Find the half-life

Question

Find the half-life of an element which decays by 3.402% each day.

Answer 1

The fraction of the element that remains after t days can be expressed as

X/X₀ = (1 – 0.03402)^t = (0.96598)^t

where X is the amount of the element that remains after t days and X₀ is the initial amount (at t = 0)

The half-life (t_½) is the time (in days) it takes for half of the element to decay:

X/X₀ = ½ at t = t_½

So we find half life by solving the following equation for t:

½ = (0.96598)^t when t = t_½

Since (0.96598)^t = e^{t ln(0.96598)} we can rewrite the previous equation as

½ = e^{t ln(0.96598)}

Taking the natural logarithm of both sides gives

ln(½) = –ln(2) = ln(e^{t ln(0.96598)}) = t ln(0.96598)

which means that

t = –ln(2)/ln(0.96598) = 20.0

answer

The half life (t_½) of the element is 20 days.

Answer 2

0.96598 = e^k

ln 0.96598 = k

-0.0034612148 = k

0.5 = e^{-0.0034612148t}

ln 0.5 = -0.0034612148t

-0.69314718 = -0.0034612148t

200.2612437 = t