calcium nitrate, \( \text{Ca(NO}_3\text{)}_2 \), dissolve in water, they undergo dissociation. This means they break apart into their individual ions.

In the case of \( \text{Ca(NO}_3\text{)}_2 \), it dissociates completely into one calcium ion (\( \text{Ca}^{2+} \)) and two nitrate ions (\( \text{NO}_3^- \)). This results in a greater number of particles in the solution compared to the original solid form. Dissociation is essential when calculating properties such as molality, as you must consider the total number of ions produced after dissociation. The more ions, the higher the effect on properties of the solution.

Ionic Solution

An ionic solution is formed when ionic compounds dissolve in a solvent like water and dissociate into ions. The result is a solution that conducts electricity due to the presence of free-moving ions.

The ability of ionic solutions to conduct electricity is called their electrolytic property. The more ions present in the solution, the better it conducts electricity. For our example with \( \text{Ca(NO}_3\text{)}_2 \), we have three ions per formula unit after dissociation. In calculating the total molality, we take into account all the ions present, not just the initial compound. Therefore, understanding the behavior of molecules in an ionic solution is crucial for various applications, including calculating the total ionic concentration.

Mole Calculation

Mole calculation is a fundamental concept in chemistry that involves determining the number of moles in a given mass of a substance. This requires knowledge of the molar mass of the substance, which is the sum of the atomic masses of all atoms in a molecule.

For our exercise, we first calculated the molar mass of \( \text{Ca(NO}_3\text{)}_2 \) which is 164.10 g/mol. Using this value, we can determine the number of moles in 2.00 g of the compound by dividing the mass by the molar mass:\[\text{moles} = \frac{2.00 \text{ g}}{164.10 \text{ g/mol}} \approx 0.0122 \text{ mol}\]
This step is essential as it forms the basis for calculating other solution properties like molality. Always ensure correct mole calculations for accurate scientific results.

Chemical Compounds

A chemical compound is a substance formed from two or more elements chemically bonded together. They have specific chemical and physical properties, such as reactivity and molar mass, that differentiate them from their constituent elements.

Our exercise emphasizes the compound calcium nitrate, \( \text{Ca(NO}_3\text{)}_2 \), which consists of calcium, nitrogen, and oxygen. The understanding of such compounds includes knowing their chemical formula and the roles played by chemical bonding and molecular structure, particularly when they interact with other substances.

In solution chemistry, compounds like \( \text{Ca(NO}_3\text{)}_2 \) behave differently than in solid form, which is critical for applications in chemical reactions and synthesis. Understanding the properties of chemical compounds helps predict their behavior and the outcome when placed in different environments or solutions.