Half Life:
We know that radioactive elements always disintegrated and as the time passes the number of united nuclei decreases. “Half life of a radioactive sample is the time to reduce the nuclei to half of its initial value.” Half life is denoted by T. Its value for an element is constant and varies for different elements. Half life does not depend on the quantity of substance. It cannot be replaced by physical and chemical effects.
The half life of some elements are given below:
| S. No. |
Symbol |
Half-life |
| 1. |
Uranium (92U238) |
4.5 × 109 year |
| 2. |
Thorium (90Th 230) |
8 × 104 year |
| 3. |
Radium (88Ra 236) |
1620 year |
| 4. |
Bismuth (83Bi218) |
3 min |
If a radioactive element has a half life of T then after T times it will have 50% of its initial value, after 2T times 25% and after 3T times 12.5%, after 4T time 6.25% remaining.
If the graph plotted between the number of nuclei and time, then the graph is shown in a figure
Let the number of nuclei in beginning is N0 that is N = 0 at t = 0. So remaining nuclei after a half life (t = T),
Similarly after three half lives (t = nT), remaining nuclear
Nn = N0 \(\left(\frac{1}{2}\right)^{n}\) ……………… (1)
∵ t = nT
Using n = \(\frac{t}{T}\), the value of n can be determined
Relation between Half-life and Decay constant
If the number of neclei in the beginning is N0 then the number of remaining nuclei after time t
N = N0e-λt
Taking log on both sides
loge = logeeλt = λt loge e = λt
or λt = loge 2
or T = \(\frac{\log _{e} 2}{\lambda}\) …………… (2)
With the help of this equation we can find out the value of λ it value of T is known or vice-versa,
∵ loge 2 = 0.6932
∴ T = \(\frac{0.6932}{\lambda}\) …………. (3)
or λ = \(\frac{0.6932}{T}\) …………….. (4)
