one gm metal `M^(+2)`was discharged by the passage of `1.81xx10^22` electrons. What is the atomic weight of metal?

Options

A. 33.35
B. 133.4
C. 66.7
D. 55

Answer

To find the atomic weight of the metal \( M^{+2} \) discharged by the passage of \( 1.81 \times 10^{22} \) electrons, we can follow these steps:

### Step 1: Determine the charge of the electrons
The charge of one electron is approximately \( 1.6 \times 10^{-19} \) coulombs. Therefore, the total charge \( Q \) for \( 1.81 \times 10^{22} \) electrons can be calculated as:

\[ Q = \text{Number of electrons} \times \text{Charge of one electron} \]

\[ Q = 1.81 \times 10^{22} \times 1.6 \times 10^{-19} \]

Calculating this gives:

\[ Q = 2.896 \times 10^{3} \text{ coulombs} \]

### Step 2: Relate charge to moles of metal deposited
Since \( M^{+2} \) requires 2 electrons to be reduced to \( M \), we can find the number of moles of metal deposited using Faraday's law of electrolysis. The total charge required to deposit one mole of \( M \) is given by:

\[ \text{Charge for 1 mole of } M = 2 \times F \]

Where \( F \) (Faraday's constant) is approximately \( 96500 \) coulombs. Therefore, the charge required to deposit one mole of \( M \) is:

\[ \text{Charge for 1 mole of } M = 2 \times 96500 = 193000 \text{ coulombs} \]

### Step 3: Calculate the moles of metal deposited
Using the total charge calculated in Step 1, we can find the moles of metal deposited:

\[ \text{Moles of } M = \frac{\text{Total charge}}{\text{Charge for 1 mole of } M} \]

\[ \text{Moles of } M = \frac{2.896 \times 10^{3}}{193000} \]

Calculating this gives:

\[ \text{Moles of } M \approx 0.0149 \text{ moles} \]

### Step 4: Calculate the atomic weight of the metal
We know that the mass of the metal discharged is 1 gram. The atomic weight \( A \) can be calculated using the formula:

\[ A = \frac{\text{Given mass}}{\text{Number of moles}} \]

Substituting the values:

\[ A = \frac{1 \text{ g}}{0.0149 \text{ moles}} \]

Calculating this gives:

\[ A \approx 67.11 \text{ g/mol} \]

### Step 5: Conclusion
The atomic weight of the metal \( M \) is approximately \( 67.11 \text{ g/mol} \). The closest answer from the options provided is \( 66.7 \text{ g/mol} \).