According to the Heisenberg uncertainty principle, if the uncertainty in the speed of an electron is 3.5 x 10(3) m/s, the uncertainty in its position is at least
a)66 m
b)17 m
c)6.6 x 10-8 m
d)1.7 x 10-8 m
e)None of the above
a. 66
To solve this problem, we need to use the Heisenberg uncertainty principle. The Heisenberg uncertainty principle states that there is a limit to how precisely we can simultaneously know the position and momentum (or speed) of a particle.
According to the uncertainty principle, the product of the uncertainty in position (∆x) and the uncertainty in momentum (∆p) is greater than or equal to a constant value called Planck's constant (h). Mathematically, this can be written as:
∆x * ∆p ≥ h
In this case, we are given the uncertainty in the speed (∆v) of an electron as 3.5 x 10^3 m/s. Since speed is the magnitude of velocity, we can use the formula:
speed = |velocity| = |momentum| / mass
Rearranging the formula, we get:
|momentum| = speed * mass
Now, the uncertainty in momentum (∆p) can be calculated as the maximum difference between the momentum and the opposite of the momentum, which is twice the momentum:
∆p = 2 * (speed * mass)
Given the speed of the electron (∆v) as 3.5 x 10^3 m/s, and the mass of an electron as approximately 9.11 x 10^-31 kg, we can substitute these values into the formula:
∆p = 2 * (3.5 x 10^3 m/s) * (9.11 x 10^-31 kg)
Calculating this expression yields:
∆p ≈ 6.37 x 10^-25 kg·m/s
Now, according to the uncertainty principle, the product of the uncertainty in position (∆x) and the uncertainty in momentum (∆p) must be greater than or equal to Planck's constant (h). So, we can rewrite the formula as:
∆x * ∆p ≥ h
To determine the uncertainty in position (∆x), we rearrange the formula:
∆x = h / ∆p
Substituting the value of Planck's constant as approximately 6.63 x 10^-34 J·s, and the calculated value of ∆p as 6.37 x 10^-25 kg·m/s, we can compute:
∆x ≈ 6.63 x 10^-34 J·s / 6.37 x 10^-25 kg·m/s
Simplifying this expression, we find:
∆x ≈ 1.04 x 10^-9 m
Therefore, the uncertainty in the position (∆x) of the electron is approximately 1.04 x 10^-9 meters.
Since none of the given options match the calculated uncertainty, e) None of the above is the correct answer.