¹⁹⁷₈₀ Hg decays to ¹⁹⁷₇₉ Au via electron capture, with a decay constant of 0.257 per day. Assuming the captured electron comes from the K shell, determine the wavelength of the K_ X-ray emitted subsequent to the electron capture. Apply Moseley's law = a(Z - b) using a = 4.95 10⁷ s ^ -1/2 and b = 1 .

Step-by-step solution

In electron capture, the parent nucleus ¹⁹⁷₈₀ Hg captures an inner shell electron and transforms into the daughter nucleus ¹⁹⁷₇₉ Au . The vacancy created in the K shell is filled by an outer electron, resulting in the emission of a K_ X-ray characteristic of the daughter element. Therefore, the atomic number to be used is Z = 79 . Using Moseley's law: = a(Z - b) Substituting a = 4.95 10^7 s ^ -1/2 , b = 1 , and Z = 79 : = 4.95 10^7 (79 - 1) = 4.95 10^7 78 = 3.861 10^9 s ^ -1/2 Squaring both sides to find the freque