Problem 117 - A Complete Resource Book in Chemistry for JEE Main

Question

The osmotic pressure of urea solution is \(500 \mathrm{~mm}\) of \(\mathrm{Hg}\) at \(10^{\circ} \mathrm{C}\). If the solution is diluted and temperature is raised to \(25^{\circ} \mathrm{C}\), the osmotic pressure decreases to \(105.3 \mathrm{~mm}\) of \(\mathrm{Hg}\), what is the extent of dilution? (a) 10 times (b) \(2.5\) times (c) 5 times (d) \(7.5\) times

Step-by-Step Solution

Answer

The extent of dilution is approximately 5 times, so the answer is (c) 5 times.

Step 1: Understanding Osmotic Pressure Formula

Osmotic pressure (60:) at constant temperature is given by the formula \(60 = iCRT\), where \(i\) is the van't Hoff factor (for urea, \(i = 1\)), \(C\) is the concentration in moles per liter, \(R\) is the gas constant \(0.0821\, \text{L atm mol}^{-1}\, \text{K}^{-1}\), and \(T\) is the temperature in Kelvin. As temperature changes, osmotic pressure should be corrected for both the change in concentration due to dilution and temperature.

Step 2: Calculate Dilution Factor Formula

We know from Van’t Hoff equation, \(\frac{P_1}{P_2} = \frac{C_1}{C_2}\), because temperature and gas constant \(R\) are constant during dilution process. Here, \(P_1 = 500\, \text{mm Hg}\) and \(P_2 = 105.3\, \text{mm Hg}\). Thus, \(\frac{C_1}{C_2} = \frac{500}{105.3}\).

Step 3: Solving for Extent of Dilution

Calculate the dilution factor which is \(\frac{500}{105.3} = 4.75\), indicating the dilution factor when going from first condition to second condition.

Key Concepts

Understanding van't Hoff Factor

When we discuss solutions, especially with regards to osmotic pressure, the term 'van't Hoff factor' often comes up. The van't Hoff factor, denoted as \( i \), is crucial because it accounts for the effect of solute particles on osmotic pressure.
It represents the number of particles the solute splits into or generates in solution. For most non-electrolytes, like urea, this factor is usually 1 because urea molecules dissolve in water without breaking into ions or extra particles.
In equations for osmotic pressure, \( i \) helps adjust the concentration to reflect the true number of particles present. For example, in our problem with urea solutions, knowing that \( i = 1 \) simplifies calculations and makes the formula more straightforward to apply.

Exploring Dilution Factor

The dilution factor is an important concept when discussing changes in solution concentration. It's essentially a measure of how much a solution has been diluted or how the concentration of solute particles has been decreased due to adding more solvent.
In the context of osmotic pressure, when a solution is diluted, this factor tells us by how many times the concentration of the solution has decreased. For instance, if a solution's pressure drops due to dilution, the dilution factor can be calculated by comparing the initial and final osmotic pressures.
Our example exercise involved calculating the dilution factor based on osmotic pressures before and after dilution. By using the formula \( \frac{P_1}{P_2} = \frac{C_1}{C_2} \), students find that the solution was diluted by approximately 4.75 times.

Urea Solution Characteristics

Understanding urea solutions is key in many chemical and biological applications. Urea is a non-electrolyte, meaning when it dissolves in water, it doesn’t break down into ions. This is why its van’t Hoff factor is 1, simplifying many osmotic pressure calculations.
In water, urea is soluble due to its polar nature, and it influences properties like osmotic pressure linearly with its concentration. This behavior makes it ideal for studying basic principles of chemical solutions and colligative properties, where changes depend on the number of particles in solution rather than their nature or type.
For students, experimenting with urea solutions offers a practical way to grasp how substances interact with solvents, enhancing understanding of solution behavior in broader scientific contexts.

Temperature Effects on Osmotic Pressure

Temperature plays a vital role in determining the osmotic pressure of a solution. According to the formula \( \pi = iCRT \), we see that osmotic pressure is directly proportional to temperature, meaning as temperature increases, so does the pressure, assuming the concentration is constant.
In our example, when the temperature was raised, the new conditions required account for the temperature in Kelvin to ensure accurate calculations. It's essential to note that real-life applications often involve adjusting for both temperature and dilution factors to anticipate and control changes in osmotic pressure effectively.
For example, increasing the temperature from \( 10^{\circ}\mathrm{C} \) to \( 25^{\circ}\mathrm{C} \) meant accounting for the temperature changes to correctly determine how these affect the final osmotic pressure, thus showcasing how intertwined temperature and solution properties are.