Quicklime as heat source and scrubber

Many years ago, I researched a system of capturing carbon dioxide from the atmosphere.  It was called quicklime.  It is a very old, very well-known process going back to Roman times at least.  It is produced from heated limestone in kilns and hardens once again into limestone after it absorbs carbon dioxide from the air.  It is a neatly reversible process.

It was probably a wildly impractical idea for carbon capture, as there is but trace amounts of the stuff in the atmosphere.  But what about a carbon dioxide rich source, like which could be obtained from heating ammonium carbonate?

Now, here's what could happen, if everything checks out ( cross your fingers ).  The quicklime will capture the carbon dioxide and leave the ammonia behind.  The output will be hot ammonia gas.

Now here's how: quicklime reacts with water and produces heat.  That is handy as a source for heat to decompose the ammonia carbonate.  Run a heat exchanging gas to capture the heat and use that heat to decompose the ammonia carbonate.  Run the resultant mixture of ammonia and carbon dioxide through the quicklime and remove the carbon dioxide.

Then, you can send the ammonia to the cracker unit and obtain hydrogen for the fuel cell that powers the auto.