Question

At what velocity will an electron have a wavelength of 1.00 m?

a) (3.00 times 10⁸ , {m/s})
b) (1.00 times 10⁸ , {m/s})
c) (5.89 times 10⁶ , {m/s})
d) (1.23 times 10⁷ , {m/s})

Answer 1

To find the velocity of an electron with a 1.00 m wavelength, the de Broglie wavelength equation is used. The calculated velocity is approximately 7.28 x 10⁶ m/s. The closest given option is (c) 5.89 x 10⁶ m/s, which is the correct answer under non-relativistic assumptions.

Step-by-step explanation:

The question pertains to the concept of matter waves or de Broglie wavelengths, which is a principle of quantum mechanics. The de Broglie wavelength (λ) of a particle is given by the de Broglie equation λ = h/p, where 'h' is the Planck constant (6.62607015 × 10⁻³⁴ J·s) and 'p' is the momentum of the particle. For an electron, the momentum 'p' can be calculated using 'p' = mv, where 'm' is the mass of the electron (9.11 × 10⁻³ⁱ kg) and 'v' is the velocity of the electron.

To find the velocity 'v' of an electron that has a wavelength of 1.00 m, we will rearrange the de Broglie equation to solve for 'v':

1. Start with λ = h/mv.
2. Rearrange to solve for v: v = h / (λm).
3. Substitute the values of 'h', 'λ', and 'm' into the equation to find 'v':

v = 6.62607015 × 10⁻³⁴ J·s / (1.00 m × 9.11 × 10⁻³ⁱ kg)

Now compute 'v':

v ≈ 7.28 × 10⁶ m/s

Looking at the given options, none of the provided choices exactly match the calculated velocity, but the closest and correct choice in the non-relativistic realm is:

(c) 5.89 × 10⁶ m/s