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Is storing carbon dioxide in limestone a good idea? Wouldn't that lead to a reaction like

$\mathrm{C}\mathrm{O}_2 + \mathrm{H}_2\mathrm{O}\to {\mathrm{C}\mathrm{O}_3}^{-2} + 2\mathrm{H}^+$

And then the (unbalanced) reaction

$\mathrm{Ca}\mathrm{C}\mathrm{O}_3 + \mathrm{H}^+\to \mathrm{C}\mathrm{O}_2 + \mathrm{Ca}(\mathrm{O}\mathrm{H})_2$

So now the the ground turns into a water caustic water solution. Is this right?

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Limestone is the natural sedimentary rock, e.g., the top of Mt. Everest that began as sea floor. "Storing" $\ce{CO2}$ in limestone requires heating calcium carbonate white hot to dissociate it to lime, then scrubbing "other" $\ce{CO2}$ with the lime. This is, to coin an adjective, smartless, since you also burned fuel to heat the limestone. Curing concrete imbibes and traps $\ce{CO2}$, but it was initially dissociated to release it.

Your second equation is terrible. You obtain a strong base by adding acid to a salt. $\ce{CaCO3}$ plus carbonic acid, $\ce{CO2}$ in water, makes water-soluble calcium bicarbonate $\ce{Ca(HCO3)2}$. That is how limestone caves are dissolved out, and why karst topography is stark (surface erosion) and unstable (sinkholes).

To vastly store carbon at a profit, plant forests, harvest, and build wood-frame buildings. The photosynthesis bottleneck, carbon fixation, depends upon the slowest least efficient enzyme on Earth, RuBisCO. A smart (at least, not corrupt) society then

1) Gene-gineers a better RuBisCO. Ten times the productivity leaves it the worst enzyme on Earth. Everybody would have a full belly.

2) Gene-gineers a 10x fast-growing forest with huge lignin content in its wood. It won't rot, it won't feed termites, it will be immensely strong. If wood-frame buildings become unfashionable, bury the wood in deserts to store carbon.

There is no problem so easily solved that social advocacy will not make it intractable, disastrous, and expensive. The unabled maximize self-esteem to be credible - effectively selecting for the deluded, hence advocacy. A society inundated with rules from religious and secular political classes ignoring productive ends will collapse. USSR to Pol Pot, reality is what it appears to be - thermodynamics. There is no cheating possible.

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Sincerely I'm not sure that carbon sequestration injecting $CO_{2}$ in the ground is a good idea. The risk is that possible leaks cause asphyxiation to the living creatures (humans included) in the surrounding see this article, furthermore injecting gas can change the equilibria of the ecosystem or cause other hydrogeological instability.

I strongly recommend you to read this article, an excerpt of a natural $CO_{2}$ emission:

The year was 1986, the scene of the tragedy was Lake Nyos in Cameroon. People were going about their daily lives when death struck; a large amount of $CO_{2}$ exploded from Lake Nyos and more than 1,700 people died. When medical examiners came upon the scene they found bodies sprawled everywhere, many with clothes shredded and half-torn, a last attempt to relieve the suffocation of asphyxiation. Cattle had died, and those who had survived had been a comatose state for hours. Skin was seared, bronchial tubes were burnt, and lungs filled with fluid. It was surely a quick way to die, but not something to be wished for. The cause of the explosion was at first, but after multiple tests and examinations it appears that Lake Nyos was connected to a volcanic chamber that slowly released $CO_{2}$ into the lake. After a large amount accumulated, it erupted and rose to the surface.

However there are others research more optimistic this paper from 2009 can give you a good review however nobody can predict all the possible implications of these injections, and the topic is still debated the conclusion of the article clarify this:

“There is consensus that $CO_{2}$ storage in deep underground geologic has great technological potential and may be deployable on a widespread basis. However, there are still substantial financial, institutional, regulatory, and technical challenges that remain. To address these challenges, multiple integrated $CO_{2}$ capture and storage system projects are needed to prove out the technology. Also needed is an array of small, and intermediate and large-scale $CO_{2}$ injection field tests in diverse geologies to adequately characterize and validate the U.S. geologic resource” (Position Paper of the Coal Utilization Research Council on the Status of Technology for Carbon Capture and Storage (CCS), 2008).

Back to the chemistry...

Your reaction (as stated Uncle Al) is wrong because:

$$\ce{CaOH + CO2 -> CaCO3 + H2O}$$

Has $K_{eq}$ of $4.60 \times 10^{19}$! So goes in the other side of your reaction. I think that you are being tricked by a typical reaction with an acid, like this:

$$\ce{CaCO3 + 2HCl -> CaCl2 + 2H2O + CO2 ^}$$

However if you substitute HCl with carbonic acid you have a series of complex equilibria: $$\ce{CaCO3 + 2H2CO3 <=> Ca(HCO3)2 + H2O + CO2}$$ but: $$\ce{H2O + CO2 <=>H2CO3}$$ However the ratio between carbonic acid and carabon dioxide is very low: $$\frac{[H_{2}CO_{3}]}{[CO_{2}]} ≈ 1.7×10^{−3}$$ In fact carbon sequestration normally take advantage of adsorption of gasses into the soils, but the carbonatation occur very slowly.

By the way with carbon sequestration you will have a soil with a lower pH, more acid... and maybe sparkling tap water too! ;-)

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