The colourless gas that turns lime water milky and is taken in by plants during photosynthesis is carbon dioxide (CO2).

Here's why:

* Turns lime water milky: Carbon dioxide reacts with calcium hydroxide (the active ingredient in lime water) to form calcium carbonate, which is a white, insoluble solid. This is what makes the lime water turn milky.

* Photosynthesis: Plants use carbon dioxide along with water and sunlight to produce glucose (sugar) and oxygen through photosynthesis. The carbon dioxide provides the carbon atoms needed to build the sugar molecules.