# Intro - Half-Life (3) *Tell me if HL3 releases in the future and this title might be void!* Radiation's one of the ways we psychotic children enter the wild world of physics - one book, tv show depicting nuclear bombs later, you're now trapped in a cubicle, somehow embedded squarely in the heart of academia. It's the sheer destructive power of the nuclear weapon that can help spur on further waves of curiosity from prospective physics students. In doing so, one term that's likely to be repeated - "half-life". So what really is "Half-Life?" No, I'm not talking about the video game series, I'm talking about **the time it takes for half of a radioactive sample to decay.** This article will cover everything decay-related that a high-schooler might need. Good luck! Don't let nuclear fallout turn you into a ghoul. # The Radioactive Decay Equation The equation to find the **number of particles left of a radioactive sample** $N$ is given as: $N = N_{0}e^{-\lambda t}$ Where: - $N_0$ is the original number of particles in the sample - $\lambda$ is the decay constant - $t$ is the time in days >[!Success]- Deriving the Equation > >We know that at the start (where $t = 0$) that: >$N = N_{0}$ >The rate of decay (or the *activity of the sample* $A$) after a certain time $dt$ is a differential equation relative to the original number of particles (as different samples will have different rates!): >$A = -\frac{dN_{0}}{dt}$ >This is measured with the **becquerel** - 1 nucleus disintegration per second. Another unit is the *Curie*, given as $3.7 \times 10^{10} Bq$. > >$\lambda$, or the *decay constant*, is the fraction of nuclei that will decay over time, so we divide the rate by the original to get it: >$\lambda = -\frac{\frac{dN_{0}}{dt}}{N_{0}}$ >Therefore the rate $R$ can be expressed as: >$A = -\frac{\frac{dN_{0}}{dt}}{N_{0}} \times N_{0} = -\lambda \times N$ >Form a differential equation using $R$: >$\frac{dN_{0}}{dt} = -\lambda \times N$ >Solve it! >$\int \frac{1}{N} \, dN = \int -\lambda \, dt$ >$\ln N = -\lambda t+c$ >To find $c$, set the time to be $0$ (see [[Boundary Value Problems (Maths)#Intro - Initial Value Problem|initial value problems]] for reference), giving us: >$C = \ln N_{0}$ >Substitute: >$\ln N_{0} - \lambda t = \ln N$ >Simplify to get: >$N = N_{0}e^{-\lambda t}$ # More about Half-Life *Go back and check out the derivation for the radioactive decay equation first!* Now that we've established that the activity of the radioactive sample is $A = \lambda N_0 e^{-\lambda t}$, we can attempt to solve for the half-life. ![[nuclei-decay-graph.png]] Odds are that you've seen a graph similar to this. This is a counts against time graph - the sample has an initial number of nuclei of $N_0$ that decays with time. To find the half-life, set the number of target nuclei to $\frac{N_0}{2}$: $\frac{N_{0}}{2} = N_{0}e^{-\lambda T^{\frac{1}{2}}}$ Where $T^{\frac{1}{2}}$ refers to the half-life. Therefore: $\frac{1}{2} = e^{-\lambda T^{\frac{1}{2}}}$ "ln" both sides of this equation... $\ln\frac{1}{2} = -\lambda T^{\frac{1}{2}}$ This equals to... $-\frac{\ln 2}{-\lambda}=T^\frac{1}{2}$ Which therefore gives us an equation for the half-life! $T^\frac{1}{2} = \frac{\ln 2}{\lambda}$