Q3(c)(ii). Photodecomposition of HI: rate, quantum yield and order

1. The rate of the primary step equals the rate at which light is absorbed, Iₐ (one HI dissociates per photon absorbed). HI is used up in steps 1 and 2: -(d[HI])/(dt)=Iₐ+k₂[H][HI]
2. H atoms are very reactive and their concentration stays small and constant (steady-state approximation): (d[H])/(dt)=Iₐ-k₂[H][HI]=0, so k₂[H][HI]=Iₐ
3. Substituting: -(d[HI])/(dt)=Iₐ+Iₐ=2Iₐ
4. Quantum yield: φ=(rate of HI decomposition)/(rate of photon absorption)=(2Iₐ)/(Iₐ)=2 Every photon removes one HI in step 1 and the H atom it makes removes a second in step 2.
5. Step 2 does not give back an H atom, so there is no chain and φ stays at 2. Step 3 removes only I atoms and uses no HI. The steady state for I, Iₐ+k₂[H][HI]-2k₃[I]²=0, gives [I]=(Iₐ/k₃)^(1/2) and d[H₂]/dt=d[I₂]/dt=Iₐ, consistent with 2HI → H₂ + I₂.
6. Kinetic order: [HI] cancels from the rate law, so the reaction is zero order in HI. It is first order in Iₐ: doubling the absorbed light doubles the rate.
7. If the light is weakly absorbed, Iₐ≈2.303 ε l I₀[HI] (Beer–Lambert law), and the rate becomes first order in HI.

Application. a quantum yield above 1 shows that secondary (dark) steps follow the primary act. φ=2 means one secondary step per photon; a chain, in which the radical is regenerated, gives φ much greater than 1. Measuring φ with a known Iₐ is how a photochemical mechanism is tested.