The distance between the centres of carbon and oxygen atoms in the carbon monoxide molecule is `1.130Å` . Locate the centre of mass of the molecule relative to the carbon atom .

Options

A. `5.428Å`
B. `1.130Å`
C. `0.6457Å`
D. `0.3260Å`

Answer

To locate the center of mass of the carbon monoxide (CO) molecule relative to the carbon atom, we can follow these steps:

### Step 1: Understand the system
We have a carbon atom (C) and an oxygen atom (O) separated by a distance of 1.130 Å (angstroms). The atomic masses are:
- Mass of Carbon (m_C) = 12 u (atomic mass units)
- Mass of Oxygen (m_O) = 16 u

### Step 2: Define the distance
Let the distance between the carbon and oxygen atoms be denoted as \( d = 1.130 \, \text{Å} = 1.130 \times 10^{-10} \, \text{m} \).

### Step 3: Use the center of mass formula
The center of mass (CM) for two point masses can be calculated using the formula:
\[ x_{CM} = \frac{m_C \cdot x_C + m_O \cdot x_O}{m_C + m_O} \]
Where:
- \( x_C \) is the position of the carbon atom (we can set this as 0 for reference).
- \( x_O \) is the position of the oxygen atom, which is \( d \).

### Step 4: Set the positions
- Let \( x_C = 0 \) (position of carbon).
- Therefore, \( x_O = d = 1.130 \times 10^{-10} \, \text{m} \).

### Step 5: Substitute the values into the formula
Substituting the known values into the center of mass formula:
\[ x_{CM} = \frac{(12 \, \text{u}) \cdot (0) + (16 \, \text{u}) \cdot (1.130 \times 10^{-10} \, \text{m})}{12 \, \text{u} + 16 \, \text{u}} \]
This simplifies to:
\[ x_{CM} = \frac{0 + 18.08 \times 10^{-10}}{28 \, \text{u}} = \frac{18.08 \times 10^{-10}}{28} \]

### Step 6: Calculate the center of mass
Calculating the above expression:
\[ x_{CM} = \frac{18.08 \times 10^{-10}}{28} \approx 0.6457 \times 10^{-10} \, \text{m} \approx 0.6457 \, \text{Å} \]

### Step 7: Conclusion
The center of mass of the CO molecule relative to the carbon atom is approximately \( 0.6457 \, \text{Å} \).