User:Inconspicuum/Physics (A Level)/Decay Questions

Exponential Relationships

1. Simplify Newton's Law of Cooling for the case when I place a warm object in a large tank of water which is on the point of freezing. Measure temperature in °C.
2. What will the temperature of an object at 40 °C be after 30 seconds? (Take r=10^-3 s^-1.)
3. A body is found in a library (as per Agatha Christie) at 8am. The temperature of the library is kept at a constant temperature of 20 °C for 10 minutes. During these 10 minutes, the body cools from 25 °C to 24 °C. The body temperature of a healthy human being is 36.8 °C. At what time was the person murdered?
4. Suppose for a moment that the number of pages on Wikibooks p can be modelled as an exponential relationship. Let the number of pages required on average to attract an editor be a, and the average number of new pages created by an editor each year be z. Derive an equation expressing p in terms of the time in years since Wikibooks was created t.
5. Wikibooks was created in mid-2003. How many pages should there have been 6 years later? (Take a = 20, z = 10 yr^-1.)
6. The actual number of pages in Wikibooks in mid-2009 was 35,148. What are the problems with this model? What problems may develop, say, by 2103?

Capacitors

1. A 2 mF capacitor is connected to a 10V DC power supply. How much charge can be stored by the capacitor?
2. What is the highest possible energy stored by this capacitor?
3. The capacitor is placed in series with a 5Ω resistor and charged to capacity. How long would it take for the charge in the capacitor to be reduced to 1 mC?
4. After this time has elapsed, how much energy is stored in the capacitor?
5. What is the capacitance of the equivalent capacitor to the following network of capacitors?

Radioactive Decay

1 mole = 6.02 x 10²³ atoms 1u = 1.66 x 10^-27kg

1. Americium-241 has a decay constant of 5.07 x 10^-11s^-1. What is the activity of 1 mole of americium-241?
2. How many g of lead-212 (λ = 18.2μs^-1) are required to create an activity of 0.8 x 10¹⁸Bq?
3. How long does it take for 2kg of lead-212 to decay to 1.5kg of lead-212?
4. Where does the missing 0.5kg go?
5. Some americium-241 has an activity of 3kBq. What is its activity after 10 years?
6. This model of radioactive decay is similar to taking some dice, rolling them once per. second, and removing the dice which roll a one or a two. What is the decay constant of the dice?
7. If you started out with 10 dice, how many dice would you have left after 10s? What is the problem with this model of radioactive decay?

Half-lives

1. Radon-222 has a decay constant of 2.1μs^-1. What is its half-life?
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?
3. How long will it be until it contains 0.5 grams of U-238?
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)?
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?
6. How long will it take for the charge in the capacitor to reach 0.677C? ($0.677={\frac {5}{e^{2}}}$)