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?

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

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:

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

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

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

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

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

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)

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

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

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?

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

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