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The reason why soda water is better than plain water for plants is it contains phosphate and other nutrients that enrich the soil and promote growth. In one study done at University of Colorado Boulder, researchers found that plants given carbonated water had shoots that grew 170 per cent of their original height, compared to just 67 per cent for plants given tap water. The soda water-fed plants also developed a healthier shade of green.
As Gardening Know How points out, you don't want to feed your plants regular soft drink because of all the sugar -- and mineral water might contain other elements that aren't helpful.
Is Soda Pop a Fertiliser? Information About Pouring Soda on Plants [Gardening Know How]