How long would an unmanned lighthouse containing 10kg of 90Sr produce at least 1kW of thermal power, which would keep a good sized light on? Don’t forget to account for any daughter products of 90Sr decay, which may also be radioactive. Give your answer to the nearest year.

Answer 1

In order to calculate the time required for the unmanned lighthouse to produce at least 1kW of thermal power from 10kg of 90Sr, we need to consider the half-life of 90Sr and the decay process.

The half-life of 90Sr is approximately 28.79 years. This means that in 28.79 years, half of the 90Sr will decay into its daughter product, which is 90Yttrium (90Y).

The decay of 90Sr to 90Y is as follows:
90Sr -> 90Y + β- + anti-neutrino

The daughter product 90Y is also radioactive and undergoes further decay. The decay of 90Y is as follows:
90Y -> stable Zirconium (90Zr) + β- + anti-neutrino

The decay of both 90Sr and 90Y will produce heat, which can be converted into thermal power for the lighthouse. The energy released in these decays is approximately 0.54 MeV for the decay of 90Sr and 2.28 MeV for the decay of 90Y.

Considering the decay energies and assuming that all decay energy is converted into heat, we can calculate the thermal power produced by the decays of 90Sr and 90Y.

First, calculate the total decay energy released per unit time:
- For 90Sr: 10kg * (6.02x10^23 atoms/mol) * (1 mol / 90 g) * (1 decay / half-life) * (0.54 MeV/decay) = 3.793x10^12 MeV/year
- For 90Y: (1/2) * decay energy of 90Sr = 1.8965x10^12 MeV/year

Convert the decay energy to watts:
- 1 MeV = 1.6x10^-13 J, therefore 1 MeV/year = 5.03361x10^-21 W
- Power from 90Sr decay = (3.793x10^12 MeV/year) * (5.03361x10^-21 W/MeV) = 1.910x10^-8 W
- Power from 90Y decay = (1.8965x10^12 MeV/year) * (5.03361x10^-21 W/MeV) = 9.527x10^-9 W

Total thermal power produced by 90Sr and 90Y decays = 2.862x10^-8 W

Now, we need to find the time required for the thermal power produced by 90Sr and 90Y decays to reach at least 1kW:
- Time = (1kW) / (2.862x10^-8 W) = 3.5006x10^10 years

Therefore, to keep a good sized light on, the unmanned lighthouse containing 10kg of 90Sr would need to operate for approximately 3.5006x10^10 years to produce at least 1kW of thermal power.