A hydrate of magnesium iodide has a formula `Mgl_(2).xH_(2)O.A 1.055 g` sample is heated to a constant weight of `0.695 g`. What is the value of `x` ?

Options

A. `2`
B. `4`
C. `6`
D. `8`

Explanation

To find the value of \( x \) in the hydrate formula \( \text{MgI}_2 \cdot x \text{H}_2\text{O} \), we can follow these steps:

### Step 1: Determine the mass of water lost
The mass of the hydrate before heating is given as \( 1.055 \, \text{g} \) and the mass after heating (the residue) is \( 0.695 \, \text{g} \).

To find the mass of water lost during heating, we subtract the mass of the residue from the mass of the hydrate:

\[ \text{Mass of water lost} = \text{Mass of hydrate} - \text{Mass of residue} = 1.055 \, \text{g} - 0.695 \, \text{g} = 0.360 \, \text{g} \]

### Step 2: Calculate the molar mass of \( \text{MgI}_2 \)
The molar mass of magnesium iodide \( \text{MgI}_2 \) can be calculated as follows:

- Molar mass of Mg = \( 24.31 \, \text{g/mol} \)
- Molar mass of I = \( 126.90 \, \text{g/mol} \)

Thus, the molar mass of \( \text{MgI}_2 \) is:

\[ \text{Molar mass of } \text{MgI}_2 = 24.31 + 2 \times 126.90 = 24.31 + 253.80 = 278.11 \, \text{g/mol} \]

### Step 3: Set up the equation for the molar mass of the hydrate
The molar mass of the hydrate \( \text{MgI}_2 \cdot x \text{H}_2\text{O} \) is given by:

\[ \text{Molar mass of hydrate} = \text{Molar mass of } \text{MgI}_2 + x \times \text{Molar mass of } \text{H}_2\text{O} \]

The molar mass of water \( \text{H}_2\text{O} \) is \( 18.02 \, \text{g/mol} \). Therefore, we can express the molar mass of the hydrate as:

\[ \text{Molar mass of hydrate} = 278.11 + 18.02x \]

### Step 4: Use the mass ratio to find \( x \)
According to the problem, the ratio of the mass of the hydrate to the mass of the residue (which is \( \text{MgI}_2 \)) should equal the ratio of their molar masses:

\[ \frac{1.055 \, \text{g}}{0.695 \, \text{g}} = \frac{278.11 + 18.02x}{278.11} \]

### Step 5: Solve for \( x \)
Cross-multiplying gives:

\[ 1.055 \times 278.11 = 0.695 \times (278.11 + 18.02x) \]

Calculating the left side:

\[ 1.055 \times 278.11 \approx 293.57 \]

Now calculating the right side:

\[ 0.695 \times 278.11 + 0.695 \times 18.02x \]

Calculating \( 0.695 \times 278.11 \):

\[ 0.695 \times 278.11 \approx 193.20 \]

So we have:

\[ 293.57 = 193.20 + 0.695 \times 18.02x \]

Subtract \( 193.20 \) from both sides:

\[ 293.57 - 193.20 = 0.695 \times 18.02x \]

\[ 100.37 = 0.695 \times 18.02x \]

Now, calculate \( 0.695 \times 18.02 \):

\[ 0.695 \times 18.02 \approx 12.52 \]

So we have:

\[ 100.37 = 12.52x \]

Now, divide both sides by \( 12.52 \):

\[ x = \frac{100.37}{12.52} \approx 8.01 \]

Since \( x \) must be a whole number, we round it to \( x = 8 \).

### Final Answer
Thus, the value of \( x \) is \( 8 \).

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