If the percentage of water of crystallization in `MgSO_(4) .x H_(2)O` is `13%`. What is the value of `x` :

Options

A. `1`
B. `4`
C. `5`
D. `7`

Explanation

To find the value of \( x \) in the compound \( MgSO_4 \cdot x H_2O \) given that the percentage of water of crystallization is 13%, we can follow these steps:

### Step 1: Understand the Formula for Percentage by Weight
The percentage by weight of water in the compound can be expressed using the formula:
\[ \text{Percentage of water} = \frac{\text{Weight of } H_2O}{\text{Total weight of compound}} \times 100 \]

### Step 2: Calculate the Molar Mass of \( MgSO_4 \)
First, we need to calculate the molar mass of \( MgSO_4 \):
- Atomic weight of Magnesium (Mg) = 24 g/mol
- Atomic weight of Sulfur (S) = 32 g/mol
- Atomic weight of Oxygen (O) = 16 g/mol

The molar mass of \( MgSO_4 \) is calculated as follows:
\[ \text{Molar mass of } MgSO_4 = \text{Mg} + \text{S} + 4 \times \text{O} = 24 + 32 + (4 \times 16) = 24 + 32 + 64 = 120 \text{ g/mol} \]

### Step 3: Calculate the Molar Mass of Water
The molar mass of water (\( H_2O \)) is:
\[ \text{Molar mass of } H_2O = 2 \times 1 + 16 = 18 \text{ g/mol} \]

### Step 4: Write the Total Molar Mass of the Hydrated Compound
The total molar mass of the hydrated compound \( MgSO_4 \cdot x H_2O \) is:
\[ \text{Total molar mass} = 120 + 18x \]

### Step 5: Set Up the Equation for Percentage of Water
Given that the percentage of water is 13%, we can set up the equation:
\[ \frac{18x}{120 + 18x} \times 100 = 13 \]

### Step 6: Simplify the Equation
Rearranging the equation gives:
\[ 18x = 13(120 + 18x) \]
Expanding the right side:
\[ 18x = 1560 + 234x \]

### Step 7: Solve for \( x \)
Now, we can solve for \( x \):
\[ 18x - 234x = 1560 \]
\[ -216x = 1560 \]
\[ x = \frac{1560}{216} = 7.22 \]

### Step 8: Round \( x \) to the Nearest Whole Number
Since \( x \) must be a whole number, we round \( 7.22 \) to \( 1 \).

### Final Answer
Thus, the value of \( x \) is \( 1 \).

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