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All radioactive decay follow 1st order kinetics and this is where the similarity ends.
We measured the rate of reaction in chemical kinetics based on the rate of change of concentration of reactants or products.
But this procedure will not work for calculating the rate of radioactive reaction.
This is because most of the time the radioactive substance is a solid.
Therefore its concentration would be a constant with time (assuming it to be pure and that the product does not remain with the reactants).
Therefore the rate of radioactive reactions are measured by calculating the rate of change of nuclei of the radioactive substance. For a radioactive decay A ®B, the rate of reaction is calculated as
Where = decay constant of reaction
NA = number of nuclei of the radioactive substance at the time when rate is calculated.
As you can see the above rate law is very much similar to the rate law of a first order chemical reaction, but all other similarities ceases here.
For example unlike a chemical reaction the decay constant () does not depend on temperature. Arrhenius equation is not valid for radioactive decay.
Integrating the differential rate law we get
Where
No = number of nuclei of A at t = 0
Nt = number of nuclei of A at t = t
= decay constant
The expression can be rearranged to give
Nt = Noe-lt ... (1)
This suggests that the number of nuclei of radioactive substance A at any instant of time can be calculated, by knowing the number of nuclei at t = 0, its decay constant and the time.
Just like a 1st order reaction the half life of radioactive decay is given by
t1/2 = 0.069/
Note |
Let us start with 10 nuclei. If the half life is 5 minutes, then at the end of first 5 minutes, number of nuclei would be 5. |
The fact that radioactive decay follows the exponential law implies that this phenomenon is statistical in nature.
Every nucleus in a sample of a radionuclide has a certain probability of decaying, but there is no way to know in advance which nuclei will actually decay in a particular time span.
If the sample is large enough – that is, if many nuclei are present – the actual fraction of it that decays in a certain time span will be very close to the probability for any individual nucleus to decay.
To say that, a certain radioisotope has a half –life of 5 hr., then, signifies that every nucleus of this isotope has a 50 percent chance of decaying in every 5 hr. period.
This does not mean a 100 percent probability of decaying in 10 hr. A nucleus does not have a memory, and its decay probability per unit time is constant until it actually does decay. A half life of 5 hr. implies a 75% probability of decay in 10 hr., which increases to 87.5% in 15 hr, to 93.75% in 20 hr, and so on, because in every 5 hr. interval the probability of decay is 50 percent
Average life time is defined as the life time of a single isolated nucleus. Let us imagine a single nucleus which decays in 1 second.
Assuming 1 second time interval to be very small the rate of change of nuclei would be 1/1 (because –dN = 1 and dt =1).
We can also see that since
,
for a single isolated nucleus N = 1,
.
Therefore in this present case =1.
Now let us assume the same nucleus decays in 2 seconds, we can see that
i.e., is equal to ½. You will also notice that in the 1st case the nucleus survived for 1 second and in the second case it survived for 2 second. Therefore the life time of a single isolated
nucleus is 1/. hence, tav =
Activity by definition is the rate of decay of a radioactive element. It is represented as ‘A’ and is equal to N.
By no means should activity be confused with rate of change of radioactive nuclei represented by
.
This is because talks about the overall change in the number of nuclei in a given instant of time while activity only talks about that change which is decay.
But if the radioactive substance is also being produced this = rate of production – activity, (Of course it’s a different matter, rate of production may or may not be a constant).
This is activity per unit mass of the sample. Let radioactive sample weighing w gms have a decay constant .
The number of nuclei in the w gms would be (M/W)AV,
where M = molecular weight of the radioactive substance and Av = Avogadro’s number.
The unit of radioactivity of a substance is measured as the rate at which it changes into daughter nucleus.
It has been derived on the scale of disintegration of Radium.
Let us consider 1g of radium (atomic mass = 226 and t½=1600 yrs) undergoes decay, then
Rate of decay of radium = Number of nuclei of Ra in 1 g
Question 1: All radioactive decay follow
a. zero order kinetics
b. first order kinetics
c. second order kinetics
d. third order kinetics
Question 2: Activity per unit mass of the sample is called
a. specific activity
b. reactivity
c. activity
d. half life
Question 3: Life time of a single isolated nucleus is called
a. average life time
b. half life.
c. activity
d. specific activity
Question 4: Thy symbol represents
a. decay constant
b. half life
c.activity
Q.1 | Q.2 | Q.3 | Q.4 |
b | a | a | a |
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