Deriving rate law from a reaction mechanism for a chain reaction using steady-state approximation I am trying to derive the overall rate equation for the photocatalysed halogenation (chlorination) of an alkane that proceed via this 4 step chain reaction:

$\ce{Cl2-&gt;2Cl^. \quad(1) initiation}$
$\ce{Cl^. +RH-&gt;R^. + HCl \quad(2) propagation}$
$\ce{R^. + Cl2-&gt; RCl + Cl^. \quad(3) propagation}$
$\ce{R^. + Cl^. -&gt;RCl \quad(4) termination}$

I have calculated the differential rate equations for $\ce{R^.}$ and $\ce{Cl^.}$ and set them to zero at steady state.
[The free radical dot symbols are omitted from this point onward for brevity]
This gives me $$\ce{For Cl ,2k1[Cl2]-k2[Cl][RH] +k3[R][Cl2] -k4[R][Cl]=0 \quad(5)}$$
$$\ce{For R, k2[Cl][RH] -k3[R][Cl2] -k4[R][Cl]=0 \quad(6)}$$
Adding these two equations,
$$\ce{[Cl]=\frac{k1}{k4}\frac{[Cl2]}{[R]}}$$
Then I tried substituting this into equation (6), which gives me a quadratic equation. The solution of the quadratic equation is of the form $$\ce{[R]=constant +\sqrt{\mathrm{constant [RH]}}}$$
The overall rate law from this expression is a very complicated equation.
This looks wrong because the rate law for radical halogenation is usually of the form $\ce{k[RH]\sqrt{\mathrm{[Cl_2]}}}$ (Wikipedia)
Where am I going wrong?
In the book that Wikipedia cites (Advanced Organic Chemistry: Reaction Mechanisms
By Reinhard Bruckner, ISBN 9780080498805), they have a different set of assumptions. They are saying the formation of chlorine radicals proceeds by a fast equilibrium. Then, they say that the organic radical is at steady state, ignoring reaction (4) given by the OP (the termination reaction).


Where am I going wrong?

I'm not sure why your steady state approximation is less appropriate than that of the textbook. I'm sure that there is some experimental data (and some fitted rate constants) that would speak to that. Also, the appropriate approximation depends on the initial conditions.

[Comment from Poutnik] The initiation rate may not depend on [Cl2]( for high enough [Cl2]), but on irradiation level only. i.e being a reaction of zero order.

The derivation in the book is for the thermal reaction, but the OP asked about the photocatalyzed reaction, so there is a discrepancy there.