A-level Physics (Advancing Physics)/Half-lives/Worked Solutions

1. Radon-222 has a decay constant of 2.1μs⁻¹. What is its half-life?

$t_{\frac {1}{2}}={\frac {\ln {2}}{2.1\times 10^{-6}}}=330070{\mbox{ s }}=3.82{\mbox{ days}}$

2. Uranium-238 has a half-life of 4.5 billion years. How long will it take for a 5 gram sample of U-238 to decay to contain 1.25 grams of U-238?

2 half-lives, since 1.25 is a quarter of 5. 2 x 4.5 = 9 billion years.

3. How long will it be until it contains 0.5 grams of U-238?

First calculate the decay constant:

$\lambda ={\frac {\ln {2}}{t_{\frac {1}{2}}}}={\frac {\ln {2}}{4.5\times 10^{9}}}=1.54\times 10^{-10}{\mbox{ yr}}^{-1}$

$0.5=5e^{-1.54\times 10^{-10}t}$

$0.1=e^{-1.54\times 10^{-10}t}$

$\ln {0.1}=-1.54\times 10^{-10}t$

$t={\frac {\ln {0.1}}{-1.54\times 10^{-10}}}=14.9{\mbox{ Gyr}}$

4. Tritium, a radioisotope of Hydrogen, decays into Helium-3. After 1 year, 94.5% is left. What is the half-life of tritium (H-3)?

$0.945=e^{-\lambda \times 1}$ (if λ is measured in yr⁻¹)

$\lambda =-\ln {0.945}=0.0566{\mbox{ yr}}^{-1}={\frac {\ln {2}}{t_{\frac {1}{2}}}}$

$t_{\frac {1}{2}}={\frac {\ln {2}}{0.0566}}=12.3{\mbox{ yr}}$

5. A large capacitor has capacitance 0.5F. It is placed in series with a 5Ω resistor and contains 5C of charge. What is its time constant?

$\tau =RC=5\times 0.5=2.5{\mbox{ s}}$

6. How long will it take for the charge in the capacitor to reach 0.677C? ($0.677={\frac {5}{e^{2}}}$)

2 x τ = 5s