Temperature sensitivity of a catalytic ozone balance ($d\ln [{\mathrm{O}}_3]/d(T^{-1})$)

Suppose a reduced steady balance is $P=A{X}^2+BX$, where $X=[{\mathrm{O}}_3]$, and hold $P$ and all specified background concentrations fixed. With $u=1/T$, $d\ln A/du=-a$ and $d\ln B/du=-b$, implicit differentiation gives $d\ln X/du=(aA{X}^2+bBX)/(2A{X}^2+BX)$. The quadratic loss term includes temperature-dependent O/O3 partitioning as well as the direct reaction coefficient. Holding the loss ratio fixed over a finite change is an additional approximation; solving the new quadratic provides a check.