The rate of decay of an Iodine-123 isotope is proportional to the mass
of isotope present at that time. The initial mass of the isotope was
200 g. Determine the mass of Iodine present after 39 days if the half
life period of the lodine isotope is approximately 13 hours.
n:...
Answers
Given : The rate of decay of an Iodine-123 isotope is proportional to the mass of isotope present at that time.
The initial mass of the isotope was 200 g.
half life period of the lodine isotope is approximately 13 hours.
To Find : Determine the mass of Iodine present after 39 days
Solution:
Let say m = mass of the substance
t = time
The rate of decay of the mass of a radio active substance = dm/dt
dm/dt = km
=> dm/m = kdt
integrating both sides
∫(1/m).dm = ∫k.dt
=> ln |m| = kt + c
at t = 0 m = 200
=> ln 200 = 0 + c
=> c = ln 200
Hence ln m = kt + ln 200
half life period of the lodine isotope is approximately 13 hours.
=> at t = 13 hrs m = 200/2 = 100
=> ln 100 = k(13) + ln200
=> k = ( ln 100 - ln200)/13
ln m = {( ln 100 - ln200)/13 }t + ln 200
t = 39 days = 39 x 24 hrs
=> ln m = {( ln 100 - ln200)/13 }39 x 24 + ln 200
=> ln m = {( ln 100 - ln200) }72 + ln 200
=> ln m = -49.9066 + 5.2983
=> ln m = -44.6083
=> m = 4.2351 x 10⁻²⁰
mass of Iodine present after 39 days = 4.2351 x 10⁻²⁰ g
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