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.
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
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To estimate the effective distance between the atoms in an oxygen molecule, we can use the given moment of inertia and the total mass of the molecule.
The formula for moment of inertia of a diatomic molecule is:
I = 2*(M/2)*(d/2)^2
We are given that the moment of inertia (I) is equal to 1.82 × 10^-46 kg·m^2.
And the total mass (M) of the oxygen molecule is 5.3 x 10^-26 kg.
Substituting these values into the formula, we get:
1.82 × 10^-46 kg·m^2 = 2*(5.3 x 10^-26 kg/2)*(d/2)^2
Simplifying the equation, we have:
1.82 × 10^-46 = (5.3 x 10^-26)*(d^2/4)
Now we can solve for 'd', the effective distance between the oxygen atoms.
d^2 = (1.82 × 10^-46)*(4/(5.3 x 10^-26))
d^2 = 13.74 x 10^-21 m^2
Taking the square root of both sides, we find:
d = √(13.74 x 10^-21) m
d ≈ 3.707 x 10^-11 m
d ≈ 3.707 x 10^-9 cm (since 1 meter is equal to 10^2 cm)
d ≈ 3.707 Angstroms (since 1 Angstrom is equal to 10^-10 m)
Therefore, the estimated effective distance between the oxygen atoms in an oxygen molecule is approximately 3.707 Angstroms.