Using excel and linear regression analysis, determine the rate constant value k

Question

Using excel and linear regression analysis, determine the rate constant value k for the nuclear decay. To calculate the actual activity of the sample, subtract the background activity from the corresponding raw activity. (Hint; nuclear decays are first-order reactions). 1. Raw Time (days) Activity Background 48249 33769 23508 15717 11280 7943 5720 4353 3058 2153 1531 1107 863 554 484 329 268 226 140 136 90 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 18004o 1900 2000 50 90 65 95 80 85 60 60 80 75 2. Calculate the half-life of the radioactive isotope. 3. Calculate the activity left after 512 day. 100 85 95 65 95 85 60 75 50

Explanation / Answer

the first order decay proces of an element is given by

-dN/dt= KN, where N is activity and K is rate constant. when integrated noting that at t=0, N= No and at t=t, N= N

lnN= lnNO-Kt (1)

so a plot of lnN vs time (t ) gives straight line with a slope of -K.

from the plot, the slope = -K= -0.003/day

half life is defined as the time requires for the activity to drop to 50% of the initial value.

i.e N=No/2

hence Eq.1 becomes ln (NO/2)= lnNO- K*half life

half life =0.693/K= 0.693/0.003 =231 days

with the rate constant value lnN =ln(48249)-0.003*t

at t= 512 days, lnN= ln(48249)-0.003*512

N= 10385

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