The equivalent conductance of a substance is 1/3rd of the molar conductance . The time required to electrolyse 3 moles the substance using a current of 9 ampere is

Options

A. 4808 min
B. 1608 min
C. 2408 min
D. 2008 min

Answer

To solve the problem step by step, we need to find the time required to electrolyze 3 moles of a substance using a current of 9 amperes, given that the equivalent conductance is one-third of the molar conductance.

### Step 1: Understand the relationship between equivalent conductance and molar conductance
We know that:
\[ \text{Equivalent Conductance} (\Lambda_{eq}) = \frac{1}{3} \times \text{Molar Conductance} (\Lambda_m) \]
This implies:
\[ \Lambda_{eq} = \frac{\Lambda_m}{3} \]

### Step 2: Relate equivalent conductance and molar conductance to normality and molarity
The formulas for equivalent conductance and molar conductance are:
\[ \Lambda_{eq} = \frac{\kappa \times 1000}{N} \quad \text{and} \quad \Lambda_m = \frac{\kappa \times 1000}{C} \]
where:
- \(\kappa\) is the specific conductance,
- \(N\) is the normality,
- \(C\) is the molarity.

### Step 3: Set up the relationship between normality and molarity
From the relationship established in Step 1, we can write:
\[ \frac{\Lambda_{eq}}{\Lambda_m} = \frac{C}{N} \]
Substituting \(\Lambda_{eq} = \frac{\Lambda_m}{3}\):
\[ \frac{\frac{\Lambda_m}{3}}{\Lambda_m} = \frac{C}{N} \implies \frac{1}{3} = \frac{C}{N} \]
This means:
\[ C = \frac{N}{3} \]

### Step 4: Determine the n-factor
Since we have established that:
\[ \Lambda_{eq} = \frac{\Lambda_m}{n} \]
and substituting \(\Lambda_{eq} = \frac{\Lambda_m}{3}\), we find:
\[ \frac{\Lambda_m}{3} = \frac{\Lambda_m}{n} \implies n = 3 \]

### Step 5: Use Faraday's first law to find time
According to Faraday's first law:
\[ \text{Weight deposited} = Z \cdot I \cdot t \]
where:
- \(Z\) is the electrochemical equivalent,
- \(I\) is the current,
- \(t\) is the time.

The electrochemical equivalent \(Z\) can be expressed as:
\[ Z = \frac{\text{Molecular weight}}{n \cdot F} \]
where \(F\) is Faraday's constant (96500 C/mol).

### Step 6: Relate moles to the equation
We know that:
\[ \text{Number of moles} = \frac{I \cdot t}{n \cdot F} \]
Rearranging gives:
\[ t = \frac{\text{Number of moles} \cdot n \cdot F}{I} \]

### Step 7: Substitute the known values
Substituting the values:
- Number of moles = 3,
- \(n = 3\),
- \(F = 96500 \, \text{C/mol}\),
- \(I = 9 \, \text{A}\):

\[ t = \frac{3 \cdot 3 \cdot 96500}{9} \]

### Step 8: Calculate the time
Calculating gives:
\[ t = \frac{3 \cdot 3 \cdot 96500}{9} = \frac{3 \cdot 96500}{3} = 96500 \, \text{seconds} \]

### Step 9: Convert seconds to minutes
To convert seconds to minutes:
\[ t = \frac{96500}{60} \approx 1608.33 \, \text{minutes} \]

### Final Answer
The time required to electrolyze 3 moles of the substance using a current of 9 amperes is approximately **1608 minutes**.