The ratio aof `Delta T_(f) [ Fe(CN)_(6)]` solution ( assuming complete ionisation ) to `DeltaT_(f)` for solution of sugar of equal concentration is

Options

A. `4:1`
B. `5:2`
C. `5:1`
D. `4:15`

Explanation

To solve the problem of finding the ratio of \(\Delta T_f\) for a solution of \([Fe(CN)_6]^{4-}\) (assuming complete ionization) to \(\Delta T_f\) for a solution of sugar of equal concentration, we can follow these steps:

### Step 1: Understand the Formula for \(\Delta T_f\)
The depression in freezing point (\(\Delta T_f\)) is given by the formula:
\[ \Delta T_f = K_f \cdot m \cdot i \]
where:
- \(K_f\) is the cryoscopic constant (which is constant for a given solvent),
- \(m\) is the molality of the solution,
- \(i\) is the Van't Hoff factor, which represents the number of particles the solute dissociates into.

### Step 2: Identify the Van't Hoff Factor for \([Fe(CN)_6]^{4-}\)
For \([Fe(CN)_6]^{4-}\), it completely dissociates into ions. The dissociation can be represented as:
\[ K_4[Fe(CN)_6] \rightarrow 4K^+ + Fe(CN)_6^{4-} \]
This means that one formula unit of \([Fe(CN)_6]^{4-}\) produces 5 ions (4 potassium ions and 1 complex ion). Therefore, the Van't Hoff factor \(i\) for \([Fe(CN)_6]^{4-}\) is:
\[ i = 5 \]

### Step 3: Identify the Van't Hoff Factor for Sugar
Sugar is a non-electrolyte, meaning it does not dissociate into ions in solution. Therefore, the Van't Hoff factor \(i\) for sugar is:
\[ i = 1 \]

### Step 4: Calculate the Ratio of \(\Delta T_f\)
Since both solutions are of equal concentration, the \(K_f\) and \(m\) will be the same for both solutions. Therefore, we can express the ratio of \(\Delta T_f\) for \([Fe(CN)_6]^{4-}\) to that for sugar as:
\[ \frac{\Delta T_f([Fe(CN)_6])}{\Delta T_f(sugar)} = \frac{K_f \cdot m \cdot i_{[Fe(CN)_6]}}{K_f \cdot m \cdot i_{sugar}} = \frac{i_{[Fe(CN)_6]}}{i_{sugar}} \]
Substituting the values of \(i\):
\[ \frac{\Delta T_f([Fe(CN)_6])}{\Delta T_f(sugar)} = \frac{5}{1} = 5 \]

### Conclusion
Thus, the ratio of \(\Delta T_f\) for \([Fe(CN)_6]^{4-}\) solution to that for sugar solution of equal concentration is:
\[ \text{Ratio} = 5 : 1 \]