Magnesium and Titanium form a crystalline oxide in which Mg appears at the eight corners , oxygen appears at face centres and titanium appears at the centre of cube. The atomic mass of magnesium , titanium and oxygen is 24, 45.9 and 16 amu respectively. A similar compound is formed by calcium with titanium and oxygen .
What would be the molar mass of unit cell of `MgTiO_3` ?

Options

A. (A) Nearly `118 "g mol"^(-1)`
B. (B) `"236 g mol"^(-1)`
C. (C) `"59 g mol"^(-1)`
D. (D) None of these

Answer

To find the molar mass of the unit cell of \( \text{MgTiO}_3 \), we will follow these steps:

### Step 1: Identify the contributions of each atom in the unit cell.
- **Magnesium (Mg)**: Appears at the eight corners of the cube. Each corner contributes \( \frac{1}{8} \) of an atom. Therefore, the total contribution from magnesium is:
\[ \text{Contribution from Mg} = 8 \times \frac{1}{8} = 1 \text{ atom} \]

- **Oxygen (O)**: Appears at the face centers of the cube. Each face contributes \( \frac{1}{2} \) of an atom, and there are 6 faces. Therefore, the total contribution from oxygen is:
\[ \text{Contribution from O} = 6 \times \frac{1}{2} = 3 \text{ atoms} \]

- **Titanium (Ti)**: Appears at the center of the cube, contributing 1 atom. Therefore:
\[ \text{Contribution from Ti} = 1 \text{ atom} \]

### Step 2: Determine the formula of the compound.
From the contributions calculated:
- Mg: 1 atom
- O: 3 atoms
- Ti: 1 atom

Thus, the empirical formula of the compound is \( \text{MgTiO}_3 \).

### Step 3: Calculate the molar mass of \( \text{MgTiO}_3 \).
Now we will use the atomic masses provided:
- Atomic mass of Mg = 24 amu
- Atomic mass of Ti = 45.9 amu
- Atomic mass of O = 16 amu

Now, we calculate the total molar mass:
\[ \text{Molar mass of } \text{MgTiO}_3 = (\text{mass of Mg}) + (\text{mass of Ti}) + 3 \times (\text{mass of O}) \]
\[ = 24 + 45.9 + 3 \times 16 \]
\[ = 24 + 45.9 + 48 \]
\[ = 24 + 45.9 + 48 = 117.9 \text{ amu} \]

### Step 4: Round to the nearest whole number.
The molar mass of \( \text{MgTiO}_3 \) is approximately:
\[ \approx 118 \text{ g/mol} \]

### Final Answer:
The molar mass of the unit cell of \( \text{MgTiO}_3 \) is **118 g/mol**.

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