Suppose the grand unified theory GUT was correct and halflif

Suppose the grand unified theory (GUT) was correct and half-life of a proton was 1.2 * 10^35 y. Estimate the decay rate of a liter of water in decays per second and curies. How does this compare to a small (by laboratory standards) radioactive source of one microcurie

Solution

molar mass of water = 18 g / mol

mass of 1 lit water = 10^3 g

moles of water = 10^3 / 18 mol = 55.6 mol

no. of molecules = 55.6 x 6.022 x 10^23

no of protons = 10 x 55.6 x 6.022 x 10^23 protons

Decay rate = lambda N

lambda = ln(2) / T = ln2 / (1.2 x 10^35 yr x 365 day/yr x 24 hr/day x 3600 sec/hr)

= 1.832 x 10^-43

Decay rate = (1.832 x 10^-43) ( 10 x 55.6 x 6.022 x 10^23)

        = 6.132 x 10^-17 decays/sec

1 Ci (Curie) = 3.7 x 10^10 decays per sec

Decay rate (in curies) = 6.132 x 10^-17 / (3.7 x 10^10)

          = 1.657 x 10^-27 Ci

(This is very very small compared to small radioactive source)