# Approximate time for pure water pH to equilibrate in open air?

I'm a complete non-chemist, so I don't know how to calculate this, or even what to google for (my searches yielded esoteric research papers). I'm doing some hobby experiments, and I'd like to know what approximate timeframe to expect for relatively pure (distilled or double distilled) water left in open air at around 24 °C/75F to reach whatever the pH equilibrium point is (I read around 5.5 somewhere).

I'm not looking for a giant equation please, just a quick approximate number. Thanks

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Assume you have poured your distilled water in a 1 liter container.

We write the flux of $n_{\ce{CO2}}$ between air towards your container, as $$J_{\ce{CO2}}=\frac{dn_{\ce{CO2}}}{dt}\times\frac{1}{A}=\frac{d[\ce{H2CO3}]^{*}}{dt}\times\frac{V}{A}$$ where $n_{\ce{CO2}}$ is the number of ${\ce{CO2}}$ moles in water, $A$ the water surface and $V$ the water volume. Assuming $\ce{CO2}$ atmospheric pressure equal to $10^{-3.5}\,atm$, we obtain the concentration $[\ce{CO2}]_{air}$ approximately equal to $10^{-5}\,M$ (the latter value obtained via $PV=nRT$).

We call $[\ce{H2CO3}]^{*}$ the sum of $[{\ce{CO2}}]_{water}$ and $[\ce{H2CO3}]_{water}$ concentrations in water: at the beginning, we set it equal to $0$. Note that $[\ce{H2CO3}]^{*}$ could be reasonably written as:

$$[\ce{H2CO3}]^{*}\approx[{\ce{CO2}}]_{water}$$

indicating that the majority of carbon dioxide in water is not converted into carbonic acid, being the kinetics for such a conversion very slow, as pointed out by @Nicolau Saker Neto, .

By expressing $J_{\ce{CO2}}$ as:

$$J_{\ce{CO2}}=\frac{D_{{\ce{CO2}}}}{Z}\times ([\ce{CO2}]_{air}-[\ce{H2CO3}]^{*})$$

with $D_{\ce{CO2}}$ the diffusion coefficient of ${\ce{CO2}}$ equal to $7.2\times10^{-4}\,dm^{2}h^{-1}$and $Z$ the thickness of superficial layer through which the exchange occurs. For the latter, we take $40\times10^{-5}\,dm$.

Putting previous equations together (with $A/V=1\,dm^{-1}$), the following is obtained:

$$\frac{d[\ce{H2CO3}]^{*}}{dt}=1.8\times10^{-5}-1.8\times[\ce{H2CO3}]^{*}$$ which yields:

$$[\ce{H2CO3}]^{*}=1\times10^{-5}(1-e^{-1.8t})$$

The graph below shows that about $2$ hours later, a plateau is established, for $[\ce{H2CO3}]^{*}= 1\times10^{-5} M$.

By considering the acid-base equilibrium ($pK_{a}=6.3$): $$\text{H}_{2}\text{C}\text{O}_{3}^{*} \ce{<=> HCO3- + H+}$$ we obtain a $\text{pH}=5.7$, at the plateau.

UPDATE 1: the value for $Z$ has of course a big impact on time. A larger value will increase the time needed to reach the plateau. I have used $40\times10^{-5}\,dm$, considered a "typical value" by the authors of this book (sorry in French, page 181, 3rd edition). Any more accurately derived value for $Z$ is welcome !

UPDATE 2: I used the notation $\text{H}_{2}\text{C}\text{O}_{3}^{*}$ to represent the two species $\ce{CO2}$ and $\ce{H2CO3}$, in water.

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Very clearly laid out, thanks! – Ranging Mar 12 '14 at 21:21
I'm not too good with diffusion and kinetics, but this answer seems to assume that acidification of pure water is limited by the diffusion of $\ce{CO2}$ into the liquid. However, the actual reaction $\ce{CO2 + H2O → H2CO3}$ is quite slow, having a rate constant of $0.039\ s^{-1}$. Do the reaction kinetics have no impact on the acidification speed? – Nicolau Saker Neto Mar 12 '14 at 23:52
@NicolauSakerNeto Definitely you are right: I should have used (and I will do) the notation $[\ce{H2CO3}]^{*}$ as suggested by en.wikipedia.org/wiki/Carbonic_acid to indicate the overall concentration of the two species, $\ce{H2CO3}$ and $\ce{CO2}$, the latter being definetely dominating, as you say. But this will not alter the final $\text{pH}$, being $\ce{CO2}$ pressure the only factor, determining the composition of the solution (en.wikipedia.org/wiki/…) – mannaia Mar 13 '14 at 7:30

Diffusion is surface area, pH is volume, and convection matters. $\ce{CO2}$ is not merely dissolving, it is also hydrating to carbonic acid, and that is dissociating into protons (lowered pH) and bicarbonate. Hydration is slow - seconds. If conductivity water is left out in the open, dust falls in and it isn't conductivity water anymore.

"Diffusion Coefficients of CO2, C2H4, C3H6 and C4H8 in Water from 6° to 65° C,"
DOI: 10.1021/je60022a043
"Diffusion Coefficients for Hydrogen Sulfide, Carbon Dioxide, and Nitrous Oxide in Water over the Temperature Range 293-368 K,"
DOI: 10.1021/je00014a031 $\ce{CO2}$ graph is pictured on the first page.
DOI: 10.1029/JC092iC10p10767
DOI: 10.1029/92JC00188

We now know the diffusion coefficient, but we do not necessarily know the interface impedance mismatch. The world is a dirty place. A monolayer of ambiphile or hydrophobe could substantially change exchange kinetics. Take a clean water surface (overflowed glass), sprinkle a little talcum powder, cornstarch, or flour; stroke your fingertip across your forehead or nose, then touch that spot to the corner of the water. Sebum is surface active. Cerumen also works.

(Interface impedance mismatch is big stuff: Lithium-doped glass helium dewar surfaces, Hi-Float for helium-filled elastomer balloons, anti-reflection coatings for optics.)

carbon dioxide diffusion coefficient water
The original literature is in Google Scholar. Google is a retelling. You do not know the quality of the work from which the numbers came, including typos.

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