Question

The mass of an electron is 9.11×10⁻³¹/1 kg. if the de broglie wavelength for an electron in a hydrogen atom is 3.31×10⁻¹⁰ m, how fast is the electron moving relative to the speed of light? the speed of light is 3.00×10⁸ m/s. express your answer numerically as a percent

Answer 1

The velocity of an electron with a given de Broglie wavelength is calculated using the de Broglie equation and then expressed as a percentage of the speed of light, resulting in approximately 0.0073%.

Step-by-step explanation:

The question involves calculating the velocity of an electron relative to the speed of light. Given that the mass of the electron is 9.11×10⁻³¹ kg, and its de Broglie wavelength is 3.31×10⁻¹⁰ m, we use the de Broglie equation λ = h / (mv) to find the velocity (v). Planck's constant (h) is 6.626×10⁻3J.s.

First, we rearrange the de Broglie equation to solve for v: v = h / (λm). Using the given values:

v = 6.626×10⁻3J.s / (3.31×10⁻¹⁰ m × 9.11×10⁻³¹ kg)

Calculating, v = (6.626×10⁻3J.s) / (3.31×10⁻¹⁰ m × 9.11×10⁻³¹ kg) = 2.18×10⁴ m/s.

To express this velocity relative to the speed of light (c = 3.00×10⁻8 m/s), we calculate the percentage: (v/c) × 100% = (2.18×10⁴ m/s) / (3.00×10⁻8 m/s) × 100% = 0.00727%, or approximately 0.0073%.