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
d 1.7 x 10-8m
To answer this question, we need to use the Heisenberg uncertainty principle, which states that there is a fundamental limit to the precision with which certain pairs of physical properties, such as position and momentum, can be known simultaneously.
The Heisenberg uncertainty principle is given by the equation:
Δx * Δp ≥ h/4π
Where Δx is the uncertainty in position, Δp is the uncertainty in momentum (which is equal to the mass times the uncertainty in velocity), and h is the Planck constant.
In this case, we are given the uncertainty in velocity, which is 3.5 x 10^3 m/s. The uncertainty in momentum can be calculated by multiplying the mass of the electron (which is approximately 9.1 x 10^-31 kg) by the uncertainty in velocity.
So, Δp = (9.1 x 10^-31 kg) * (3.5 x 10^3 m/s) = 3.185 x 10^-27 kg•m/s
Now we can rearrange the uncertainty principle equation to solve for Δx:
Δx ≥ h/(4πΔp)
Substituting the values we have:
Δx ≥ (6.63 x 10^-34 J•s)/(4π * 3.185 x 10^-27 kg•m/s)
Δx ≥ 6.6 x 10^-8 m
Therefore, the uncertainty in the position of the electron is at least 6.6 x 10^-8 m. The correct option is c) 6.6 x 10^-8 m.
According to the Heisenberg uncertainty principle, the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) must be greater than or equal to the reduced Planck's constant h/2π.
Mathematically, this can be represented as Δx × Δp ≥ h/2π
Given that the uncertainty in speed of the electron (Δv) is 3.5 x 10^3 m/s, we can calculate the uncertainty in momentum (Δp) using the formula:
Δp = m × Δv
where m is the mass of the electron.
The mass of an electron (m) is approximately 9.11 x 10^(-31) kg.
Plugging in the values:
Δp = (9.11 x 10^(-31) kg) × (3.5 x 10^3 m/s)
Δp = 3.19 x 10^(-27) kg·m/s
Now, we need to find the uncertainty in position (Δx). Rearranging the uncertainty principle equation, we have:
Δx ≥ h/2πΔp
Plugging in the known values:
Δx ≥ (6.626 x 10^(-34) J·s) / (2π × 3.19 x 10^(-27) kg·m/s)
Δx ≥ 6.56 x 10^(-8) m
Therefore, the uncertainty in the position of the electron is at least 6.56 x 10^(-8) m.
The correct answer is c) 6.6 x 10^(-8) m.