where k is the rate constant, [X] is the molar concentration of the reactant, and rate measures the disappearance of X in units of M/s (or M/min or M/hour…)
If this equation is integrated, it can be reworked to give either of these equations:
where [X] is the current concentration, [X]0 is the initial concentration, k is the rate constant and t is the amount of time that has passed.
Both of these are mathematically identical. Which you prefer is personal choice (or perhaps the choice of your mathematics instructor). I will be using the first of the two on this page.
As we know, k is a constant. That means that if you put in [X] = 1M and [X]0 = 2 M and solve for t, you will get the same value as if you put in [X] = 4000M and [X]0 = 8000M. In fact, no matter what the actual values are put in the equation, any ration of 1:2 will always give the same value for t. THIS is the half-life of the reaction.
Another way to think about the equation is that, once you know the value of k, the equation contains 3 variables ([X], [X]0 and t). That means that given the initial and current concentrations you can determine the amount of time that has passed. (This is how radio-isotope dating is done). Or given the initial concentration and the amount of time, you could determine the amount remaining. You can even figure out the initial amount if you are given the current amount and the time that has passed.
Let’s try a sample problem:
A bone is discovered that has 37% on the normal amount of 14C. Given that the
half-life of 14C is 5260 years, how old is the bone?
First, given the half-life we can determine the value of k.
We can start with the equation
which rearranges into the equation
Substituting in numerical values gives us
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