Is this possible as an inexhaustible supply of rocket fuel

The molecules put in the rocket are the same that shoot out the back, but the chemical bonds have changed. It would take energy to reset the chemical bonds from the exhaust into what they were for the fuel.

"the collecting is done by a nozzle which has no detrimental effect on the thrust of the rocket"

Not possible. The thrust is generated by pushing the particles behind the rocket. If the rocket is catching the particles, it's pushing against itself and there is no thrust.

Ignoring that, it still won't work. The energy required to recombine the burned particles back into their original form so that they can be burned again must be greater than the energy gained from burning them.

So there's a physics reason and a chemistry reason why it won't work.

Algae fuels. you may construct a greenhouse form of farm interior the barren region to enhance algae in great shallow swimming pools of water to enhance a crop of algae a week or so year around non-give up. turn algae into biodiesel for use in any diesel motor vehicle. you may recycling the carbon so which you does not be including any carbon into the ambience nor will you ought to be based on the middle east for oil, thereby reducing the reasons to get ourselves in contact in that mess and because no one lives interior the barren region (rarely no one) and it particularly is does not fertile farmlands, it won't impression nutrition ingredients. Politicians and their hobbies prevents this from all happening, does not remember in the event that they are democrats or republicans, they are all went to the comparable regulation faculties. some inner maximum companies have already finished this and could make gas for about $a million.50 a gallon as quickly as mass production is underway are prevented from advertising this gas to the favourite public by the government stating "further learn" aspiring to be finished.

Violates conservation of momentum. To stop these high speed particles for recycling causes a thrust in the wrong direction.

Nope, that's a perpetual motion machine. It requires energy from somewhere to collect the exhaust. Where is that energy coming from?

Sounds too much like a perpetual motion machine.

It won't work because there's a "law with which all energy must agree. Whenever it changes form, it loses quality (in other words...) Damn that rising entropy!"