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What is the difference in kinetics between a 0th, 1st and 2nd order reaction?

The rate of every reaction, R, relies on many different things including pressure, temperature or concentration of the reactants. To simplify this everything expect for concentration are included in the rate constant, k.

Let's now go through the different reaction orders:

In a 0th order reaction R only depends on k, not on the concentration of any of the reactants:
R = k

In a 1st order reaction R depends on k and on the concentration of one reactant, A:
R = k*[A]

In a 2nd order reaction R depends on k and on the concentration of two reactants, A and B:
R = k*[A]*[B]

In this reaction it is also possible that A and B are two of the same molecules, let’s say A and A, which collide to form the products:
R = k*[A]*[A] which is the same as R = k*[A]2

From the different rate of reaction equations there are some things you can note:

In a 0th order reaction the concentration of the reactants does not matter. When you change the concentration the rate of the reaction will stay the same.

In a 1st order reaction the rate relies on the concentration of one of the reactants. When you increase this concentration the reaction will go faster and when you decrease it the reaction will go slower.

In a 2nd order reaction the rate relies on two different concentrations. When you increase either one the rate of the reaction will increase and when you decrease either one the rate will decrease. Things get a bit trickier when you increase one and decrease the other and it will depend on how much you increase/decrease them whether the rate goes up or down.

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