`A 500mL` sample of a `0.1M Cr^(3+)` is electrolyzed with a current of `96.5A`. If the remaining `[Cr^(3+)]` is `0.04M` then the duration of process is:-

Explanation

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the Initial Moles of Cr³⁺
The initial concentration of Cr³⁺ is given as 0.1 M, and the volume of the solution is 500 mL (which is 0.5 L).

\[ \text{Initial moles of Cr}^{3+} = \text{Molarity} \times \text{Volume} = 0.1 \, \text{mol/L} \times 0.5 \, \text{L} = 0.05 \, \text{mol} \]

**Hint:** Remember to convert mL to L when calculating moles.

### Step 2: Calculate the Final Moles of Cr³⁺
The remaining concentration of Cr³⁺ after electrolysis is given as 0.04 M. We will use the same volume (0.5 L) to find the final moles.

\[ \text{Final moles of Cr}^{3+} = 0.04 \, \text{mol/L} \times 0.5 \, \text{L} = 0.02 \, \text{mol} \]

**Hint:** Use the same volume for both initial and final calculations to maintain consistency.

### Step 3: Calculate the Change in Moles (Δn)
Now, we can find the change in moles of Cr³⁺ during the electrolysis process.

\[ \Delta n = \text{Initial moles} - \text{Final moles} = 0.05 \, \text{mol} - 0.02 \, \text{mol} = 0.03 \, \text{mol} \]

**Hint:** The change in moles indicates how much Cr³⁺ was consumed during the process.

### Step 4: Use Faraday's Law to Relate Charge, Current, and Time
According to Faraday's law, the relationship between charge (Q), current (I), and time (t) is given by:

\[ Q = I \times t \]

We also know that the moles of electrons (n) transferred can be related to the charge using Faraday's constant (F = 96500 C/mol). The equivalent weight can be expressed as:

\[ \text{Charge (Q)} = n \times F \]

Where \( n \) is the number of moles of electrons. For Cr³⁺, since it gains 3 electrons to become Cr, we have:

\[ n = \Delta n \times 3 = 0.03 \, \text{mol} \times 3 = 0.09 \, \text{mol} \]

Thus, the total charge is:

\[ Q = 0.09 \, \text{mol} \times 96500 \, \text{C/mol} = 8685 \, \text{C} \]

**Hint:** Remember that the number of moles of electrons transferred depends on the valency of the ion being reduced.

### Step 5: Calculate the Time (t)
Now, we can rearrange the formula \( Q = I \times t \) to find time:

\[ t = \frac{Q}{I} = \frac{8685 \, \text{C}}{96.5 \, \text{A}} \approx 90 \, \text{s} \]

**Hint:** Ensure that the current is in amperes and the charge is in coulombs to maintain unit consistency.

### Final Answer
The duration of the electrolysis process is approximately **90 seconds**.