## Half-Life ($T_{1/2}$) The time required for half of the original amount of a radioactive substance to decay into a more stable form. It is a characteristic property of each radioactive element and describes the rate at which a radioactive substance decays. A longer half-life implies a slower rate of decay and less radiation emitted over time. ## Decay Constant ($\lambda$) A constant that represents the probability per unit time for a nucleus to decay. It is directly related to the rate of decay. ## Relation between Decay Constant and Half-Life The number of un-decayed nuclei ($N$) at time ($t$) from an initial number ($N_o$) is given by the exponential decay law: $$N = N_o e^{-\lambda t}$$ At one half-life ($t = T_{1/2}$), the number of nuclei remaining is $N = N_o/2$. Substituting these values into the decay law: $$ \begin{aligned} \frac{N_o}{2} &= N_o e^{-\lambda T_{1/2}} \\ \frac{1}{2} &= e^{-\lambda T_{1/2}} \\ 2 &= e^{\lambda T_{1/2}} \end{aligned} $$ Taking the natural logarithm of both sides: $$\ln 2 = \lambda T_{1/2}$$ Thus, the relationship is: $$T_{1/2} = \frac{\ln 2}{\lambda} \quad \text{or} \quad T_{1/2} = \frac{0.693}{\lambda}$$ ## Graphical Representation of Nuclear Decay Figure 22.9 illustrates the exponential decrease in the number of un-decayed nuclei over time. The half-life can be visually determined from such a graph by finding the time it takes for the number of nuclei to drop to half of its initial value. ## Activity ($A$) The rate of decay of a radioactive sample. It is given by the formula $A = \lambda N$, where $\lambda$ is the decay constant and $N$ is the number of radioactive nuclei present at that instant. Units of activity include Becquerel (Bq) and Curie (Ci). ## Possible Questions and Answers **Q:** Cesium-137, a beta emitter, has a half-life of $9.5 \times 10^8$ s. What is its decay constant? **A:** Given $T_{1/2} = 9.5 \times 10^8$ s. Using the formula $\lambda = \frac{0.693}{T_{1/2}}$: $$\lambda = \frac{0.693}{9.5 \times 10^8 \text{ s}} = 7.3 \times 10^{-10} \text{ s}^{-1}$$ **Q:** For the Cesium-137 in the previous question, if we have one mole ($N_o = 6.02 \times 10^{23}$) at the start, what will be its initial activity? **A:** We found $\lambda = 7.3 \times 10^{-10}$ s$^{-1}$. Using the formula $A = \lambda N$: $$A = 7.3 \times 10^{-10} \text{ s}^{-1} \times 6.02 \times 10^{23} = 4.395 \times 10^{14} \text{ Bq}$$ (approximately $1.188 \times 10^4$ Ci) **Q:** The half-life of strontium (Sr-91) is 9.70 hours. Find its decay constant. **A:** Given $T_{1/2} = 9.70$ hours. Using the formula $\lambda = \frac{0.693}{T_{1/2}}$: $$\lambda = \frac{0.693}{9.70 \text{ hours}} \approx 0.0714 \text{ hours}^{-1}$$ Alternatively, converting to seconds: $T_{1/2} = 9.70 \times 3600 = 34920$ s: $$\lambda = \frac{0.693}{34920 \text{ s}} \approx 1.98 \times 10^{-5} \text{ s}^{-1}$$ ## Main Ideas - Half-life ($T_{1/2}$) is the time required for half of a radioactive substance to decay, acting as a characteristic property of the element. - The decay constant ($\lambda$) quantifies the rate or probability of nuclear decay. - Half-life and decay constant are inversely proportional, related by the formula $T_{1/2} = \frac{\ln 2}{\lambda}$. - Radioactive decay follows an exponential decrease in the number of un-decayed nuclei over time. - Activity ($A$) is the rate of radioactive decay, calculated as the product of the decay constant and the number of nuclei present ($A = \lambda N$). | Formula | Description | | :------ | :----------- | | $N = N_o e^{-\lambda t}$ | Exponential decay law (number of nuclei) | | $T_{1/2} = \frac{\ln 2}{\lambda}$ | Relation between half-life and decay constant | | $T_{1/2} = \frac{0.693}{\lambda}$ | Practical form of half-life relation | | $A = \lambda N$ | Activity of a radioactive sample | **Significance:** Understanding half-life is crucial for predicting the longevity and decay behavior of radioactive materials. It has wide applications, from medical treatments and industrial measurements to geological dating and managing nuclear waste.

22.4.2 Relation Between Decay Constant And Half Life | Nuclear Physics | Physics 12