An oxygen molecule consists of two oxygen atoms whose total mass is 5.3 x 10-26 kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is 1.82 × 10-46 kg·m2. From these data, estimate the effective distance between the atoms.
For each atom, more than 99.9% of the mass is in the nucleus, which can be treated as a point, when compared to the internuclear separation, d. Let M be the total mass of the molecule.
The moment of inertia of the O2 molecule, though an axis that passes midway between the atoms, is
I = 2*(M/2)*(d/2)^2 = M*d^2/4
= 1.82*10^-46 kg*m^2
d^2 = 7.28^10^-46/5.3*10^-26
= 13.74*10^-21 m^2
d = 1.172*10^-10 m = 1.172*10^-8 cm
= 1.172 Angstroms
The actual value is somewhat higher (1.21 Angstroms), but this is what the formula says.
To estimate the effective distance between the atoms in the oxygen molecule, we can use the formula for calculating the moment of inertia of a diatomic molecule:
I = μ * d^2
Where:
- I is the moment of inertia
- μ is the reduced mass of the molecule
- d is the distance between the atoms
From the problem, we are given:
- Total mass of the oxygen molecule (m) = 5.3 x 10^-26 kg
- Moment of inertia about the axis (I) = 1.82 x 10^-46 kg·m2
The reduced mass (μ) can be calculated as:
μ = (m1 * m2) / (m1 + m2)
Since we have two oxygen atoms with the same mass, m1 = m2 = m. Therefore:
μ = (m * m) / (m + m)
= m / 2
Plugging in the given values:
μ = (5.3 x 10^-26 kg) / 2
= 2.65 x 10^-26 kg
Now, we can rearrange the equation for moment of inertia to solve for d:
d = sqrt(I / μ)
Plugging in the given values:
d = sqrt((1.82 x 10^-46 kg·m^2) / (2.65 x 10^-26 kg))
= sqrt(6.88 x 10^-21 m^2)
≈ 2.62 x 10^-11 m
Therefore, the estimated effective distance between the atoms in the oxygen molecule is approximately 2.62 x 10^-11 meters.
To estimate the effective distance between the atoms in an oxygen molecule, we can make use of the concept of Reduced Mass.
The reduced mass (μ) of a system is defined as:
μ = m1 * m2 / (m1 + m2)
where m1 and m2 are the masses of the individual particles (in this case, oxygen atoms).
Given that the total mass of the oxygen molecule is 5.3 x 10^-26 kg, we can assume that the masses of the two oxygen atoms are the same. Therefore, we can write:
m1 = m2 = (5.3 x 10^-26 kg) / 2 = 2.65 x 10^-26 kg
Now, let's calculate the reduced mass:
μ = (2.65 x 10^-26 kg) * (2.65 x 10^-26 kg) / ((2.65 x 10^-26 kg) + (2.65 x 10^-26 kg))
= (7.0225 x 10^-52) kg^2 / (5.3 x 10^-26 kg)
= 1.3255 x 10^-26 kg
Next, we can use the concept of Moment of Inertia (I) to estimate the effective distance between the oxygen atoms.
The Moment of Inertia (I) of a molecule rotating about an axis perpendicular to the line joining the atoms can be given as:
I = μ * R^2
where R is the effective distance between the atoms.
Rearranging the equation, we get:
R = √(I / μ)
Substituting the given values:
R = √((1.82 x 10^-46 kg·m^2) / (1.3255 x 10^-26 kg))
= √(1.37 x 10^-20 m^2)
≈ 3.70 x 10^-11 m
Therefore, the estimated effective distance between the oxygen atoms in an oxygen molecule is approximately 3.70 x 10^-11 meters.