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 find the equivalent mass of metal M, we can follow these steps:

### Step 1: Determine the formula of the metallic chloride
Let the valency of metal M be β. The formula of the metallic chloride can be represented as \( MCl_β \).

### Step 2: Set up the percentage equation
Given that the metallic chloride contains 47.23% of metal M, we can express this as:
\[ \frac{M}{M + 35.5β} \times 100 = 47.23 \]
This can be rearranged to:
\[ \frac{M}{M + 35.5β} = 0.4723 \]

### Step 3: Rearranging the equation
Cross-multiplying gives us:
\[ M = 0.4723(M + 35.5β) \]
Expanding this, we get:
\[ M = 0.4723M + 16.79β \]
Rearranging the terms leads to:
\[ M - 0.4723M = 16.79β \]
\[ 0.5277M = 16.79β \]
Thus, we can express β in terms of M:
\[ \beta = \frac{0.5277M}{16.79} \]

### Step 4: Use the displacement information
We know that 1.0 g of metal M displaces 0.88 g of another metal X. The relationship between the equivalent masses can be expressed as:
\[ \frac{1}{E_m} = \frac{0.88}{E_x} \]
Where \( E_m \) is the equivalent mass of metal M and \( E_x \) is the equivalent mass of metal X.

### Step 5: Substitute the values
From the previous step, we can express:
\[ \frac{1}{E_m} = \frac{0.88}{E_x} \]
We also know that the equivalent mass of metal X can be calculated using its atomic mass and valency. Assuming the valency of X is 1 (for simplicity), we can find \( E_x \):
\[ E_x = \text{Atomic mass of X} = 27.96 \text{ g} \]

### Step 6: Calculate equivalent mass of M
Now substituting \( E_x \) into the equation:
\[ \frac{1}{E_m} = \frac{0.88}{27.96} \]
Calculating gives:
\[ E_m = \frac{27.96}{0.88} = 31.77 \text{ g} \]

### Conclusion
The equivalent mass of metal M is **31.77 g**.

---