In the HCl molecule, the separation between the nuclei of the two atoms is about 1.27 Å (1 Å = 10⁻¹⁰ m). Find the approximate location of the CM of the molecule, given that a chlorine atom is about 35.5 times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.
The centre of mass of the HCl molecule is located at approximately 1.235 Å from the hydrogen nucleus (or about 0.035 Å from the chlorine nucleus).
Let the hydrogen nucleus be at position x = 0 and the chlorine nucleus be at position x = 1.27 Å. Let the mass of hydrogen atom be m. Then, the mass of chlorine atom is 35.5m.
Using the formula for centre of mass for a two-particle system: X = (m₁x₁ + m₂x₂)/(m₁ + m₂)
X = (m × 0 + 35.5m × 1.27)/(m + 35.5m)
X = (35.5m × 1.27)/(36.5m) = 45.085/36.5 = 1.235 Å
Since the chlorine atom is much heavier (35.5 times), the centre of mass lies very close to the chlorine nucleus, at a distance of approximately 1.235 Å from the hydrogen nucleus.
Explanation
This question applies the centre of mass formula for a two-particle system given in the textbook. The key insight is that since nearly all mass is concentrated in the nucleus, we can treat the atoms as point masses at their nuclear positions. The CM position is weighted by mass, so the heavier chlorine atom pulls the CM closer to itself. The calculation shows the CM is only about 0.035 Å from the chlorine nucleus, demonstrating that for systems with highly unequal masses, the CM lies very close to the heavier mass.
Solution Steps
Step 1: Assign positions - Hydrogen at x = 0, Chlorine at x = 1.27 Å
Step 2: Assign masses - Hydrogen = m, Chlorine = 35.5m
Step 3: Apply CM formula: X = (m₁x₁ + m₂x₂)/(m₁ + m₂)
Step 4: Substitute values: X = (m×0 + 35.5m×1.27)/(m + 35.5m)
Step 5: Simplify: X = 45.085/36.5 = 1.235 Å from hydrogen nucleus