= Nighttime Chapman ozone
With photolysis switched off, constant molecular oxygen and collider reservoirs give $\dot x=-x(a+by)$ and $\dot y=x(a-by)$ for atomic oxygen $x$, ozone $y$, $a=k_1[\mathrm O_2][M]$ and $b=k_3$. Dividing the equations and integrating gives
$$
x-x_0=y-y_0+\frac{2a}{b}\ln\frac{a-by}{a-by_0}.
$$
For $y_0<a/b$, ozone grows toward a finite-reservoir endpoint below $a/b$. Its value at sunrise depends on the dark-interval duration; $a/b$ is an approximate quasi-steady balance while atomic oxygen remains available.
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