A total of 69500 C of electricity was required to reduced 37.7 g of `M^(3+)` to the metal. What is M ?

Answer

To find the metal \( M \) that was reduced from \( M^{3+} \), we will follow these steps:

### Step 1: Calculate the Equivalent Weight
We know that the equivalent weight can be calculated using the formula:
\[ E = \frac{W \times F}{Q} \]
Where:
- \( W \) = mass of the metal reduced = 37.7 g
- \( F \) = Faraday's constant = 96500 C/mol
- \( Q \) = total charge = 69500 C

Substituting the values:
\[ E = \frac{37.7 \, \text{g} \times 96500 \, \text{C/mol}}{69500 \, \text{C}} \]

### Step 2: Perform the Calculation
Calculating the equivalent weight:
\[ E = \frac{37.7 \times 96500}{69500} \approx 52.35 \, \text{g/equiv} \]

### Step 3: Relate Equivalent Weight to Molecular Weight
The molecular weight \( M \) of the metal can be calculated using the relationship:
\[ M = n \times E \]
Where \( n \) is the number of electrons transferred in the reduction process. Since \( M^{3+} \) accepts 3 electrons:
\[ M = 3 \times 52.35 \, \text{g/equiv} \]

### Step 4: Calculate the Molecular Weight
Calculating the molecular weight:
\[ M = 3 \times 52.35 \approx 157.05 \, \text{g/mol} \]

### Step 5: Identify the Metal
Now, we need to identify the metal with a molecular weight close to 157.05 g/mol. The metal that corresponds to this molecular weight is likely to be **Lanthanum (La)**, which has an atomic weight of approximately 138.9 g/mol, or **Gadolinium (Gd)**, which has an atomic weight of approximately 157.25 g/mol.

Thus, the metal \( M \) is **Gadolinium (Gd)**.

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