Question

Using the fact that the CO molecule absorbs infrared light with 10⁻5 m, estimate the quantum mechanical uncertainty of the distance between C and O in the ground state of this molecule.

a) 5 x 10⁻6 m
b) 1 x 10⁻5 m
c) 2 x 10⁻5 m
d) 5 x 10⁻5 m

Answer 1

Using Heisenberg's Uncertainty Principle, the estimated quantum mechanical uncertainty of the distance between C and O in the CO molecule is smaller than the wavelength of absorbed light (10⁻⁵ m), therefore the most reasonable estimate is option a) 5 x 10⁻⁶ m.

Step-by-step explanation:

To estimate the quantum mechanical uncertainty of the distance between C and O in the ground state of a CO molecule, we can use the Heisenberg Uncertainty Principle. This principle states that it is impossible to simultaneously know the exact position and momentum of a particle. In relation to the wavelength of light absorbed, some degree of position uncertainty is implied for the molecule's vibrational state.

Since the CO molecule absorbs infrared light with a wavelength of 10⁻⁵ meters, we might consider this wavelength to be related to the positional uncertainty. However, the actual position uncertainty Δx is generally much smaller than the wavelength of the light involved in the transition because the absorption involves quantized energy levels rather than a continuous spectrum.

Answer 2

The quantum mechanical uncertainty of the distance between C and O in the CO molecule is c) 2 x 10⁻⁵ m.

Step-by-step explanation:

To estimate the quantum mechanical uncertainty of the distance between C and O in the ground state of the CO molecule, we can use the equation Δx × Δp ≥ ℏ/2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ℏ is the reduced Planck's constant.

Since we are given the absorption of infrared light with a wavelength of 10⁻⁵ m, we can use the de Broglie wavelength equation λ = h/p, where λ is the wavelength, h is the Planck's constant, and p is the momentum.

By substituting the values, we can solve for Δx, which is the uncertainty in position:

Δx = ℏ/2Δp = (ℏ/(2p))Δp = (ℏ/(2h/λ))Δp = λ/2πΔp

Since λ = 10⁻⁵ m, we can estimate the quantum mechanical uncertainty of the distance between C and O to be:

Δx = (10⁻⁵ m)/(2π)

Therefore, the correct answer is (c) 2 x 10⁻⁵ m.