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The half-life chemistry also known as the reaction half-life, t1/2, is defined as the time taken for a reactant concentration to decrease by half in comparison to its initial concentration. In chemistry and medicine, the half-life application is used to anticipate the concentration of a chemical over time. Half-life is used to determine how quickly a chemical decreases in the target once it has been absorbed over a period of time (sec, minute, day) or the elimination rate constant 'k' (minute-1, hour-1, day-1).
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Key Terms: Half-life chemistry, Half-life formula, Rate constant 'k',effect of half-life, Initial concentration, Zero-Order Reaction, First-order reactions, Second order Reaction, Arrhenius Equation
Half-Life Formula
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It's important to remember that a reaction's half-life formula changes depending on the order of the reactions. The formula for the half-life of different reactions is given below.
- The half-life of a zero-order reaction, the formula is given as t1/2 = R0/2k
- The half-life of a first-order reaction is given as t1/2 = 0.693/k.
- The half-life of a second-order reaction is given by the formula 1/kR0.
The video below explains this:
Half-Life Formula Detailed Video Explanation:
Where,
The half-life of a reaction is referred to as t1/2 (unit - seconds)
The initial reactant concentration is referred to as R0 (in mol.L-1 or M) is [R0], and
The reaction's rate constant is given as k (unit - M(1-n)s-1, where 'n' is the reaction's order)
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Derivation of Half-Life Formula for Zero Order Reaction
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The rate constant for a zero-order reaction is measured in mol.L-1.s-1.
The following is the formula for a zero-order rate constant:
k=[R]0−[R ]/ t
By substituting t = t1/2, [R] = [R]0/2 is obtained (at the half-life of a reaction, reactant concentration is half of the initial concentration):
k=[R]0−[R]0/2 / t1/2
The half-life of a zero-order reaction can be calculated by rearranging the equation:
t1/2 = [R]0 / 2k
Derivation of Half-Life Formula for First Order Reaction
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The following is the formula for a first-order rate constant expressed mathematically:
k = 2.303/t log [R]0/[R]
At t = t1/2, [R] = [R]0/2, according to the reaction half-life definition. The following equation is obtained by substituting these values in the expression for the first-order rate constant:
k=2.303/t1/2 log[R]0 / [R]0/2
Rearranging the expression:
t1/2 = 2.303 log(2) / k = 0.693/k
Thus, the half-life of a first-order reaction is obtained as 0.693/k.
Effect of Half-Life
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- The radioisotope component has a physical Half-life, which means that the amount of radioactivity drops by half with each Half-life. Tc99m, a common radioisotope with a half-life of 6 hours, is an example of a common radioisotope. If it holds 100 MBq of this isotope right now, it will contain 50 MBq in 6 hours and 25 MBq in the next 6 hours.
- The patient can also excrete a little amount of the radiopharmaceutical, usually through the urine and kidneys, thus there is a certain amount that the body gets rid of, and so the radiation can be considered outside the body. If this occurs, that portion of the patient's radiation dose is no longer affected. The biological half-life is the term for this.
Things to Remember
- The half-life chemistry or reaction half-life, t1/2, is defined as the time it takes for a reactant concentration to decrease by half when compared to its initial concentration.
- Half-life is used to determine how quickly a chemical decreases in the target once it has been absorbed over a period of time (sec, minute, day) or the elimination rate constant 'k' (minute-1, hour-1, day-1).
- A reaction's half-life formula changes depending on the order of the reactions.
- The half-life of a zero-order reaction, the formula is given as t1/2 = R0/2k, The half-life of a first-order reaction is given as t1/2 = 0.693/k, The half-life of a second-order reaction is given by the formula 1/kR0.
- The rate constant for a zero-order reaction is measured in mol.L-1.s-1.
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Sample Questions
Ques. Half life period of a radioactive element is 10 years. Calculate its disintegration constant & average life? (2 marks)
Ans: Given, t1/2 =10 years
k = 0.693 / t1/2 =0.693 /10
K= 0.0693
Average life = 1.44 × t1/2=1.44×10
Average life = 14.4 years
Ques. The Half-life period of U234 is 2.5×105 years. In how many years it will remain 25% of its original amount? (3 marks)
Ans: Given t1/2 =2.5×105 years
No = 100 gm
N = 25 grams
N= N0(1/2)n
25 = 100 (1/2)n
25 / 100 =(1/2)n
(1/4 ) =(1/2)n
(1/2)2 =(1/2)n
n=2
T=n×t1/2 = 2×2.5×105
T=5×105 years
Ques. 10-gram thorium remains 5 grams in 24 days. How much of thorium remains in 48 days? (2 marks)
Ans: t1/2=24 days
T=n×t1/2
n =48/24 =2
N= N0(1/2)n
N= 10 (1/2)2
N = 2.5 grams
Ques. Half life of 83 I125 is 60 days.How much of its radioactivity remains after 180 days? (2 marks)
Ans: Given,
t1/2 = 60 days
T = 180 days
Let initial reactivity ‘No’ = 100 gm
T=n×t1/2
n = T / t1/2
n =180 / 60 = 3
N= N0(1/2)n
N = 100 (1/2)3 =100 (1/8) = 12.5
N = 12.5 %
Ques. The activity of a radioactive element remains 12.5% in 90 days. Calculate the disintegration constant of the element? (3 marks)
Ans: Given,
T = 90 days
No = 100 gm
N = 12.5 grams
N= N0(1/2)n
12.5 = 100 (1/2)n
12.5 / 100 =(1/2)n
(1/8 ) =(1/2)n
(1/2)3 =(1/2)n
n=3
T=n×t1/2
90 = 3 × t1/2
t1/2 =90/3 = 30 days
K =0.693/t1/2 = 0.693 / 30
K = 0.0231 days-1
Ques. Starting with 16 atoms of a radioactive element, how many atoms are left after 4 half-lives? (2 marks)
Ans: No = 16 atoms
N = ?
n = 4
N= N0(1/2)n
N = 16 (1/2)4 = 16 (1/16) = 1
N = 1 atom
Ques. Half life of 1H3 is 13.3 years. How much of 1H3 should be taken initially so that it remains 4 Kg after 26.6 years ? (3 marks)
Ans: Given,
T = 26.6 years
t1/2 = 13.3 years
T=n×t1/2
n = T / t1/2 = 26.6 / 13.3 = 2 half lives
N = 4 Kg
N= N0(1/2)n
4 = N0 (1/2)2
4 = N0 /4
N0 = 16 kg
Ques. How much of a radioactive element is left after 5 half lives? (2 marks)
Ans: Initial amount = N0
n = 5
N= N0(1/2)n
N = N0 (1/2)5 = N0 (1/32) = N0 / 32
N = N0 / 32
The amount left after 5 half-lives is 1/32 of the initial amount.
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