= Solution
{c}
The constants $R_a,R_b,R_d$ are effective reservoir capacities: stored <carbon dioxide> <mass> anomaly per unit effective concentration anomaly, in GtCO$_2$ per ppmv. Only $C_a$ is a literal atmospheric mole fraction; $C_b,C_d$ are scaled reservoir variables chosen to share its equilibrium value. The exchange coefficients $r,s$ are conductances, in GtCO$_2$ per ppmv per year: $r(C_a-C_b)$ is transfer from the <atmosphere> to the fast reservoir and $s(C_b-C_d)$ transfer from that reservoir to the deep <ocean>. They are not by themselves inverse times; division by a reservoir capacity gives an inverse-time rate.
<Hydrostatic balance> estimates total atmospheric <mass> as $M_{\rm air}=4\pi R_E^2\bar p/g$. One ppmv of a dilute <ideal gas> is a mole fraction $10^{-6}$, so, with dry-air <molar mass> $M_{\rm mol,air}\simeq28.97$ g/mol and <carbon dioxide> <molar mass> $44.01$ g/mol,
$$
R_a=\frac{4\pi(6.371\times10^6)^2(98500)}{9.81}
\frac{44.01}{28.97}\frac{10^{-6}}{10^{12}}
\simeq\boxed{7.8\,\mathrm{GtCO_2/ppmv}.}
$$
The final factor converts kilograms to gigatonnes. This is <carbon dioxide> <mass>, not the smaller <carbon> <mass>.
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