The equivalent weight of an element is `4`. Its chloride has a vapour density `59.25`. Find the valency of element.

Explanation

To find the valency of the element based on the given information, we can follow these steps:

### Step 1: Understand the relationship between vapor density and molecular weight.
The vapor density (VD) of a gas is related to its molecular weight (M) by the formula:
\[ \text{Vapor Density} = \frac{M}{2} \]
From the problem, we know the vapor density of the chloride is 59.25.

### Step 2: Calculate the molecular weight of the chloride.
Using the formula, we can rearrange it to find the molecular weight:
\[ M = 2 \times \text{Vapor Density} \]
Substituting the given vapor density:
\[ M = 2 \times 59.25 = 118.5 \]

### Step 3: Write the formula for the chloride.
Let the element be represented as \( M \). The chloride of the element can be represented as \( MCl_n \), where \( n \) is the number of chlorine atoms.

### Step 4: Determine the molecular weight of the chloride.
The molecular weight of the chloride can be expressed as:
\[ \text{Molecular Weight of } MCl_n = \text{Molecular Weight of } M + n \times \text{Molecular Weight of Cl} \]
The molecular weight of chlorine (Cl) is approximately 35.5 g/mol.

### Step 5: Set up the equation.
From the previous steps, we have:
\[ 118.5 = \text{Molecular Weight of } M + n \times 35.5 \]

### Step 6: Use the equivalent weight to find the molecular weight of the element.
The equivalent weight (EW) of an element is given by:
\[ \text{Equivalent Weight} = \frac{\text{Molecular Weight}}{\text{Valency}} \]
Given that the equivalent weight is 4, we can express the molecular weight of the element as:
\[ \text{Molecular Weight of } M = \text{Valency} \times 4 \]

### Step 7: Substitute into the equation.
Now we can substitute this expression into our earlier equation:
\[ 118.5 = (\text{Valency} \times 4) + n \times 35.5 \]

### Step 8: Rearranging the equation.
Rearranging gives:
\[ 118.5 - n \times 35.5 = \text{Valency} \times 4 \]

### Step 9: Solve for valency.
To find the valency, we need to find suitable values for \( n \). The valency of chlorine is typically 1, so we can start with \( n = 1 \):
\[ 118.5 - 1 \times 35.5 = \text{Valency} \times 4 \]
\[ 118.5 - 35.5 = \text{Valency} \times 4 \]
\[ 83 = \text{Valency} \times 4 \]
\[ \text{Valency} = \frac{83}{4} = 20.75 \quad \text{(not possible)} \]

Trying \( n = 2 \):
\[ 118.5 - 2 \times 35.5 = \text{Valency} \times 4 \]
\[ 118.5 - 71 = \text{Valency} \times 4 \]
\[ 47.5 = \text{Valency} \times 4 \]
\[ \text{Valency} = \frac{47.5}{4} = 11.875 \quad \text{(not possible)} \]

Trying \( n = 3 \):
\[ 118.5 - 3 \times 35.5 = \text{Valency} \times 4 \]
\[ 118.5 - 106.5 = \text{Valency} \times 4 \]
\[ 12 = \text{Valency} \times 4 \]
\[ \text{Valency} = \frac{12}{4} = 3 \]

### Conclusion:
The valency of the element is **3**.

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