Radioactive Decay

The radioactive decay law is an exponential function.

N = N_o e ^{- λ t}

N Number of radioactive nuclei at time t

N_o Number of radioactive nuclei at start

t time/ s

λ decay constant/ s ^{- 1} Bq (Becquerel)

m = m_o e ^{- λ t}

m mass of radioactive nuclei at time t

m_o mass of radioactive nuclei at start

N = N_o e ^{- λ t}

Differentiate with respect to t (find gradient):

ΔN = - λ N_o e ^{- λ t}

Δt

but N = N_o e ^{- λ t}

ΔN = - λ N

Δt

ΔN/Δt rate of decay/ s^-1 (Bq)

λ decay constant/ s^-1 (Bq)

N number of radioactive nuclei at time t

The rate of decay is proportional to the number of radioactive nuclei present.

N = N_o e ^{- λ t}

Multiply by –λ

–λ N = –λ N_o e ^{- λ t}

ΔN = ΔN_o e ^{- λ t}

Δt Δt

ΔN/Δt rate of decay at time t/ s^-1 (Bq)

ΔN_o/Δt rate of decay at start/ s^-1 (Bq)

ΔN = - λ N

Δt

ΔN/N = - λ

Δt

λ = -ΔN/N

Δt

The decay constant (λ) is defined as the probability of decay (-ΔN/N) of a nucleus per unit time (Δt).

ΔN/N Probability of decay

Δt time/ s^-1 Bq

λ decay constant/ s^-1 Bq