Final answer:
The uncertainty in the velocity of an electron confined to a space of 1.0 x 10^-9 m is calculated using the Heisenberg Uncertainty Principle and is found to be approximately 5.8 x 10^4 m/s.
Step-by-step explanation:
The student is asking about determining the uncertainty in the velocity (Δv) of an electron when it is confined to a known position (Δx).
The question is related to the Heisenberg Uncertainty Principle in quantum mechanics, which tells us that the more precisely we know the position of a particle, the less precisely we can know its velocity, and vice versa.
This principle can be mathematically represented by the inequality 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 (ℏ = h / (2π)).
To calculate the uncertainty in velocity (Δv) for an electron, we first need to find the uncertainty in momentum (Δp) using the formula Δp ≥ ℏ / (2Δx), where Δx has been given as 1.0 x 10-9 m.
Plugging in the values, we get:
Δp ≥ (1.055 x 10-34 kg m²/s) / (2 * 1.0 x 10-9 m) ≥ 5.275 x 10-26 kg m/s.
Now, knowing the mass of the electron (me = 9.1 × 10-31 kg),
we can calculate Δv by dividing Δp by the mass of the electron (me):
Δv = Δp / me
≥ (5.275 x 10-26 kg m/s) / (9.1 × 10-31 kg) ≥ 5.8 x 104 m/s.
Therefore, the uncertainty in the velocity of the electron is approximately 5.8 x 104 m/s.