We have four aqueous solutions labelled as:

A. \(0.1\, \ M\, \ NaCl\)

B. \(0.01\, \ M\, \ NaCl\)

C. \(0.01\, \ M\, \ BaCl_2\)

D. \(0.01\, \ M\, \ Sucrose\)

Choose the correct increasing order of van't Hoff factor from the options  given below:

\(i_D < i_A = i_B < i_C\)

The correct answer is option 3. \(i_D < i_A = i_B < i_C\).

To determine the increasing order of the van't Hoff factor (\(i\)) for each solution, we need to understand what the van't Hoff factor represents and how it applies to each solution.

van't Hoff Factor (\(i\))

The van't Hoff factor (\(i\)) is a measure of the number of particles into which a solute dissociates in solution. It can be defined as:

For a non-electrolyte (like sucrose), \(i\) is 1, as it does not dissociate in solution.

For an electrolyte, \(i\) depends on the number of ions produced from the dissociation of the electrolyte in solution.

Analysis of Each Solution

1. Solution A: \(0.1\, M\, \text{NaCl}\)

NaCl is an electrolyte and dissociates into 2 ions: Na\(^+\) and Cl\(^-\).

van't Hoff factor \(i_A = 2\).

2. Solution B: \(0.01\, M\, \text{NaCl}\)

Similar to Solution A, NaCl dissociates into 2 ions.

van't Hoff factor \(i_B = 2\).

3. Solution C: \(0.01\, M\, \text{BaCl}_2\)

BaCl\(_2\) dissociates into 3 ions: Ba\(^{2+}\) and 2 Cl\(^-\).

van't Hoff factor \(i_C = 3\).

4. Solution D: \(0.01\, M\, \text{Sucrose}\)

Sucrose is a non-electrolyte and does not dissociate in solution.

van't Hoff factor \(i_D = 1\).

Increasing Order of Van't Hoff Factors

To find the correct order of \(i\), we arrange the values from smallest to largest:

Solution D (Sucrose): \(i_D = 1\)

Solution A (0.1 M NaCl): \(i_A = 2\)

Solution B (0.01 M NaCl): \(i_B = 2\)

Solution C (0.01 M BaCl\(_2\)): \(i_C = 3\)

Note that the concentration affects the magnitude of colligative properties but does not change the van't Hoff factor. Hence, the van't Hoff factor only depends on the type of solute and its dissociation, not its concentration.

The increasing order of van't Hoff factors is: \(i_D < i_A = i_B < i_C\)