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

'''1. Radon-222 has a decay constant of 2.1&mu;s−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 &lambda; 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 5&Omega; 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 &tau; = 5s