In the lungs there are tiny sacs of air, which are called alveoli. An oxygen molecule (mass = 5.30 × 10-26 kg) is trapped within a sac, and the uncertainty in its position is 0.120 mm. What is the minimum uncertainty in the velocity of this oxygen molecule?
To determine the minimum uncertainty in the velocity of the oxygen molecule, we can use the Heisenberg's uncertainty principle, which states that the product of the uncertainties in position and velocity of a particle is greater than or equal to Planck's constant divided by 4π.
The formula for the uncertainty principle is:
Δx * Δv ≥ h / (4π)
where
Δx is the uncertainty in position,
Δv is the uncertainty in velocity,
h is Planck's constant (approximately 6.626 x 10^-34 J·s).
Given:
Δx = 0.120 mm = 0.120 x 10^-3 m
m = mass of oxygen molecule = 5.30 × 10^-26 kg
Let's substitute these values into the formula and solve for Δv:
(0.120 x 10^-3) * Δv ≥ (6.626 x 10^-34) / (4π)
Now, we can rearrange the equation to solve for Δv:
Δv ≥ (6.626 x 10^-34) / (4π * 0.120 x 10^-3)
Calculating this value using a calculator:
Δv ≥ 3.49 x 10^4 m/s
Therefore, the minimum uncertainty in the velocity of the oxygen molecule trapped within the sac is approximately 3.49 x 10^4 m/s.