"Carbon capture into fuel" - where does the energy come from? I recently found an article titled "Scientists find a way to convert Carbon Dioxide from air into fuel" (here), and my immediate reaction was "surely not - the energy balance would kill you right away". It seems, however, that this is a serious proposal, with a piece in Phys.org and a legitimate journal article,

Conversion of CO2 from Air into Methanol Using a Polyamine and a Homogeneous Ruthenium Catalyst. J. Kothandaraman et al. J. Am. Chem. Soc. 138 (3), pp 778–781 (2016).

The InterestingEngineering and Phys.org pieces talk mostly about the new catalysts found by Kothandaraman and co-workers, and the journal article goes rather above my head - which I suspect is also the case for most of the general public.
So, in the interest of having someone somewhere make it clear that energy is always conserved and that miracle schemes always have their cost: where does the energy come from in the carbon-capture scheme proposed by Kothandaraman et al.? In what resources is that energy stored, how much energy do those resources have, and how would those be created?
 A: This is very similar to how plants capture CO2 and form "fuels" (sugars) to feed themselves.  In the natural case, the energy comes from sunlight captured by the chlorophylls in the plant cells, and the chemical reaction is carried out by a group of enzymes (Photosystem I and Photosystem II).
A lot of scientists are trying to replicate this process industrially using artificial catalysts, and this seems to be what your source is talking about.  In this case, there are several possible sources for the energy.  The most common are sunlight or electricity - ultimately these chemical reactions are due to the transfer of electrons, and an applied electric potential can drive this.  This is what the sunlight approach does as well - an electron is excited by a photon, and ultimately this leads to the application of a potential that drives the reaction.
As far as how much energy is needed... well, you have to consider the whole picture to understand that.  There's another side to the process, and that is the conversion of water into oxygen gas (this is why plants release oxygen into the atmosphere!).  The potential difference across these two processes turns out to be +1.23 V - this assumes 100% thermodynamic efficiency.  Most real systems are significantly higher (+1.7 to +1.8 V) because some of the energy is lost as heat, and so a lot of current research is focused on lowering this gap to the ideal +1.23 V.
EDIT: I apologize, the +1.23 V is for the reduction of H+ to H2.  The reduction of CO2 to CH3OH is closer to +1.6 V.
