The cathodic reaction of a dry cell is represented as `2MnO_(2)(s)+Zn^(2+)+2e^(-) rarr ZnMn_(2)O_(4) (s)`
If there are 8 g `MnO_(2)` in the cathodic compartment then the time for which the dry cell will continue to give current of 2 milliampere, is

Options

A. 25.675 day
B. 51.35 day
C. 12.8 day
D. 6.423 day

Explanation

To solve the problem, we need to determine the time for which the dry cell will continue to give a current of 2 mA, given that there are 8 g of MnO₂ in the cathodic compartment. We will follow these steps:

### Step 1: Calculate the Molar Mass of MnO₂
The molar mass of MnO₂ can be calculated as follows:
- Molar mass of Mn = 54.94 g/mol
- Molar mass of O = 16.00 g/mol

Thus, the molar mass of MnO₂ = 54.94 + (2 × 16.00) = 54.94 + 32.00 = 86.94 g/mol.

### Step 2: Calculate the Number of Moles of MnO₂
Using the mass of MnO₂ provided (8 g), we can calculate the number of moles:
\[ \text{Number of moles} = \frac{\text{mass}}{\text{molar mass}} = \frac{8 \text{ g}}{86.94 \text{ g/mol}} \approx 0.092 \text{ moles} \]

### Step 3: Determine the Number of Electrons Transferred
From the cathodic reaction:
\[ 2 \text{MnO}_2 + \text{Zn}^{2+} + 2e^- \rightarrow \text{ZnMn}_2\text{O}_4 \]
We see that 2 moles of MnO₂ consume 2 moles of electrons. Therefore, 1 mole of MnO₂ will consume 1 mole of electrons.

Thus, the number of moles of electrons transferred for 0.092 moles of MnO₂ is also 0.092 moles.

### Step 4: Convert Moles of Electrons to Charge
Using Faraday's constant (approximately 96500 C/mol), we can calculate the total charge (Q) transferred:
\[ Q = \text{number of moles of electrons} \times \text{Faraday's constant} = 0.092 \text{ moles} \times 96500 \text{ C/mol} \approx 8874.8 \text{ C} \]

### Step 5: Calculate Time Using Current
We know that current (I) is defined as charge (Q) per unit time (t):
\[ I = \frac{Q}{t} \implies t = \frac{Q}{I} \]
Given that the current is 2 mA (which is 0.002 A), we can calculate the time:
\[ t = \frac{8874.8 \text{ C}}{0.002 \text{ A}} = 4437400 \text{ seconds} \]

### Step 6: Convert Time from Seconds to Days
To convert seconds into days:
\[ \text{Time in days} = \frac{4437400 \text{ seconds}}{86400 \text{ seconds/day}} \approx 51.3 \text{ days} \]

### Final Answer
The dry cell will continue to give a current of 2 mA for approximately **51.3 days**.

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