# 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