Problem 100 - Chemistry and Chemical Reactivity

Question

Which substance would have the greater influence on the vapor pressure of water when added to \(1000 .\) g of the liquid: \(10.0 \mathrm{g}\) of sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)\) or \(10.0 \mathrm{g}\) of ethylene glycol \(\left[\mathrm{HOCH}_{2} \mathrm{CH}_{2} \mathrm{OH}\right]\) ?

Step-by-Step Solution

Answer

Ethylene glycol has a greater influence on water's vapor pressure.

Step 1: Understand the Concept of Vapor Pressure

The vapor pressure of a liquid is influenced by the presence of a solute, which can lower the vapor pressure by reducing the number of molecules of the solvent (in this case, water) that can escape into the vapor phase. This is due to the colligative property known as vapor pressure lowering.

Step 2: Define the Key Formula

The lowering of vapor pressure is related to the number of solute particles in solution. This is expressed by Raoult's Law, where the relative lowering of vapor pressure is proportional to the mole fraction of the solute. The formula is:\[\Delta P = X_s P^0\]where \( \Delta P \) is the change in vapor pressure, \( X_s \) is the mole fraction of the solute, and \( P^0 \) is the vapor pressure of the pure solvent.

Step 3: Calculate Moles of Each Solute

Calculate the moles of sucrose and ethylene glycol using their molar masses. 1. Molar mass of sucrose \( C_{12}H_{22}O_{11} = 342.3\, \text{g/mol} \)2. Moles of sucrose: \[\text{Moles of sucrose} = \frac{10.0\, \text{g}}{342.3\, \text{g/mol}} \approx 0.0292\, \text{mol}\]3. Molar mass of ethylene glycol \( HOCH_2CH_2OH = 62.07\, \text{g/mol} \)4. Moles of ethylene glycol:\[\text{Moles of ethylene glycol} = \frac{10.0\, \text{g}}{62.07\, \text{g/mol}} \approx 0.1611\, \text{mol}\]

Step 4: Determine Mole Fraction of Each Solute

Assume the mass of water \( m_{water} = 1000\, \text{g} \) with molar mass \( M_{water} = 18.015\, \text{g/mol} \), thus moles of water are:\[\text{Moles of water} = \frac{1000\, \text{g}}{18.015\, \text{g/mol}} \approx 55.5\, \text{mol}\] Calculate the mole fractions:- Mole fraction of sucrose \[X_{sucrose} = \frac{0.0292}{0.0292 + 55.5} \approx 0.000526\]- Mole fraction of ethylene glycol \[X_{ethylene\ glycol} = \frac{0.1611}{0.1611 + 55.5} \approx 0.00289\]

Step 5: Compare the Mole Fractions

The substance with the greater mole fraction has a greater effect on vapor pressure lowering. Ethylene glycol has a greater mole fraction (0.00289) compared to sucrose (0.000526). Therefore, ethylene glycol exerts a greater influence on the vapor pressure of water.

Key Concepts

Colligative Properties

Colligative properties are fascinating because they depend solely on the number of solute particles in a solution, not their identity. This means that characteristics such as vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure are directly related to the amount of solute present. These properties are called "colligative" because they "collectively" focus on the concentration of solute particles.

Let's break it down; when you dissolve a solute in a solvent like water, several properties of the solvent change. For instance, adding sugar or salt to water affects the vapor pressure of the water due to the addition of more particles in the solution, which disrupts the solvent's natural behavior.

One of the reasons that colligative properties are useful is because they allow chemists to determine the amount of solute in a solution without knowing exactly what the solute is. This can be particularly valuable in experimental chemistry and practical applications where the identity of the solute might not be as crucial as how much is present.

Raoult's Law

Raoult's Law provides a way to predict how a solute will influence the vapor pressure of a solvent. It states that the vapor pressure of the solvent above a solution is equal to the mole fraction of the solvent times the vapor pressure of the pure solvent.

The formula for Raoult's Law is given by the equation:

• \( P = X_{solvent} \times P^0 \)

Where \( P \) is the vapor pressure of the solution, \( X_{solvent} \) is the mole fraction of the solvent, and \( P^0 \) is the vapor pressure of the pure solvent.

When you add a solute to a solvent, the mole fraction of the solvent decreases. Consequently, according to Raoult's Law, the vapor pressure of the solvent is also reduced. This is known as vapor pressure lowering.

One of the main points of using Raoult's Law is to calculate the change in vapor pressure when a non-volatile solute is added to a volatile solvent. It helps us understand how the presence of a solute influences molecular interactions in the solution, thereby affecting various colligative properties.

Mole Fraction

Mole fraction is a useful concept when dealing with solutions. It indicates the ratio of moles of a particular substance to the total moles of all substances present. When considering Raoult's Law, the mole fraction allows you to express the proportion of the solvent or solute in the solution.

Mathematically, the mole fraction of a component in a solution is given by:

• \( X_i = \frac{n_i}{n_{total}} \)

Where \( X_i \) is the mole fraction of substance \( i \), \( n_i \) is the moles of substance \( i \), and \( n_{total} \) is the total moles of all substances in the solution.

In the context of the exercise, we looked at the mole fraction of sucrose and ethylene glycol, separately in water. Even though equal masses of solute were added, their different molar masses resulted in different mole fractions. Ethylene glycol had a higher mole fraction due to its smaller molar mass, leading to it exerting a greater influence on the vapor pressure of water.

Understanding mole fractions is crucial because it provides a baseline for calculation of colligative properties and allows for comparison of how different solutes affect the overall solution.