In the electrolysis of an aqueous `Cr_(2)(SO_(4))_(3)` solution using a current of 2 amp, the mass of cathode is increased by 8 g. How long was electrolysis conducted ?

Answer

To solve the problem of how long the electrolysis of an aqueous `Cr_(2)(SO_(4))_(3)` solution was conducted, we can follow these steps:

### Step 1: Determine the moles of chromium deposited
We know that during electrolysis, chromium ions are reduced at the cathode. The molar mass of chromium (Cr) is approximately 52 g/mol. Given that the mass of the cathode increased by 8 g, we can calculate the moles of chromium deposited.

\[ \text{Moles of Cr} = \frac{\text{mass of Cr}}{\text{molar mass of Cr}} = \frac{8 \, \text{g}}{52 \, \text{g/mol}} \approx 0.1538 \, \text{mol} \]

### Step 2: Calculate the moles of electrons used
The reduction of chromium ions to metallic chromium involves the transfer of 3 electrons per chromium atom (n-factor = 3). Therefore, the total moles of electrons required can be calculated as follows:

\[ \text{Moles of electrons} = \text{Moles of Cr} \times 3 = 0.1538 \, \text{mol} \times 3 \approx 0.4614 \, \text{mol} \]

### Step 3: Convert moles of electrons to charge in coulombs
Using Faraday's constant (approximately 96500 C/mol), we can calculate the total charge (Q) in coulombs:

\[ Q = \text{Moles of electrons} \times \text{Faraday's constant} = 0.4614 \, \text{mol} \times 96500 \, \text{C/mol} \approx 44538.6 \, \text{C} \]

### Step 4: Use the current to find the time
We know that the relationship between charge (Q), current (I), and time (t) is given by the formula:

\[ Q = I \times t \]

Rearranging this formula to solve for time gives us:

\[ t = \frac{Q}{I} \]

Substituting the values we have:

\[ t = \frac{44538.6 \, \text{C}}{2 \, \text{A}} = 22269.3 \, \text{s} \]

### Step 5: Convert time from seconds to hours
To convert seconds into hours, we divide by the number of seconds in an hour (3600 s):

\[ t \approx \frac{22269.3 \, \text{s}}{3600 \, \text{s/h}} \approx 6.18 \, \text{h} \]

### Final Answer
The electrolysis was conducted for approximately **6.18 hours**.

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