Question

The molal boiling point elevation constant of water is \(0.513^{\circ} \mathrm{C} \mathrm{kg} \mathrm{mol}^{-1}\). When \(0.1\) mole of sugar is dissolved \(200 \mathrm{~g}\) of water, the solution boils under a pressure of 1 atm at (a) \(100.513^{\circ} \mathrm{C}\) (b) \(102.565^{\circ} \mathrm{C}\) (c) \(100.256^{\circ} \mathrm{C}\) (d) \(101.025^{\circ} \mathrm{C}\)

Short answer

The solution boils at \(\100.256^{\text{\textdegree}} \mathrm{C}\) under a pressure of 1 atm, so the correct answer is (c) \(\100.256^{\text{\textdegree}} \mathrm{C}\).

Step-by-step solution

Understand the concept

The boiling point elevation is a colligative property, meaning it depends on the amount of solute particles in a solution regardless of their identity. The elevation \(\Delta T_b\) in the boiling point of a solvent can be calculated using the formula \(\Delta T_b = i \cdot K_b \cdot m\), where \(\Delta T_b\) is the boiling point elevation, \(\K_b\) is the molal boiling point elevation constant, \(\m\) is the molality of the solution, and \(\i\) is the van't Hoff factor which represents the number of particles the solute splits into or forms in solution. For sugar, \(\i\) is 1 since it does not dissociate in solution.

Calculate molality of the solution

Molality \(\m\) is defined as moles of solute per kilogram of solvent. We have \(\0.1\) mole of sugar dissolved in \(\200 \mathrm{g}\) or \(\0.2 \mathrm{kg}\) of water, so the molality is \(\m = \frac{\text{moles of solute}}{\text{kilograms of solvent}} = \frac{0.1}{0.2} = 0.5 \, \mathrm{mol} \cdot \mathrm{kg}^{-1}\).

Calculate boiling point elevation

Use the formula \(\Delta T_b = i \cdot \K_b \cdot \m\) to calculate the boiling point elevation. Since sugar does not dissociate, \(\i = 1\). Therefore, \(\Delta T_b = 1 \cdot 0.513^{\text{\textdegree}} \mathrm{C} \cdot \mathrm{mol}^{-1} \cdot \text{kg} \cdot 0.5 \, \mathrm{mol} \cdot \mathrm{kg}^{-1}\) which equals \(\0.2565^{\text{\textdegree}} \mathrm{C}\).

Determine new boiling point

Add the boiling point elevation to the normal boiling point of water, which is \(\100^{\text{\textdegree}} \mathrm{C}\). The new boiling point is \(\100^{\text{\textdegree}} \mathrm{C} + 0.2565^{\text{\textdegree}} \mathrm{C} = 100.2565^{\text{\textdegree}} \mathrm{C}\). Since we need to choose the closest answer, option (c) \(\100.256^{\text{\textdegree}} \mathrm{C}\) is the correct boiling point.

Key concepts

Colligative Properties

Colligative properties are fascinating characteristics of solutions that depend solely on the ratio of solute particles to solvent particles, regardless of the solute's chemical identity. Imagine adding salt to water; the specific type of salt doesn't matter as much as how much you've added. This is because colligative properties are all about the count of particles, not what they are.

Boiling point elevation is a classic example of a colligative property. When a non-volatile solute, like sugar, is dissolved in a solvent, such as water, it raises the boiling point of the solvent. This happens because the solute particles hinder the solvent molecules from entering the vapor phase, requiring more energy (and thus a higher temperature) to boil. Other colligative properties include freezing point depression, where adding a solute lowers the freezing point, and osmotic pressure, which is the force required to prevent the solvent from passing through a semipermeable membrane to dilute the solute.

Molal Boiling Point Elevation Constant

The molal boiling point elevation constant, often denoted as \(K_b\), is a key player in the equation for boiling point elevation. It is a property unique to each solvent and reflects how sensitive the solvent's boiling point is to the addition of a solute. For water, this constant is \(0.513^{\text{\textdegree}}C \text{kg mol}^{-1}\), which means for every one molal (one mole per kilogram of solvent) increase in the concentration of a solute, the boiling point of water goes up by about half a degree Celsius.

In the realm of boiling point elevation, \(K_b\) represents the temperature increase per molal concentration of a solute in a solution. If you know the \(K_b\) value for your solvent and the molality of your solute in that solvent, you've got a direct route to calculate how much higher your solution will boil compared to the pure solvent.

Van't Hoff Factor

The van't Hoff factor, symbolized as \(i\), punches in an extra layer of consideration when you're dealing with solutions. Not all solutes are created equal—in terms of their effect on a solution's boiling point. The van't Hoff factor quantifies how many particles a compound will form when dissolved. For instance, table salt (NaCl) dissociates into two particles (Na+ and Cl-), giving it an \(i\) value of 2.

Role in Boiling Point Elevation

When calculating boiling point elevation, \(i\) becomes vital. It adjusts the boiling point elevation formula to account for the actual number of particles that a solute creates in solution. A solute that doesn't dissociate like sugar, where \(i = 1\), will have less impact than an ionic compound like magnesium chloride (MgCl2), which would typically have an \(i\) of 3. Thus, the van't Hoff factor lets us tweak our calculation to the specific behavior of the solute, making our predictions much more accurate.

Molality

When we talk about molality, we're focused on the concentration of a solution. Uniquely, molality measures moles of solute per kilogram of solvent—not volume, like molarity does. Why the distinction? Well, molality has the upside of being temperature-independent because the mass doesn't change with temperature.

Calculating molality is straightforward: \(molality (m) = \frac{moles \text{ of solute}}{kilograms \text{ of solvent}}\). In our problem, we had 0.1 mole of sugar and 0.2 kilograms of water, resulting in a molality of 0.5 mol/kg. This figure is pivotal in our boiling point elevation equation because it directly affects how much the boiling point will go up. The precision of measuring molality makes it a reliable and consistent way to calculate colligative properties like boiling point elevation and freezing point depression.