Half-life

Quick
Half-life is a characteristic property of a nuclide.


Equations
(Eq1)    
t1/2 =
0.693
k
 Half-life of a reaction


Nomenclature
t1/2time required for one-half of the quantity of reactant originally present to react; half-life of a reaction
krate constant for the reaction
Ntfinal quantity amount
N0original quantity amount
tif a certain amount of a quantity is left to be reacted, t is the amount of time it took to get to that point


Details

The half-life of a reaction is the time required for one-half of the quantity of reactant originally present to react. The half-life is an important quantity for first-order reactions because it is constant. For first-order reactions, the half-life does not depend on the concentration of the reactant.

The half-life and the value of the rate constant for first-order reactions are related:

(Eq1)    
t1/2 =
0.693
k

Eq1 can be used to calculate k from the half-life of a first-order reaction.

Because the half-life for a first-order reaction does not depend on initial concentration, the concentration (or amount) remaining after a whole number of half-lives can be calculated very simply.

Calculating the number of half-lives needed to reach a given concentration, such as the minimum concentration for a drug to be effective, is equally simple.

If the time you are interested in is not a whole number of half-lives, you can use the integrated rate law. From the integrated rate law, given the value of the rate constant, you can calculate:
1. The time required for an initial concentration to decrease to a specified concentration.
2. The concentration that will be reached in a given time.
3. The initial concentration if the concentration after some period of time is known.
The rate constant k for a radioactive decay is a measure of the probability that a nucleus will decay in a unit of time. Information about the rates of radioactive decay reactions is usually tabulated in the form of the half-lives; half-life is a characteristic property of a nuclide. Because radioactive decay is a first-order process, the value of the rate constant k can be calculated from the half-life by means of Eq1. The average lifetime of a radionuclide is the reciprocal of k:

average lifetime of a radionuclide =
1
 k

If the sample is large, that is, if many nuclei are present, the amount of radionuclide Nt left after time t has passed can be calculated by means of the integrated rate equation for first-order processes:

(Eq2)    
ln(
Nt
N0
)  =  −kt

In Eq2, N0 is the amount of radionuclide present at the start of the experiment and the quotient Nt /N0 is the fraction of the original quantity of radionuclide left after time t. As long as the same units are used at time t as at time 0, units cancel from the ratio Nt /N0. The quantity of radionuclide can be expressed in any one of a variety of units: number of atoms, counts, mass, moles, molarity.

Another useful equation can be obtained by combining Eq1 and Eq2. First rearrange Eq1 so that:

k  =  
0.693
t1/2
  =  
−ln 0.5
t1/2

Adding a −t to both sides:

(Eq3)    
−kt  =  
t ln 0.5
t1/2

Now setting the left side of Eq2 to the right side of Eq3:

ln(
Nt
N0
)  =  
t ln 0.5
t1/2

From natural logarithm properties:

Nt
N0
  =  exp(
t ln 0.5
t1/2
)

Remember exp(x) is the same as ex. Now, simplifying the right-hand side:

Nt
N0
  =  exp(
t ln 0.5
t1/2
)  =  (exp(ln 0.5))
t
 t1/2
 
  =  (
1
2
)
t
 t1/2
 

Then:

(Eq4)    
Nt
N0
  =  (
1
2
)
t
 t1/2
 

This can be a useful equation if you are given three of the four quantities in Eq4 and need to find the remaining one. See the example problem by following the link below.


Related
▪ P - Half-life Time