Problem 140 - Introductory Chemistry Atoms First

Question

A student dissolves \(36.9 \mathrm{~g}\) of calcium nitrate in \(500.0 \mathrm{~g}\) of water. How many grams of glucose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\), would he have to dissolve in \(500.0 \mathrm{~g}\) of water to make the glucose solution have the same boiling point as the calcium nitrate solution?

Step-by-Step Solution

Answer

The student would need to dissolve approximately \(121.6 \mathrm{~g}\) of glucose in \(500.0 \mathrm{~g}\) of water to make the glucose solution have the same boiling point as the calcium nitrate solution.

Step 1: Calculate molality of calcium nitrate solution

First, we need to find the molality of the calcium nitrate solution. Molality can be calculated using the formula: m = (moles of solute) / (mass of solvent in kg) Molecular weight of calcium nitrate, Ca(NO3)2, is: \(1 \times 40.08 \text{(Ca)} + 2 \times (1 \times 14.01 \text{(N)} + 3 \times 16.00 \text{(O)}) = 164.1 \mathrm{~g/mol}\) Moles of calcium nitrate = (36.9 g) / (164.1 g/mol) = 0.225 mol And we know that the mass of water (solvent) is 500.0 g (which is 0.500 kg). Now, we can calculate the molality: m = (0.225 mol) / (0.500 kg) = 0.450 mol/kg

Step 2: Calculate boiling point elevation of calcium nitrate solution

Now that we have the molality, we can find the boiling point elevation using the formula ΔT = Kb * m * i. Since calcium nitrate completely dissociates into 3 ions (Ca²⁺ and 2NO₃⁻), the van't Hoff factor i = 3. The boiling point elevation constant for water, Kb, is 0.52 °C/mol/kg. ΔT = Kb * m * i = (0.52 °C/mol/kg) * (0.450 mol/kg) * (3) = 0.702 °C

Step 3: Calculate molality needed for glucose solution

To get the same boiling point elevation, we can find the molality needed for the glucose solution (since glucose doesn't dissociate, i = 1): m_glucose = ΔT / (Kb * i) = 0.702 °C / (0.52 °C/mol/kg * 1) = 1.35 mol/kg

Step 4: Calculate mass of glucose needed

Now that we have the required molality for the glucose solution, we can find the mass of glucose needed. The molecular weight of glucose, C6H12O6, is: \(6 \times 12.01 \text{(C)} + 12 \times 1.01 \text{(H)} + 6 \times 16.00 \text{(O)} = 180.16 \mathrm{~g/mol}\) We know that the mass of water is still 500.0 g (or 0.500 kg). Therefore, the mass of glucose needed can be calculated using the molality: mass of glucose = molality * (mass of solvent in kg) * molecular weight = (1.35 mol/kg) * (0.500 kg) * (180.16 g/mol) = 121.6 g The student would need to dissolve approximately 121.6 grams of glucose in 500.0 grams of water to make the glucose solution have the same boiling point as the calcium nitrate solution.

Key Concepts

Calcium Nitrate

Calcium nitrate, known chemically as \( ext{Ca(NO}_3 ext{)}_2\), is a soluble inorganic salt. When dissolved in water, calcium nitrate fully dissociates into calcium ions \( ext{Ca}^{2+}\) and nitrate ions \( ext{NO}_3^-\). This dissociation is crucial for understanding colligative properties, like boiling point elevation, which depend on the number of particles in a solution rather than the type.

• Formula: \( ext{Ca(NO}_3 ext{)}_2\)
• Molar Mass: 164.1 g/mol
• Ionization: It dissociates completely into three ions in water (1 calcium ion and 2 nitrate ions)

In the context of boiling point elevation, the presence of more ions from calcium nitrate contributes to a greater increase in the boiling point of the solution.

Glucose Solution

Glucose, represented by the formula \( ext{C}_6 ext{H}_{12} ext{O}_6\), is a simple sugar and an important source of energy for cells. Unlike ionic compounds such as calcium nitrate, glucose does not dissociate in solution, meaning it remains as whole molecules when dissolved.

• Formula: \( ext{C}_6 ext{H}_{12} ext{O}_6\)
• Molar Mass: 180.16 g/mol
• Dissolution: Glucose remains as a single molecule in water

When comparing with ionic compounds, it’s important to note that each glucose molecule counts as a single particle. Hence, in boiling point elevation calculations, glucose has a van't Hoff factor of 1, indicating no dissociation. This affects how its molality contributes to changes in boiling points, making it necessary to adjust calculations accordingly.

Molality

Molality is a measure of the concentration of a solute in a solution, specifically defined as the number of moles of solute per kilogram of solvent. It is different from molarity, which is moles per liter of solution, because molality is concerned solely with the mass of the solvent.The formula for molality (\(m\)) is:\[ m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \]Molality is beneficial in calculations involving colligative properties like boiling point elevation because it doesn’t change with temperature, as mass remains constant unlike volume.In this problem, we calculated the molality of calcium nitrate to predict its effect on boiling point elevation. Similarly, we adjusted this to find the needed molality for the glucose solution parallel to this effect, emphasizing the difference and independence of molality from temperature and volume changes.

Van't Hoff Factor

The van't Hoff factor (\(i\)) accounts for the number of particles a solute yields in solution. It plays an essential role in calculating colligative properties, such as boiling point elevation and freezing point depression.For non-electrolytes like glucose, which do not dissociate, the van’t Hoff factor is 1, signifying that each molecule counts as a single particle.For ionic compounds like calcium nitrate, which dissociate fully into multiple ions, the van't Hoff factor equals the total number of resultant particles. For example, the van't Hoff factor for \( ext{Ca(NO}_3 ext{)}_2\) is 3, as it produces one calcium ion and two nitrate ions upon dissolution.Understanding the van’t Hoff factor is crucial for accurately determining how a solute will affect the physical properties of a solution, particularly in quantifying the effects on boiling point through the formula:\[ \Delta T = K_b \times m \times i \]Here, \(\Delta T\) represents the boiling point elevation, \(K_b\) is the boiling point elevation constant, \(m\) is molality, and \(i\) is the van’t Hoff factor.