Question

Starting from the statement that decay rate is proportional to the number of nuclei in a sample, derive the exponential equation that relates the number of nuclei in a sample to the initial number and the elapsed time. Define all variables used. [3 marks] c) A 1 megaton nuclear bomb produces 0.45 kg of an isotope of 90Sr and the fallout spreads uniformly over an area of 110 km2 . 90Sr decays with a half-life of 29 years. Determine the ground area that holds an amount of radioactivity equal to 9.2 x104 Bq. Define variables used.

ESPECIALLY NEED ANS TO C THANKS

Answer 1

part 1:

at any time, number of nuclei is N and initial number of nuclei is N0.

then dN/dt=-k*N

where k is a constant.

(-ve sign is because of the fact that with increasing time, number of nuclei decreases)

hence dN/N=-k*dt

integrating both sides,

ln(N)=-k*t+c

at t=0, N=N0

==>ln(N0)=c

then ln(N)=-k*t+ln(N0)

==>ln(N/N0)=-k*t

==>N/N0=exp(-k*t)

==>N=N0*exp(-k*t)

part c:

mass of isotope produced=0.45 kg

molar mass of 90Sr=90 grams

number of moles produced=total mass/molar mass=0.45/(0.09)=

=5 moles

=5*6.022*10^23 nuclei

number of nuclei per m^2 of area=total number of nuclei/total area

=5*6.022*10^23/(110*10^6 m^2)

=2.7373*10^16

half life=29 years

then decay constant=ln(2)/half life=ln(2)/(29*365*24*3600 seconds)

=7.5792*10^(-10) s^(-1)

activity=number of nuclei*decay constant

==>9.2*10^4=number of nuclei*7.5792*10^(-10)

==>number of nuclei=1.214*10^14

then area=number of nuclei/nuclei per m^2

=1.214*10^14/(2.7373*10^16)

=0.004435 m^2