Atoms & Nuclei — Advanced Multi-Step Practice (Hard)

Quick Recap — Advanced Atoms & Nuclei (Multi-step)

• Hydrogen-like transitions: $\Delta E=13.6\,Z^2\left(\dfrac{1}{n_1^2}-\dfrac{1}{n_2^2}\right)$ eV, then $\lambda=\dfrac{1240}{\Delta E}$ nm.
• de Broglie: $\lambda=\dfrac{12.27}{\sqrt V}$ Å (electron through $V$ volts); photon momentum $p=\dfrac{E}{c}$.
• Decay maths: $N=N_0\left(\tfrac12\right)^{t/T}$; the number of half-lives $=\dfrac{t}{T}$; activity ratio equals the surviving-fraction.
• Q-value: $Q=[\text{(mass of reactants)}-\text{(mass of products)}]\,c^2$; positive $Q$ means energy is released.
• Series ratios: the longest Balmer line ($3\to2$, $656$ nm) versus the Lyman-$\alpha$ line ($2\to1$, $122$ nm) illustrate the $Z^2$ and level-difference scaling.

Worked mini-example. A sample decays to $\tfrac{1}{16}$ of its initial amount in 40 minutes. Since $\tfrac{1}{16}=\left(\tfrac12\right)^4$, that is 4 half-lives, so $T_{1/2}=\dfrac{40}{4}=10$ minutes.