The moment of inertia of NaCl molecule with bond length r about an axis perpendicular to the bond and passing through the centre of mass is

Options

A. `(m_(Na) + m_(CI))r^2`
B. `(m_(Na) + m_(CI))/(m_(Na)-m_(CI))r^2`
C. `(m_(Na)xxm_(CI))/(m_(Na)+m_(CI))r^`
D. `(m_(Na)+m_(CI))/(m_(Na)-m_(CI))r^2`

Answer

To find the moment of inertia \( I \) of a NaCl molecule with bond length \( r \) about an axis perpendicular to the bond and passing through the center of mass, we can follow these steps:

### Step 1: Define the System
We have a NaCl molecule consisting of a sodium ion (Na\(^+\)) and a chloride ion (Cl\(^-\)). The bond length between the two ions is \( r \).

### Step 2: Determine the Center of Mass
The center of mass \( x_{cm} \) of the system can be calculated using the formula:

\[ x_{cm} = \frac{m_{Na} \cdot 0 + m_{Cl} \cdot r}{m_{Na} + m_{Cl}} \]

Here, we assume the sodium ion is at position 0 and the chloride ion is at position \( r \).

### Step 3: Calculate Distances from Center of Mass
Let \( R_{Na} \) be the distance from the center of mass to the sodium ion and \( R_{Cl} \) be the distance from the center of mass to the chloride ion.

Using the center of mass position, we can express these distances as:

\[ R_{Na} = x_{cm} = \frac{m_{Cl} \cdot r}{m_{Na} + m_{Cl}} \]

\[ R_{Cl} = r - R_{Na} = r - \frac{m_{Cl} \cdot r}{m_{Na} + m_{Cl}} = \frac{m_{Na} \cdot r}{m_{Na} + m_{Cl}} \]

### Step 4: Write the Moment of Inertia Formula
The moment of inertia \( I \) about the center of mass is given by:

\[ I = m_{Na} \cdot R_{Na}^2 + m_{Cl} \cdot R_{Cl}^2 \]

### Step 5: Substitute the Distances
Substituting the expressions for \( R_{Na} \) and \( R_{Cl} \) into the moment of inertia formula:

\[ I = m_{Na} \left( \frac{m_{Cl} \cdot r}{m_{Na} + m_{Cl}} \right)^2 + m_{Cl} \left( \frac{m_{Na} \cdot r}{m_{Na} + m_{Cl}} \right)^2 \]

### Step 6: Simplify the Expression
Now we can simplify the expression:

\[ I = m_{Na} \cdot \frac{m_{Cl}^2 \cdot r^2}{(m_{Na} + m_{Cl})^2} + m_{Cl} \cdot \frac{m_{Na}^2 \cdot r^2}{(m_{Na} + m_{Cl})^2} \]

Factoring out the common terms:

\[ I = \frac{r^2}{(m_{Na} + m_{Cl})^2} \left( m_{Na} \cdot m_{Cl}^2 + m_{Cl} \cdot m_{Na}^2 \right) \]

This can be further simplified to:

\[ I = \frac{m_{Na} \cdot m_{Cl} \cdot r^2}{m_{Na} + m_{Cl}} \]

### Final Result
Thus, the moment of inertia \( I \) of the NaCl molecule about the specified axis is:

\[ I = \frac{m_{Na} \cdot m_{Cl} \cdot r^2}{m_{Na} + m_{Cl}} \]