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Sources state that the concentration of $\ce{H2CO3}$ in rain is of a few µmol; I don't know if that's per ml. The buffering effect of most soils is >1000 µmol $\ce{H+}$/kg.

Is the difference from low acid rain water versus lab distilled water on a pH test for soil less than 0.1 in pH values?

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Pure water (rain as well as distilled water) in equilibrium with the atmosphere ($p_{\ce{O2}}=10^{-3.5}\ \mathrm{atm}$) can be calculated to contain about $$\begin{align} \mathrm{pH}=-\log[\ce{H+}]&=5.65\\ -\log[\ce{HCO3-}]&=5.65\\ -\log[\ce{CO3^2-}]&=10.3\\ -\log[\ce{H2CO3^*}]&=5.0\\ -\log[\ce{CO2}]&=5.0\\ -\log[\ce{H2CO3}]&=7.8\\ \end{align}$$

(The calculation can be found in: Stumm, W.; Morgan, J. J. Aquatic Chemistry, Third Edition; John Wiley & Sons: New York, NY, 1996; pp 159–161.)

Note that $[\ce{H2CO3^*}]$ is the total analytical activity of dissolved $\ce{CO2}$, i.e. $[\ce{H2CO3^*}]=[\ce{CO2(aq)}]+[\ce{H2CO3}]$. $[\ce{H2CO3}]$ is the concentration of true $\ce{H2CO3}$.

Thus, $[\ce{H2CO3^*}]=10^{-5}$ and $c(\ce{H2CO3^*})=10^{-5}\ \mathrm{mol/l}$; i.e. a few µmol/l.

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