How Is This Acid Base Map Established?

Question

In the Acid-base map -shown below- each line denoted as an isohydric line shows the pairs of $ PCO_{2}$ & $ [HCO_{3}^{-}] $ that give a certain $ pH $. So each line connects the values of $ PCO_{2}$ & $ [HCO_{3}^{-}] $ that give a value of $ pH $ that is constant across that line..

The 7.4 line shows the isohydric line of physiological pH,the central ellipse shows the normal range.

I understand the Henderson–Hasselbalch equation:

$\mathrm{pH}=\mathrm{p} K_{\mathrm{a}}+\log _{10}\left(\frac{[\mathrm{Base}]}{[\mathrm{Acid}]}\right)$

What I would like to know is how this map established either experimentally or mathematically or both preferably.

Answer 1

The curves are rather obtained clinically, statistically evaluating diagnose, symptoms and results of clinical lab tests.

Metabolic curve follows malfunctioning metabolism keeping too low ( m. acidosis ) or too high ( m. alkalosis ) level of bicarbonate.

Respiratury curves follows states of too intense breathing ( r. alkalosis ) or too shallow breathing ( r. acidosis ), because of too low respectively high level of carbon dioxide partial pressure, caused by abnormal breathing.

Note that the $K_{\ce{H, CO2}}$ Henry constant or CO2 is $\pu{29.41 atm/(mol/L)}$. the value will be higher because of the body temperature, can be corrected by temperature dependent $\ce{CO2}$ solubility.

$$\mathrm{pH}=\mathrm{p} K_{\mathrm{a}}+\log\left(\frac{[\mathrm{Base}]}{[\mathrm{Acid}]}\right)=\mathrm{p} K_{\mathrm{a}}+\log\left(\frac{[\ce{HCO3-}] \cdot K_{\ce{H, CO2}}}{p_{\ce{CO2}} }\right)$$

Where pKa of carbonic acid is 6.35, if counting the total $\ce{CO2}$ concentration.

There are 3 variables ( $\mathrm{pH}$, $p$, $\ce{[HCO3-]}$ ) and 2 degrees of freedom. $\mathrm{p}$ is controlled by breathing, $\ce{[HCO3-]}$ is controlled by metabolism. $\mathrm{pH}$ is result.