If protons decay with a half-life of 10³⁰ years,how long will it take for 1% of the protons in the universe to decay?

Question

Quizplus · Accepted Answer

The time it takes for 1% of protons to decay can be found using the formula for exponential decay:

N(t) = N0 * (1/2)^(t/T)

where N(t) is the number of protons at time t, N0 is the initial number of protons, T is the half-life, and t is the time elapsed.

Let N(t)/N0 = 0.01, which corresponds to 1% decay.

0.01 = (1/2)^(t/T)

Taking the natural logarithm of both sides,

ln(0.01) = ln(1/2)^(t/T)

ln(0.01) = -(t/T) * ln(2)

t/T = -ln(0.01) / ln(2)

t/T = 6.64

Thus, the time it takes for 1% of protons to decay is 6.64 times the half-life, or

t = (6.64)(10^17 years)

t = 1.5 x 10^18 years

The closest answer choice is C.

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