Question #44188

A sample of a radioactive metal $M_{(s)}$ (Work function $\Phi = 4.2$ eV) undergoes $\alpha$-decay with a half-life of $120$ days. The sample is placed in a vacuum chamber at $298$ K and continuously irradiated with monochromatic light of wavelength $\lambda = 248.4$ nm and power $P = 2.56 \times 10^{-3}$ W. The photoelectric quantum efficiency (ratio of emitted photoelectrons to incident photons) is $2.5 \times 10^{-5}$.

The emitted $\alpha$-particles completely strike a $\ce{^9Be_{(s)}}$ target, triggering the nuclear reaction: $\ce{^9Be + ^4He -> ^{12}C + ^1n}$ ($100\%$ yield).

Given the initial $\alpha$-activity of the sample is $1.28 \times 10^{12}$ Bq, and assuming the nuclei are stable against photon-induced transmutation (i.e., the nuclear decay constant is independent of the electronic ionization state), after how many days will the rate of neutron production exactly equal the rate of photoelectron emission?

(Given: $hc = 1242$ eV nm, $1$ eV $= 1.6 \times 10^{-19}$ J)