Half Life — Revision Notes

⚡ 30-Second Revision

Definition: — Time for half of radioactive nuclei to decay.
Symbol: — $T_{1/2}$
Decay Law: — $N(t) = N_0 e^{-lambda t}$
Remaining after 'n' half-lives: — $N = N_0 \left(\frac{1}{2}\right)^n$
Relation to Decay Constant: — $T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$
Relation to Mean Life: — $T_{1/2} = \tau \ln 2 \approx 0.693 \tau$
Activity: — $A = \lambda N$, also decays exponentially $A = A_0 \left(\frac{1}{2}\right)^n$
Independence: — Unaffected by T, P, chemical state.

2-Minute Revision

Half-life ($T_{1/2}$) is the fundamental measure of how quickly a radioactive substance decays. It's the specific time period after which half of the initial radioactive nuclei in a sample will have transformed into other elements.

This process follows an exponential decay law, $N(t) = N_0 e^{-lambda t}$, where $N_0$ is the initial number of nuclei, $N(t)$ is the number at time $t$, and $lambda$ is the decay constant. A key relationship for NEET is $T_{1/2} = \frac{ln 2}{lambda}$, meaning half-life is inversely proportional to the decay constant.

For problems involving integer half-lives, the formula $N = N_0 (1/2)^n$ is very useful, where $n = t/T_{1/2}$. Remember that activity ($A = lambda N$) also halves with each half-life. Crucially, half-life is an intrinsic nuclear property, unaffected by external factors like temperature or pressure.

Don't confuse it with mean life ($au = 1/lambda$), which is the average lifetime of a nucleus and is longer than half-life.

5-Minute Revision

To master half-life for NEET, start with its core definition: the time it takes for half of a radioactive sample to decay. This isn't about the total mass disappearing, but specifically the number of *unstable nuclei* reducing by half.

The underlying principle is the law of radioactive decay, which is exponential. The number of nuclei remaining at time $t$ is given by $N(t) = N_0 e^{-lambda t}$, where $lambda$ is the decay constant.

The half-life is directly linked to $lambda$ by the formula $T_{1/2} = \frac{ln 2}{lambda}$. Since $ln 2 \approx 0.693$, this simplifies to $T_{1/2} \approx \frac{0.693}{lambda}$.

For practical problem-solving, especially when the total time is an integer multiple of the half-life, use the formula $N = N_0 (1/2)^n$, where $n$ is the number of half-lives ($n = t/T_{1/2}$). This formula is also applicable to activity, so $A = A_0 (1/2)^n$. For instance, if a sample has a half-life of 5 years and you want to know how much remains after 15 years, $n = 15/5 = 3$. So, $N = N_0 (1/2)^3 = N_0/8$.

Remember the distinction between half-life and mean life ($au$). Mean life is the average lifetime of a nucleus, $au = 1/lambda$, making it approximately $1.443$ times the half-life. Half-life is a nuclear property, meaning it's constant for a given isotope and unaffected by external conditions like temperature, pressure, or chemical bonding.

These conceptual points are common traps in MCQs. Practice problems involving both number of nuclei and activity, and ensure you're comfortable with logarithmic calculations.

Prelims Revision Notes

Half-Life ($T_{1/2}$) - NEET Revision

1. Definition: Time required for half of the radioactive nuclei in a sample to decay.

2. Key Formulas:

* Radioactive Decay Law: $N(t) = N_0 e^{-lambda t}$ * $N_0$: Initial number of nuclei * $N(t)$: Number of nuclei remaining at time $t$ * $lambda$: Decay constant (probability of decay per unit time) * Half-life and Decay Constant: $T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.

693}{\lambda}$* **Number of Half-lives:**$n = \frac{t}{T_{1/2}}$* **Remaining Nuclei after 'n' half-lives:**$N = N_0 \left(\frac{1}{2}\right)^n$* **Activity ($A$):** Rate of decay,$A = -\frac{dN}{dt} = \lambda N$* **Activity Decay:**$A(t) = A_0 e^{-lambda t} = A_0 \left(\frac{1}{2}\right)^n$* **Mean Life ($ au$):** Average lifetime of a nucleus,$\tau = \frac{1}{\lambda}$* **Relation between Half-life and Mean Life:**$T_{1/2} = \tau \ln 2 \approx 0.

3. Important Points:

* Constant: $T_{1/2}$ is constant for a given radionuclide. * Independence: $T_{1/2}$ is independent of temperature, pressure, chemical state, and initial amount. * Exponential Decay: The amount of radioactive substance never truly reaches zero; it approaches asymptotically. * Units: Ensure consistency (e.g., if $lambda$ is in $s^{-1}$, $T_{1/2}$ will be in seconds).

4. Common Traps:

* Confusing $T_{1/2}$ with $\tau$. * Assuming complete decay after a few half-lives. * Errors in logarithmic or exponential calculations. * Incorrect unit conversions.

5. Application: Carbon dating, medical imaging, nuclear power.

Vyyuha Quick Recall

Half-life Links Nuclei Decay Constantly.

Half-life: $T_{1/2}$
Links: $\ln 2$
Nuclei: $N = N_0 (1/2)^n$
Decay Constantly: $T_{1/2} = \frac{\ln 2}{\lambda}$ (Decay Constant $\lambda$)