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Straight off I want to say, I never took chemistry classes, so my understanding is very limited. My question is considering the equations below, why is it so hard to reduce greenhouse gases? Is it simply a matter of conversion requiring too much energy?
Can't we remove carbon dioxide & methane from the atmosphere by converting them into water and carbon?
$$\ce{CO2 + CH4 -> 2H2O + 2C}$$
Or by converting them into water and propane?
$$\ce{CO2 + 5CH4 -> 2H2O + 2C3H8}$$
There are a few problems with what you're proposing:
The energy problem can be solved by using surplus energy from renewable source like solar and wind, but, as airhuff also mentiond in a comment, there is much more CO2 in the atmosphere than methane. Also capturing CO2 from the air isn't simple, and finally why would you use a valuable fuel like methane to create carbon or propane?
In any case I think a much more useful reaction is $$\ce{CO2 + 2H2O -> CH4 + 2O2}$$
Water is split into oxygen and hydrogen using electrolysis. The hydrogen reacts with carbon-dioxide to form methane. There are already small plants that do this, but AFAIK the idea hasn't really caught on yet. This may have something to do with the cost and/or the availability of carbon dioxide or surplus energy.
More recently scientists did something even cooler; they captured CO2 directly from the atmosphere and turned it into methanol (CH3OH) using Ruthenium as a catalyst. A drawback is that the reaction works at high temperatures (about 150 degrees C).
I've been hearing about companies able to extract and concentrate carbon dioxide from ambient air, and was thinking about how it might be sequestered on sufficient scale to remove all excess carbon dioxide from the atmosphere. Pumping the carbon dioxide underground or at the bottom of the ocean would probably work but raises concerns about stability and leakage. It could be reacted with lime or other minerals underground to form stable minerals but the completeness of the reaction would be difficult to monitor. Converting it into methane makes the problem bigger, since then the methane has to be sequestered and it is a more potent greenhouse gas than carbon dioxide.
The carbon dioxide could be converted into carbon, such as by turning organic matter into biochar, and then it would be stable. But why reduce carbon dioxide all the way to carbon ($\ce{CO2 + 2 H2 -> C + 2 H2O}$) when it could be left at oxalic acid ($\ce{2CO2 + H2 -> C2O4H2}$), another stable solid? This could be accomplished electrochemically.
If we were to return the carbon dioxide in our atmosphere to pre-industrial levels (i.e., $\pu{280 ppm}$), it would require the removal of
$$(\pu{410 ppm} - \pu{280 ppm}) \cdot \frac{\pu{2.13 Gt C}}{\pu{1 ppm CO_{2}}} \cdot \frac{\pu{3.664 Gt CO_{2}}}{\pu{1 Gt C}} = \pu{1.01 Tt}$$
of carbon dioxide. This carbon dioxide could be converted into $\pu{1.04 Tt}$ of oxalic acid, with a volume of
$$\pu{1037797146168 t}\cdot\pu{907185 g/t}\cdot\frac{\pu{1 cm3}}{\pu{1.9 g}}\cdot\frac{\pu{1 m3}}{\pu{1000000 cm3}}\cdot\frac{\pu{1 km3}}{\pu{1000000000 m3}} = \pu{496 km3}$$
of clean, beautiful former coal. If we assume that the shape of this mountain approximates that of a pile of salt (where $\pu{1000 t}$ of salt forms a conical pile $67’1”$ in diameter and $40’$ along its slope having a volume of $\pu{25000 ft3}$), then we can calculate the height of that mountain using the fact that the ratio of the radius of a cone of a given shape to its height ($h^2 = \text{slope}^2 – r^2$) is constant, which is
$$\frac{r}{h} = \frac{33.54’}{21.80’} = 1.54.$$
The equation for a cone is
$$V = \frac{πr^2h}{3},$$
or
$$V = \frac{π(1.54h)^2h}{3} = 2.48h^3.$$
Solving for $h$ where $V = \pu{496 km3}$ gives $h = \pu{5.85 km}$, with $r = \pu{9.00 km}$. For comparison, Mt. Fuji is $\pu{3.78 km}$ high and a radius of about $\pu{22 km}$ at its base. Once paved to prevent it from dissolving in the rain, it could provide a convenient skiing location.
Of course, this still leaves the problems of building the factory to do this and supplying renewable energy to run the factory. This article proposes electrochemically reducing carbon dioxide into stable, storable forms to be sequestered, with a potential to reduce $\pu{967 g}$ of carbon dioxide to oxalic acid per kilowatt-hour of electricity. That comes out to at least:
$$\pu{1037797146168 t oxalic acid} \cdot \frac{\pu{1000000 g}}{\pu{1 t}} \cdot \frac{\pu{1 g CO_{2}}}{\pu{1.023 g oxalic acid}} \cdot \frac{\pu{1 kWh}}{\pu{967 g CO_{2}}} = \pu{1.05 * 10^15 kWh} = \pu{1.05 EWh}$$
of electricity needed to sequester the above excess carbon dioxide from the air (not counting the oceans) as oxalic acid.
