Problem 68 - A Complete Resource Book in Chemistry for JEE Main

Question

If at certain temperature the vapour pressure of pure water is \(25 \mathrm{~mm}\) of \(\mathrm{Hg}\) and that of a very dilute aqueous urea solution is \(24.5 \mathrm{~mm}\) of \(\mathrm{Hg}\), the molality of the solution is (a) \(0.02\) (b) \(1.2\) (c) \(1.11\) (d) \(0.08\)

Step-by-Step Solution

Answer

(c) 1.11

Step 1: Identify the Known Values

The problem gives us the vapor pressures of pure water and the urea solution. The vapor pressure of pure water \(P^0_{\text{water}}\) is 25 mm Hg, and the vapor pressure of the urea solution \(P_{\text{solution}}\) is 24.5 mm Hg.

Step 2: Use Raoult's Law to Find Mole Fraction

Raoult's Law for a solution states that \(P_{\text{solution}} = \text{X}_{\text{water}} \times P^0_{\text{water}}\), where \(\text{X}_{\text{water}}\) is the mole fraction of water. We can rearrange this to find \(\text{X}_{\text{water}}\): \[ \text{X}_{\text{water}} = \frac{P_{\text{solution}}}{P^0_{\text{water}}} = \frac{24.5}{25} \approx 0.98 \]

Step 3: Calculate the Mole Fraction of Solute (Urea)

Since there are only two components, water and urea, their mole fractions must add up to 1. Thus, the mole fraction of urea \(\text{X}_{\text{urea}} = 1 - \text{X}_{\text{water}} = 1 - 0.98 = 0.02\).

Step 4: Relate Mole Fraction to Molality

Molality (\(m\)) is defined as moles of solute per kilogram of solvent. To use the mole fraction to find molality, consider 1 mole of the solution. At \(\text{X}_{\text{urea}} = 0.02\), the moles of urea are \(0.02\) and moles of water are \(0.98\). Calculate water's mass: 1 mole of water is \(18 \text{g}\), thus \(0.98 \text{mol} \approx 0.98 \times 18 \text{g} = 17.64 \text{g} = 0.01764 \text{kg}\). Molality is therefore: \[ m = \frac{0.02 \text{ mol urea}}{0.01764 \text{ kg water}} \approx 1.13 \text{ mol/kg} \].

Step 5: Round the Molality to Compare with Given Options

Round 1.13 mol/kg to find the closest value corresponding to the given options, which is approximately 1.11 mol/kg.

Key Concepts

Vapor Pressure

Vapor pressure is a crucial concept in understanding how different substances behave when they are mixed. It refers to the pressure exerted by the vapor of a liquid when it is in equilibrium with its liquid state. This means that equal amounts of the substance are evaporating and condensing at the same rate.
When you introduce a solute, like urea, into a solvent, like water, the presence of this solute affects the vapor pressure of the solution. According to Raoult's Law, the vapor pressure of the solution is lower than that of the pure solvent. This is because the solute particles occupy space at the surface of the liquid and reduce the number of solvent molecules that can escape into the vapor phase.
Understanding vapor pressure helps predict how a solution will behave under different conditions, and it's especially important when dealing with solutions in closed systems or varying temperatures.

Mole Fraction

Mole fraction is a way to express the concentration of a component in a mixture, specifically a solution. It is defined as the ratio of the number of moles of a component to the total number of moles of all components in the mixture. In symbols, for component A, the mole fraction \( \text{X}_A \) is given by:

• \( \text{X}_A = \frac{n_A}{n_{\text{total}}} \)

Here, \( n_A \) represents the moles of component A and \( n_{\text{total}} \) is the total moles in the mixture.
In the context of Raoult's Law, understanding mole fraction is essential because it directly influences the vapor pressure of a solution. The mole fraction of the solvent is used to determine how the vapor pressure changes when a solute is added. For example, if \( \text{X}_{\text{water}} \) decreases when urea is dissolved in water, this change signifies a lowering of vapor pressure compared to pure water.
Mole fraction is a useful concept because it doesn't depend on temperature or pressure, making it versatile for solving various chemical problems.

Molality

Molality is another way to measure the concentration of a solution. It is defined as the number of moles of solute per kilogram of the solvent. This is particularly useful in colligative properties calculations, where temperature changes might affect other concentration measures, like molarity.
To calculate molality, follow these steps:

• Determine the number of moles of the solute.
• Find the mass of the solvent in kilograms.
• Use the formula: \( m = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \).

In the given problem, the molality is calculated to find the concentration of urea in a very dilute solution. This helps determine vital properties of the mixture, such as boiling or freezing point changes.
Molality is particularly valuable in solutions exposed to varying temperatures, as it remains constant while other concentration metrics, like molarity, can fluctuate.

Dilute Solution

A dilute solution is one where there is a relatively small amount of solute compared to the solvent. It's important to understand this concept because it affects how solutions behave, especially in terms of properties like vapor pressure and boiling point.
In the exercise, the aqueous urea solution is described as very dilute. This means that the concentration of urea is low, which is why the total change in properties, like vapor pressure, from the pure solvent is small. However, even in dilute solutions, the presence of solute can significantly affect certain colligative properties due to the behavior described by Raoult's Law.
Dilute solutions are often easier to work with in labs because their properties are typically more predictable. They are essential in industries like pharmaceuticals and food processing, where precise control over solution concentration is crucial.