# 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?

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

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:

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

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

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

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

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

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)?

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

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

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

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?

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

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

2 x τ = 5s