Concentration-Time Graphs & Half-Life

The half life, t_1/2, of a reaction is the time taken for the concentration of the reactant to decrease to half its initial concentration
- It is a convenient way to determine how fast a reaction is, especially for first order reactions
For a first order reaction, t_1/2, is dependent only on rate constant (k) and given as:

t_1/2 = 0.693/k

It remains constant throughout the reaction
- For example, the half-life of the decomposition of dinitrogen pentoxide, which follows a first order kinetics, can be illustrated graphically

half-life

A graph showing the constant half-life of a first order reaction.

Radioactive decays are good examples of reactions that follow the first order kinetics
- The half-life of radioisotopes can be used to interpret the level of radiation of the radioisotope, for example:
  - Uranium- 238 has a very long half-life (4.5 × 10⁹ yrs), so it gives off radiation very slowly
  - While fermium-258 decays with a half-life of 3.8 × 10²⁴ s has a very high decay rate

For zero order and second order reactions, the half-life changes as the reaction progresses
- Half-life depends on the initial concentration of reactant
  - The lower the initial concentration, the longer the half-life
- For a zero order reaction, the relationship between t_1/2 and initial concentration is given as:

t_1/2 = [A]_initial / 2k

- For a second order reaction, the relationship between t_1/2 and initial concentration is given as:

t_1/2 = [A]_initial / k

half-life-for-zero-and-second-order-reactions

The value for the half life of zero and second order reaction change and are dependent on the initial concentration

The equation for the decomposition of dinitrogen pentoxide when heated in carbon tetrachloride is given below:

N₂O₅(g) → 2NO₂(g) + ½ O₂(g)

If the rate constant for the decomposition of N₂O₅ is 6.2 ×10^-4 /min, what is the half-life?

Answer:

Analyze:

We are given the rate constant for reaction and asked to determine the half-life

Plan:

First, we need to determine the order of the reaction from the unit of the rate constant.
- For first order reactions, the units of k include s^-1, min^-1 and hr^-1)
Substitute the k value into the t_1/2 expression for a first order reaction

Solution:

t_1/2 = 0.693/k

t_1/2 = 0.693/6.2 ×10^-4

t_1/2 = 1.12 × 10³ mins

Watch out for the unit of rate constant to help determine the order of the reaction when you’re not expressly told.

Remember: Rate is measured per unit time, e.g. per second, s^-1

So, when the rate is measured per second:
- Rate constant, k, units of a zero order reaction are M s^-1
- Rate constant, k, units of a first order reaction are s^-1
- Rate constant, k, units of a second order reaction are M^-1 s^-1

Concentration-Time Graphs & Half-Life (College Board AP Chemistry)