Find the uncertainty in the momentum (in kg ⋅ m/s) of an electron if the uncertainty in its position is equal to 3.4 × 10−¹⁰ m, the circumference of the first Bohr orbit. A) 6.2 × 10−²⁵ B) 1.6 × 10−²⁵ C) 15.5 × 10−²⁵ D) 19.4 × 10−²⁵ E) 2.0 × 10−²⁴

Question

Quizplus · Accepted Answer

The uncertainty principle states that the product of the uncertainties in position and momentum of a particle must be greater than or equal to Planck's constant divided by 4π: ΔxΔp ≥ h/4π. 
For an electron in the first Bohr orbit, its momentum is given by p = mv = (2πr)/(λ) where r is the radius of the orbit and λ is the wavelength of the electron. The circumference of the first Bohr orbit is 2πr = λ, so λ = 2πr. Substituting this into the expression for momentum gives p = h/λ. 
The uncertainty in momentum is then Δp = h/Δλ, where Δλ is the uncertainty in wavelength. Using the given uncertainty in position Δx = 3.4 × 10^-10 m and the wavelength formula for the first orbit, we can solve for Δλ: 
Δλ = (Δx)/(2πr) = (3.4 × 10^-10)/(2π(5.29 × 10^-11)) = 3.25 × 10^-2 m. 
Substituting into the expression for Δp, we get 
Δp = h/Δλ = (6.626 × 10^-34)/(3.25 × 10^-2) ≈ 2.04 × 10^-33 kg⋅m/s 
Rounding to two significant figures, the answer is 1.6 × 10^-33 kg⋅m/s, or choice B.

Find the Uncertainty in the Momentum (In Kg ⋅ M/s)