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 solution was diluted 5 times.

Step 1: Understanding the Osmotic Pressure Formula

Osmotic pressure can be calculated using the formula: \( \pi = CRT \), where \( \pi \) is the osmotic pressure, \( C \) is the concentration, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

Step 2: Converting Temperatures to Kelvin

First, convert the given temperatures from Celsius to Kelvin: \(10^{\circ} \mathrm{C} = 283 \mathrm{~K} \) and \(25^{\circ} \mathrm{C} = 298 \mathrm{~K} \).

Step 3: Setting Initial Conditions

At \(10^{\circ} \mathrm{C}\), the initial osmotic pressure \( \pi_1 = 500 \mathrm{~mmHg} \). The initial temperature \( T_1 = 283 \mathrm{~K} \).

Step 4: Setting Final Conditions

At \(25^{\circ} \mathrm{C}\), the final osmotic pressure \( \pi_2 = 105.3 \mathrm{~mmHg} \) and the final temperature \( T_2 = 298 \mathrm{~K} \).

Step 5: Finding the Initial Concentration

Using \( \pi_1 = C_1 \cdot R \cdot T_1 \), we find \( C_1 = \frac{\pi_1}{R \cdot T_1} \).

Step 6: Finding the Final Concentration

Similarly, using \( \pi_2 = C_2 \cdot R \cdot T_2 \), we find \( C_2 = \frac{\pi_2}{R \cdot T_2} \).

Step 7: Ratio of Initial to Final Concentration

Since dilution changes the concentration, the extent of dilution is given by the initial concentration divided by the final concentration: \( \frac{C_1}{C_2} = \frac{\pi_1 \cdot T_2}{\pi_2 \cdot T_1} \).

Step 8: Calculating Extent of Dilution

Substitute the values: \( \frac{500 \cdot 298}{105.3 \cdot 283} = 5 \).

Step 9: Conclusion

The osmotic pressure data shows that the solution was diluted by a factor of 5.

Key Concepts

Dilution

Dilution involves adding more solvent to a solution which reduces the concentration of the solute. This is a key concept in chemistry when adjusting the strength of solutions. In dilution, the amount of solute remains constant, but the total volume of the solution increases. Thus, the concentration of the solution decreases.
For example, when you add water to a salt solution, the salt is distributed over a larger volume, decreasing its concentration.

• Initial concentration is noted as \( C_1 \).
• After dilution, the concentration is noted as \( C_2 \).

The factor by which the concentration changes can be found from the comparison of the initial and final states of the solution. In the context of osmotic pressure, dilation affects the concentration directly and thereby changes the pressure of the solution.

Concentration

Concentration refers to the amount of solute present in a given quantity of solvent or solution. It's often expressed in terms of molarity, which is moles of solute per liter of solution. This is crucial when discussing solutions like urea because concentration influences properties like osmotic pressure.

Higher concentrations mean more solute particles are present in a solution, which can lead to higher osmotic pressures. This is because osmotic pressure depends on the number of solute particles, as described by the formula \( \pi = C \cdot R \cdot T \). Here, \( C \) stands for concentration. By determining the initial and final concentrations of a solution, you can understand how much a solution has been diluted during an experiment.

Temperature Conversion

When working with scientific equations involving temperature, it’s often necessary to convert Celsius to Kelvin. This is because Kelvin is the SI unit for temperature and is widely used in scientific formulas and calculations.

To convert from Celsius to Kelvin, you simply add 273.15 to the Celsius temperature.

• For example, \( 10^{\circ} \mathrm{C} \) becomes \( 10 + 273.15 = 283 \mathrm{~K} \).
• Similarly, \( 25^{\circ} \mathrm{C} \) becomes \( 25 + 273.15 = 298 \mathrm{~K} \).

Using Kelvin helps ensure the temperatures used in calculations, such as those for osmotic pressure, maintain consistency and accuracy across similar scientific contexts.

Ideal Gas Constant

The ideal gas constant, denoted as \( R \), is a key figure in equations relating to gases, including the osmotic pressure formula. It provides a link between different units and measurement systems when dealing with equations like \( \pi = C \cdot R \cdot T \).

The value of the ideal gas constant is approximately \( 0.0821 \, ext{L atm} \, ext{mol}^{-1} \, ext{K}^{-1} \).

• This constant allows scientists to connect pressure, volume, temperature, and concentration in predictions and experimental setups.
• When solving osmotic pressure problems, using the correct value of \( R \) is crucial for accurate results.

The ideal gas constant ensures that calculations involving gases and their transformations are unified and coherent.