A man suffering from untreated diabetes mellitus is admitted to a hospital. Doctors fear that his blood $\mathrm{pH}$ may have dropped because of ketoacidosis. Analysis of his blood reveals that $[\ce{HCO3-}] = \pu{16 mM}$ and $p_\ce{CO2} = 30.$ If $\mathrm{p}K_\mathrm{a}$ of $\ce{HCO3-}$ is $6.1,$ determine whether the patient runs a risk of acidotic coma. (Note: in plasma under physiologic conditions, concentration of $\ce{CO2}$ and $p_\ce{CO2}$ are related by the solubility constant for $\ce{CO2}$ in plasma which is $\pu{0.03 mM/mm Hg}.$
We use the Henderson-Hasselbalch equation, then
$$\mathrm{pH} = \mathrm{p}K_\mathrm{a} + \log[\ce{HCO3−}][\ce{H2CO3}]$$
Moreover, here $\mathrm{p}K_\mathrm{a}$ of $\ce{HCO3-}$ is given whereas the acid is $\ce{H2CO3},$ and also if we consider second dissociation that is $\mathrm{p}K_\mathrm{a2}$ then also it is wrong because $\ce{HCO3-}$ will dissociate into $\ce{H+}$ and $\ce{CO3^2-}$ so in this dissociation there is no relation of $\ce{CO2}.$ And also why solubility constant is given, how can we find out $\ce{CO2}$ from solubility constant?