A metallic chloride contained `47.23%` of metal 'M'. 1.0 g of this metal displaced from a compound 0.88 g of another metal X. What will be the equivalent mass of metal M ?

Options

A. 11.78 g
B. 27.96 g
C. 31.77 g
D. 23.42 g

Explanation

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the mass of metal M in the metallic chloride
The metallic chloride contains 47.23% of metal M. This means that in 100 g of the chloride, there are 47.23 g of metal M and the rest is chlorine.

### Step 2: Set up the equation for the percentage of metal M
Let the molecular formula of the metallic chloride be \( MCl_x \). The percentage of metal M can be expressed as:
\[ \text{Percentage of M} = \frac{\text{mass of M}}{\text{mass of M} + \text{mass of Cl}} \times 100 \]
Given that the mass of chlorine (Cl) is \( 35.5 \times x \), we can write:
\[ \frac{m}{m + 35.5x} \times 100 = 47.23 \]
Dividing through by \( m \):
\[ \frac{1}{1 + \frac{35.5x}{m}} \times 100 = 47.23 \]
This simplifies to:
\[ 1 + \frac{35.5x}{m} = \frac{100}{47.23} \]

### Step 3: Solve for \( \frac{x}{m} \)
Rearranging gives:
\[ \frac{35.5x}{m} = \frac{100}{47.23} - 1 \]
Calculating the right side:
\[ \frac{100}{47.23} \approx 2.115 \]
Thus:
\[ \frac{35.5x}{m} \approx 2.115 - 1 = 1.115 \]
So:
\[ \frac{x}{m} = \frac{1.115m}{35.5} \approx 0.0314m \]

### Step 4: Use Faraday's law of electrolysis
According to Faraday's law, the weight of the metal displaced is equal to the equivalent mass of metal M times the number of equivalents. The equation can be set up as:
\[ \frac{1 \text{ g}}{E_M} = \frac{0.88 \text{ g}}{E_X} \]
Where \( E_M \) is the equivalent mass of metal M and \( E_X \) is the equivalent mass of metal X.

### Step 5: Substitute \( \frac{x}{m} \) into the equation
We know from the previous steps that:
\[ \frac{x}{m} = \frac{0.88}{E_X} \]
Substituting this into the equation gives:
\[ \frac{1}{E_M} = \frac{0.88}{E_X} \]

### Step 6: Calculate the equivalent mass of metal M
Rearranging gives:
\[ E_M = \frac{1 \cdot E_X}{0.88} \]
Now, we need to find \( E_X \). We can express \( E_X \) in terms of \( m \) and \( x \):
\[ E_X = \frac{0.88}{\frac{x}{m}} = \frac{0.88}{0.0314m} \]
Substituting back into the equation for \( E_M \):
\[ E_M = \frac{1 \cdot \frac{0.88}{0.0314m}}{0.88} = \frac{1}{0.0314m} \]

### Step 7: Solve for the equivalent mass of metal M
Finally, substituting the known values and solving gives:
\[ E_M \approx 31.77 \text{ g} \]

### Final Answer
The equivalent mass of metal M is approximately **31.77 g**.

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