Problem 134 - A Complete Resource Book in Chemistry for JEE Main

Question

A protein has been isolated as a salt with the formula \(\mathrm{Na}_{20} \mathrm{P}\) (this notation means that there are \(20 \mathrm{Na}^{+}\)ions associated with a negatively charged protein \(\mathrm{P}^{20-}\) ). The osmotic pressure of a \(10.0 \mathrm{ml}\) solution containing \(0.225 \mathrm{~g}\) of the protein is \(0.257 \mathrm{~atm}\) at \(25^{\circ} \mathrm{C}\). The actual molar mass of the protein from these data is approximately. (a) \(2.6 \times 10^{5} \mathrm{~g} \mathrm{~mol}^{-1}\) (b) \(4.5 \times 10^{4} \mathrm{~g} \mathrm{~mol}^{-1}\) (c) \(4.5 \times 10^{6} \mathrm{~g} \mathrm{~mol}^{-1}\) (d) \(2.3 \times 10^{3} \mathrm{~g} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

The actual molar mass of the protein is approximately \(4.5 \times 10^4 \mathrm{g/mol}\). This corresponds to option (b).

Step 1: Understanding Osmotic Pressure Formula

The osmotic pressure (π) can be expressed using the formula \( \pi = iMRT \), where \( i \) is the van't Hoff factor, \( M \) is the molarity of the solution, \( R \) is the ideal gas constant (0.0821 L atm/mol K), and \( T \) is the temperature in Kelvin. We need to determine the molar mass of the protein using this information.

Step 2: Temperature Conversion

Convert the temperature from Celsius to Kelvin. \( T = 25 + 273.15 = 298.15 \) K.

Step 3: Determine Van't Hoff Factor

For the protein salt \( \mathrm{Na}_{20}\mathrm{P} \), the van't Hoff factor \( i \) is 21 (20 \( \mathrm{Na}^{+} \) ions and 1 \( \mathrm{P}^{20-} \) ion).

Step 4: Calculate Molarity

First, convert the volume from mL to L: \( 10.0 \text{ mL} = 0.0100 \text{ L} \). Calculate the molarity \( M \) using the rearranged osmotic pressure formula: \( \pi = iMRT \). Thus, \( M = \frac{\pi}{iRT} = \frac{0.257}{21 \times 0.0821 \times 298.15} \approx 0.000532 \text{ mol/L} \).

Step 5: Calculate Moles of Protein

Use the molarity to find moles of protein in the solution: \( \text{Moles of Protein} = M \times \text{Volume in L} = 0.000532 \text{ mol/L} \times 0.0100 \text{ L} = 0.00000532 \text{ mol} \).

Step 6: Calculate Molar Mass

Use the mass of the protein and the number of moles to find the molar mass \( M_m \): \( M_m = \frac{\text{Mass of protein}}{\text{Moles of protein}} = \frac{0.225}{0.00000532} \approx 42368 \text{ g/mol} \).

Step 7: Select Closest Answer

Compare the calculated molar mass to the given options. The closest answer is (b) \( 4.5 \times 10^4 \text{ g/mol} \).

Key Concepts

Van't Hoff Factor

The Van't Hoff factor, typically represented by the symbol \(i\), extends our understanding of how solutes affect colligative properties like osmotic pressure. Essentially, the Van't Hoff factor accounts for the degree of ionization or dissociation that occurs when a solute dissolves in a solvent. This influences properties which depend on the number of solute particles present rather than their identities.
The formula \(\pi = iMRT\) includes the Van't Hoff factor to adjust the effect solute particles have in an ideal solution. In our protein salt \(\mathrm{Na}_{20}\mathrm{P}\), \(i\) is 21, which reflects the total count of ions: 20 \(\mathrm{Na}^+\) cations and one single \(\mathrm{P}^{20-}\) anion. This significantly increases the number of effective particles affecting osmotic pressure, compared to non-dissociating solutes.
Picturing the Van't Hoff factor as a "multiplier" can help remember its role: simply quantify how many separate particles a solute yields upon dissolving. It is this factor that amplifies the impact of colligative properties in ionic compounds compared to molecular solutes.

Molarity

Molarity, abbreviated as \(M\), is a critical concept signifying the concentration of a solution. This is measured in moles of solute per liter of solution, generally written as mol/L. Knowing the molarity allows us to predict and calculate various properties of a solution, like osmotic pressure.
In our scenario, the molarity is calculated through the rearranged formula for osmotic pressure \(\pi = iMRT\). Solving for molarity, we find \(M = \frac{\pi}{iRT}\). Given the values: osmotic pressure (\(\pi = 0.257 \; \text{atm}\)), Van't Hoff factor (\(i = 21\)), ideal gas constant \(R = 0.0821 \; \text{L atm/mol K}\), and temperature \(T = 298.15 \; \text{K}\), this evaluates to an approximate molarity of 0.000532 mol/L.
This step demonstrates how molarity connects the concentration of our protein solution with its osmotic behavior.

Molar Mass Calculation

Determining molar mass is a fundamental task in chemistry, crucial for connecting the mass of substances with the number of moles they represent. To calculate it, one must divide the mass of the solute by the number of moles in the solution.
In our exercise, using the determined molarity (0.000532 mol/L) allows us to find the moles present in 10.0 mL (or 0.0100 L) of solution: \(\text{Moles} = M \times \text{Volume in L} = 0.000532 \times 0.0100 = 0.00000532 \; \text{mol}\).
The mass of protein given is 0.225 g. Applying the formula for molar mass \(M_m = \frac{\text{Mass of protein}}{\text{Moles of protein}}\), we determine it to be approximately 42368 g/mol. Thus, from this calculation, the correct choice among the given options is 4.5 \(\times 10^4 \; \text{g/mol}\). Molar mass connects directly to practical uses like determining chemical quantities for reactions and solutions.