(a) Write Einstein's photoelectric equation and identh: each component of the equation. (b) For a photocell, star; one factor each that is responsible fo: t...

(a) Einstein's photoelectric equation

\[ hf = W_{0} + \tfrac{1}{2}mv_{\max}^{2} \]

• \(hf\) = energy of the incident photon (\(h\) = Planck constant, \(f\) = frequency of the incident light).
• \(W_{0}\) = work function of the metal (minimum energy to free an electron from the surface).
• \(\tfrac{1}{2}mv_{\max}^{2}\) = maximum kinetic energy of the emitted photoelectron.

(b) Factors for a photocell

• (i) Emission: the frequency of the incident light must be at or above the threshold frequency (photon energy at least equal to the work function).
• (ii) Rate of emission: the intensity (brightness) of the incident light (number of photons arriving per second).
• (iii) Energy of photoelectrons: the frequency of the incident light (and the work function of the metal).

(c)(i) Nuclear equations

Equation A: \(^{222}_{p}\text{Rn} \to\ ^{218}_{84}\text{Po} +\ ^{q}_{2}\text{He}\). Balancing charge: \(p = 84 + 2 = 86\). Balancing mass: \(q = 222 - 218 = 4\). So \(p = 86\), \(q = 4\) (an alpha particle is emitted).

Equation B: \(^{214}_{83}\text{Bi} \to\ ^{214}_{84}\text{Po} +\ ^{m}_{n}\text{X}\). Balancing mass: \(m = 214 - 214 = 0\). Balancing charge: \(n = 83 - 84 = -1\). So \(m = 0\), \(n = -1\), and X is a beta particle (electron), \(^{0}_{-1}e\).

(ii) Radioactive waste emits ionising radiation that is harmful to living cells (it can cause cancer, genetic damage and death) and can persist for a long time, so it must be disposed of safely to protect people and the environment from radiation exposure.

(d)(i) Energy of the photon

\(E = \dfrac{hc}{\lambda} = \dfrac{6.6\times10^{-34}\times3.0\times10^{8}}{2.4\times10^{-7}} = \dfrac{19.8\times10^{-26}}{2.4\times10^{-7}} = 8.25\times10^{-19}\ \text{J}\).

This corresponds to the option \(8.2\times10^{-19}\ \text{J}\).

(ii) On the energy-level diagram, the photon of energy \(8.2\times10^{-19}\ \text{J}\) is emitted when the atom makes a downward transition (electron falling) between the two levels whose energy difference equals \(8.2\times10^{-19}\ \text{J}\); the arrow is drawn pointing downwards from the higher to the lower of those two levels. The exact levels can only be marked on the actual figure provided.