Find the center of mass of the carbon monoxide molecule , given that carbon has a mass of 12 u, oxygen has a mass of 16 u , and the two atoms are 0.111 nm apart.
To find the center of mass of a molecule, we need to calculate the average position of its constituent particles weighted by their respective masses. In this case, we have a carbon atom and an oxygen atom.
Step 1: Assign coordinates to the atoms
Let's assume the carbon atom is located at the origin (0, 0, 0) and the oxygen atom is at position (0.111 nm, 0, 0).
Step 2: Calculate the mass-weighted coordinates
To calculate the center of mass, we need to find the average x, y, and z coordinates of the atoms, weighted by their masses.
For carbon atom:
- Mass of carbon, m1 = 12 u
- Coordinate of carbon, x1 = 0, y1 = 0, z1 = 0
For oxygen atom:
- Mass of oxygen, m2 = 16 u
- Coordinate of oxygen, x2 = 0.111 nm, y2 = 0, z2 = 0
Step 3: Calculate the center of mass
The center of mass coordinates can be calculated using the following formulas:
x̄ = (m1 * x1 + m2 * x2) / (m1 + m2)
ȳ = (m1 * y1 + m2 * y2) / (m1 + m2)
z̄ = (m1 * z1 + m2 * z2) / (m1 + m2)
Substituting the values we have:
x̄ = (12 u * 0 + 16 u * 0.111 nm) / (12 u + 16 u)
ȳ = (12 u * 0 + 16 u * 0) / (12 u + 16 u)
z̄ = (12 u * 0 + 16 u * 0) / (12 u + 16 u)
Note: nm means nanometers. We need to convert it to meters for consistency.
Step 4: Convert the result to SI units
To convert nanometers to meters, we divide by 10^9.
x̄ = (12 u * 0 + 16 u * 0.111 nm) / (12 u + 16 u) / (10^9)
ȳ = (12 u * 0 + 16 u * 0) / (12 u + 16 u) / (10^9)
z̄ = (12 u * 0 + 16 u * 0) / (12 u + 16 u) / (10^9)
Now we can calculate the values:
x̄ = (0 + 1.776 nm * u) / 28 u / (10^9)
ȳ = (0) / 28 u / (10^9)
z̄ = (0) / 28 u / (10^9)
Step 5: Simplify the expression
Dividing each term by 28u, we have:
x̄ = 1.776 nm / (28 * (10^9 u))
ȳ = 0
z̄ = 0
So, the center of mass of the carbon monoxide molecule is located at (1.776 nm / (28 * (10^9 u)), 0, 0).