In a carbon monoxide molecule, the carbon and the oxygen atoms are separted by a distance ` 1.12 xx 10^(10)` m. The distance of the centre of mass from the carbon atom is

Options

A. ` 0.64 xx 10^(-10)m`
B. `0.56xx10^(-6) `m
C. `0.51 xx 10^(-10) m`
D. `0.48 xx 10^(-10)` m

Explanation

To find the distance of the center of mass from the carbon atom in a carbon monoxide (CO) molecule, we can follow these steps:

### Step 1: Identify the masses of the atoms
- The mass of carbon (C) is approximately \( m_1 = 12 \, \text{u} \) (atomic mass units).
- The mass of oxygen (O) is approximately \( m_2 = 16 \, \text{u} \).

### Step 2: Determine the total distance between the two atoms
- The distance between the carbon and oxygen atoms is given as \( d = 1.12 \times 10^{-10} \, \text{m} \).

### Step 3: Use the center of mass formula
The formula for the center of mass \( r_{cm} \) of a two-particle system is given by:

\[ r_{cm} = \frac{m_1 \cdot r_1 + m_2 \cdot r_2}{m_1 + m_2} \]

Where:
- \( r_1 \) is the distance of the carbon atom from the center of mass (which we will take as 0, since we are measuring from the carbon atom).
- \( r_2 \) is the distance of the oxygen atom from the center of mass, which is equal to \( d \).

### Step 4: Substitute the known values
Since \( r_1 = 0 \) (distance from carbon to itself), the equation simplifies to:

\[ r_{cm} = \frac{m_2 \cdot d}{m_1 + m_2} \]

Substituting the values:

\[ r_{cm} = \frac{16 \cdot (1.12 \times 10^{-10})}{12 + 16} \]

### Step 5: Calculate the total mass
The total mass \( m_1 + m_2 = 12 + 16 = 28 \, \text{u} \).

### Step 6: Calculate \( r_{cm} \)
Now substituting the values into the equation:

\[ r_{cm} = \frac{16 \cdot (1.12 \times 10^{-10})}{28} \]

### Step 7: Perform the multiplication
Calculating the numerator:

\[ 16 \cdot (1.12 \times 10^{-10}) = 17.92 \times 10^{-10} \]

### Step 8: Divide by the total mass
Now divide by 28:

\[ r_{cm} = \frac{17.92 \times 10^{-10}}{28} \approx 0.639 \times 10^{-10} \, \text{m} \]

### Step 9: Convert to scientific notation
This can be expressed as:

\[ r_{cm} \approx 6.39 \times 10^{-11} \, \text{m} \]

### Final Answer
Thus, the distance of the center of mass from the carbon atom is approximately:

\[ r_{cm} \approx 6.39 \times 10^{-11} \, \text{m} \]

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