Final answer:
To find the velocity of an electron with a given wavelength, use the de Broglie equation, λ = h / (mv), and rearrange to solve for velocity. Given the wavelength and known constants, the velocity v can be calculated, ensuring to account for physical limitations like the speed of light.
Step-by-step explanation:
To solve the mathematical problem completely of finding the velocity of an electron with a given wavelength, we employ the de Broglie equation: λ = h / (mv), where λ is the wavelength, h is Planck's constant (6.62607015 × 10⁻³⁴ J·s), m is the mass of the electron (9.11 × 10⁻³ⁱ kg), and v is the velocity of the electron we are looking to find.
Given the wavelength (λ) of 5.00 × 10²⁷ m, we rearrange the de Broglie equation to solve for v: v = h / (λ · m). Plugging in the given values, we calculate the velocity:
v = (6.62607015 × 10⁻³⁴ J·s) / ((5.00 × 10²⁷ m) · (9.11 × 10⁻³ⁱ kg))
As this calculation involves values that are far from ordinary scales and could potentially lead to relativistic speeds, it's important to note any physical limitations such as the speed of light and the non-relativistic assumption in the de Broglie hypothesis.