Question

Some nonelectrolyte solute (molar mass \(=142 \mathrm{~g} / \mathrm{mol}\) ) was dissolved in \(150 . \mathrm{mL}\) of a solvent \(\left(\right.\) density \(\left.=0.879 \mathrm{~g} / \mathrm{cm}^{3}\right)\). The elevated boiling point of the solution was \(355.4 \mathrm{~K}\). What mass of solute was dissolved in the solvent? For the solvent, the enthalpy of vaporization is \(33.90 \mathrm{~kJ} / \mathrm{mol}\), the entropy of vaporization is \(95.95 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol}\), and the boiling-point elevation constant is \(2.5 \mathrm{~K} \cdot \mathrm{kg} / \mathrm{mol}\).

Short answer

The mass of the solute dissolved in the solvent is approximately 16.46 g.

Step-by-step solution

Calculate the change in boiling point

First, we need to determine the change in boiling point, which is the difference between the elevated boiling point of the solution and the normal boiling point of the solvent. For the solvent, the enthalpy and entropy of vaporization can be used to calculate the normal boiling point. Normal boiling point, T_b = ΔHvap / ΔSvap Plug in the given values: \(T_b = \frac{33.90 \times 10^{3}\,\mathrm{J/mol}}{95.95\,\mathrm{J/(K\cdot mol)}}\) \(T_b = 353.2\,\mathrm{K}\) Now, we need to calculate the change in boiling point, ΔT: ΔT = elevated boiling point - normal boiling point ΔT = 355.4 K - 353.2 K ΔT = 2.2 K

Calculate molality using boiling-point elevation constant

Next, we need to use the boiling-point elevation constant, K_b, to calculate the molality (m) of the solution. The formula for boiling-point elevation is: ΔT = K_b * m We are given that K_b = 2.5 K⋅kg/mol. Rearrange the formula to solve for molality (m): m = ΔT / K_b Plug in the given values: m = 2.2 K / (2.5 K⋅kg/mol) m = 0.88 mol/kg

Calculate the mass of the solvent

We need to determine the mass of the solvent in order to find the mass of solute. We are given the volume and density of the solvent. The formula to calculate the mass of the solvent is: mass (solvent) = volume * density Plug in the given values: mass (solvent) = 150 mL * 0.879 g/cm³ mass (solvent) = 131.85 g Now, we need to convert the mass of the solvent to kilograms: mass (solvent) = 131.85 g * (1 kg / 1000 g) = 0.13185 kg

Calculate the moles of solute in the solution

Now that we have the molality and the mass of solvent, we can calculate the moles of solute, using the formula: moles of solute = molality * mass (solvent) Plug in our values: moles of solute = 0.88 mol/kg * 0.13185 kg moles of solute = 0.11596 mol

Calculate the mass of the solute

Finally, we can determine the mass of the solute by multiplying the moles of solute by its molar mass, as given: mass (solute) = moles of solute * molar mass Plug in the given values: mass (solute) = 0.11596 mol * 142 g/mol mass (solute) = 16.46 g Therefore, the mass of the solute dissolved in the solvent is approximately 16.46 g.

Key concepts

Boiling Point Elevation

Boiling point elevation is a fascinating colligative property that refers to the phenomenon where the boiling point of a liquid (solvent) increases when a solute is dissolved in it. This occurs because the solute particles interfere with the formation of vapor bubbles within the liquid, effectively requiring a higher temperature to allow the solvent to reach the gaseous state.

In practical terms, boiling point elevation can be quantified using the equation \(\Delta T_b = K_b \cdot m\), where \(\Delta T_b\) is the change in boiling point of the solution, \(K_b\) is the boiling-point elevation constant (also known as the ebullioscopic constant), and \(m\) is the molality of the solution, which indicates the moles of solute per kilogram of solvent.

For students aiming to understand this concept, it's crucial to grasp that the magnitude of the boiling point elevation is directly proportional to the molality of the solution. Thus, the more solute particles present, the greater the boiling point elevation will be.

Molality Calculation

The molality of a solution is a measure of its concentration, specifically defining the number of moles of solute per kilogram of solvent. Unlike molarity, which measures moles per volume of solution, molality is exclusively concerned with the mass of the solvent, making it temperature-independent (since mass does not change with temperature).

To calculate molality, you can use the formula \( \m = \frac{\text{moles of solute}}{\text{mass of solvent in kilograms}} \). It's key for students to understand that correct units are crucial when performing such calculations. Remember to convert mass to kilograms and volume to liters when necessary, as standard scientific calculations require consistent units.

This concept is particularly relevant when working with colligative properties because these properties depend on the quantity, not the identity, of solute particles. By using molality, we ensure that our calculations reflect this principles.

Vaporization Enthalpy and Entropy

Vaporization enthalpy (\(\Delta H_{vap}\)) and entropy (\(\Delta S_{vap}\)) are crucial thermodynamic properties that describe the energy required to transform a liquid into a gas, and the disorder change associated with this process, respectively.

Enthalpy of vaporization represents the heat needed to vaporize a molecule from its liquid phase without a change in temperature. It is usually expressed in joules per mole (J/mol). Higher values indicate a stronger intermolecular force keeping the molecules together in the liquid state.

The entropy of vaporization, on the other hand, represents the increase in disorder that occurs when a liquid becomes a gas. This value is also usually expressed in Joules per mole per Kelvin (J/(mol·K)).

Together, these values can give the normal boiling point of a liquid using the relation \( T_b = \frac{\Delta H_{vap}}{\Delta S_{vap}} \), where \( T_b \) is the boiling point. This equation represents the balance between energy input (enthalpy) and increased disorder (entropy) at the phase transition. Students should pay attention to the interplay between these two properties as it underscores many phenomena in physical chemistry.

Non-Electrolyte Solutions

Non-electrolyte solutions are those which contain solutes that, when dissolved in a solvent, do not dissociate into ions. These substances tend to be organic compounds, such as sugars, that dissolve in water but do not affect the water's ability to conduct electricity.

In the context of colligative properties, non-electrolyte solutions are simpler to analyze than electrolyte solutions because the solute does not break apart into multiple particles. Therefore, the number of particles in solution can be directly calculated from the amount of solute added—no need to account for the dissociation into ions as in electrolyte solutions.

It is essential for students to differentiate between electrolyte and non-electrolyte solutions since the colligative properties, such as boiling point elevation or freezing point depression, will differ depending on the type of solute and its behavior in the solvent. Recognizing this distinction helps in predicting how the solution will react to changes in temperature and concentration.