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What is the energy input needed to break the phosphor bond during ATP Dephosphorylation? How and when this will occur?

Many thanks for your answers.

Edit 1: I know how much free energy (∆G) is released with the hydrolysis of ATP, but I dont know how much energy is needed to start the hydrolysis of ATP.

Updated Question/s: To break the phosphorus bond of ATP, must there be some input energy? What is the amount of this energy? Else how can break any bond of ATP without the supplement any energy?

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It sounds like you are asking about the activation energy of the ATP hydrolysis reaction. This paper estimates the activation at 35--39 kcal / mol, depending on details of the reaction mechanism. So yes, energy is needed to break the phosphate bond; this ensures that ATP does break down spontaneously (which would make it a rather useless molecule for cells).

Note that this concerns ATP hydrolysis without enzymatic catalysis. Enzymes that break the phosphate bond of ATP to drive reactions lower this reaction energy substantially.

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