Calculating Radioactive DecayCalculating Radioactive Decay

Honors ChemistryHonors Chemistry

Dr. YagerDr. Yager

The basic concept:The basic concept:

At any given time t, the At any given time t, the rate of change of massrate of change of mass

of radioisotope is of radioisotope is proportional to the massproportional to the mass of of

the radioisotope.the radioisotope.

Let M(t) = radioisotope mass at time t Let M(t) = radioisotope mass at time t

MMoo = M(0) = initial mass at t = 0 = M(0) = initial mass at t = 0

k = constantk = constant

From calculus, we can write dM(t)/dt = k M(t)From calculus, we can write dM(t)/dt = k M(t)

((rate of change of mass rate of change of mass is proportional tois proportional to mass) mass)

which has the solution:which has the solution: M(t) = Mo e-kt

M(t) = MM(t) = Moo e e-kt-kt

Definition of half life (tDefinition of half life (t1/21/2):):

In one half life, half the mass has decayed, half is left.In one half life, half the mass has decayed, half is left.

or at time t = tor at time t = t1/21/2, M = M, M = Moo/2/2

M(t) = M(tM(t) = M(t1/21/2) = M) = Moo/2 = M/2 = Moo e e-kt-kt1/2 1/2

½ = e½ = e-kt-kt1/2 1/2

ln(½) = ln(eln(½) = ln(e-kt-kt1/2 1/2 ) = -kt) = -kt1/21/2

-0.693 = -kt-0.693 = -kt1/21/2

k = 0.693/tk = 0.693/t1/21/2

M(t) = MM(t) = Moo e e-kt -kt = M= Moo e e-(0.693/t-(0.693/t1/21/2

)t)t

Manganese-56 is a beta emitter with a half-life Manganese-56 is a beta emitter with a half-life of of 2.60 h2.60 h. What is the mass of a . What is the mass of a 1.0 mg1.0 mg sample of Mn-56 after sample of Mn-56 after 10.4 h10.4 h??

Use: M = Use: M = MMooee--kktt

so; so; kk = (0.693/ = (0.693/2.60 h2.60 h)) = 0.266 h= 0.266 h-1-1

M = M = 1.0 mg 1.0 mg (e(e-0.266 x -0.266 x 10.410.4) = 0.063 mg) = 0.063 mg

2

1t

0.693k where

Thorium-234 has a half-life of 24.1 days. Thorium-234 has a half-life of 24.1 days. How much of a 1.0 mg sample will be left How much of a 1.0 mg sample will be left after 48.2 days?after 48.2 days?

kk = (0.693/24.1 days) = 0.0290 days = (0.693/24.1 days) = 0.0290 days-1-1

M = 1.0 mg (eM = 1.0 mg (e-0.0290x48.2-0.0290x48.2) = 0.25 mg) = 0.25 mg

M(t) = MM(t) = Moo e e-kt-kt

ln M = ln (Mln M = ln (Moo e e-kt-kt))

= ln M= ln Moo + ln e + ln e-kt-kt

= ln M= ln Moo - kt - kt

( y = b + mx )( y = b + mx )

How many protons are in 1 mole of ?How many protons are in 1 mole of ?

mole

protons10 x 3.61

mole

atoms6.02x10 x

atom 1

protons 6 2423

C146

1 mole of anything is equal to 6.02 x 1023 of those items.

For example:

1 mole of atoms is 6.02x1023 atoms

1 mole of cookies is 6.02 x 1023 cookies

How many protons are left after one half-life of How many protons are left after one half-life of ββ decay of one mole of C-14 ? decay of one mole of C-14 ?

6 protons6 protons + 8 neutrons+ 8 neutrons = 14 nucleons; n = 14 nucleons; n → p→ p++ + e + e--

The neutrons decay to protons; therefore, there are: The neutrons decay to protons; therefore, there are:

(3.61x 10(3.61x 102424) protons = 3.61 x 10) protons = 3.61 x 102424 protons protons

half of the neutrons change to protonshalf of the neutrons change to protons + 2.41 x 10+ 2.41 x 102424 new protons new protons

Total protons 6.02 x 10Total protons 6.02 x 1024 24 protons protons

The new material formed is N-14. The new material formed is N-14.

mole

neutrons10 x 4.82

mole

atom6.02x10 x

atom

neutrons 8 2423