Radioactive decay equation derivation

The rate at which a radioactive sample decay determines by counting the number of alpha, beta, and gamma rays emitted in a given time but half-life is the period of time when half of the radioactive elements undergo disintegrates. Thus it helps to determine the age of carbon species.

Radioactivity is one important natural phenomenon obeying the first-order kinetics. The rate of these reactions depends only on the single power of the concentration of the reactant.

\therefore \frac{dN}{dt}=-kN

where N = number of the atoms of the disintegrating radio-element, dt = time over which the disintegration is measured, and k = rate constant.

k=-\frac{\frac{dN}{dt}}{N}

The rate constant defined as the fraction decomposing in the unit time interval provided the concentration of the reactant kept constant by adding from outside during this time interval. The negative sign shows that N decreases with time.

Let N₀ = number of the atoms present at the time t = 0 and N = number of atom present after the t time interval. Rearranging and integrating over the limits N0 and N and time, 0 and t.

\therefore \frac{dN}{N}=-kdt

or,\int_{N_{0}}^{N}\frac{dN}{dt}=-\int_{0}^{t}kdt

or,\int_{N_{0}}^{N}d\left ( lnN \right )=-k\int_{0}^{t}dt

\therefore {\color{DarkBlue} ln\frac{N}{N_{0}}=-kt}

or,\frac{N}{N_{0}}=e^{-kt}

Radioactive decay and half-life

After a certain period of time, the value of (N0/N ) becomes one-half and half of the radioactive elements have undergone disintegration. This period is called the half-life of radioactive decay.

Radioactive decay half life
How to calculate the half-life of radio-element?

If radioactivity of an element 100% and the half-life period of this element 4 hours.

But after four hours, it decomposes 50% and the remaining 50%.

Thus after 8 hours it decomposes 75% and reaming 25% and the process continued.

From the laws of radioactive decay,

ln\frac{N}{N_{0}}=-kt

or, 2.303log\frac{N_{0}}{N}=kt

when t = t½, N = N₀/2

\therefore 2.303log\left ( \frac{N_{0}}{\frac{N_{0}}{2}} \right )=kt_{\frac{1}{2}}

or, t_{\frac{1}{2}}=\frac{0.693}{k}

{\color{DarkBlue} \therefore k=\frac{0.693}{t_{\frac{1}{2}}}}

This relation shows that both the half-life and radioactive decay rate constants are independent of the amount of the radio-element present at a given time.

t½ for polonium – 213 = 4.2 × 10-6 sec and bismuth – 209 = 3 × 107 years.

The average life period of radio-elements

The average life period of an atom of the radio-element tells us the average span of time after which the atom will disintegrate.

The length of time a radio-element atom can live before it disintegrates may have values from zero to infinity. This explains the gradual decay of the radio-element instead of the decay of all the atoms at the same time.

t_{av}=\frac{total\, life}{total\, number\, of\, atoms}

Let N0 atoms of a radioactive element are present in an aggregate of large no of atoms at time zero. Now in the small-time interval t to (t+dt), dN atoms are found to disintegrate.

Since dt, small-time period, we can take dN as the number of atoms disintegrating at the time t.

So the total lifetime of all the dN atoms = t dN.

Again the total number of atoms N0 is composed of many such small numbers of atoms d1, dN2, dN3, etc, each with its own life span t1, t2, t3, etc.

The average life of radio-element is reciprocal of its radioactive disintegration constant. This result can also be derived in a very simple way.

Radioactive atoms may be regarded as having an average life, then the product of the fractions of atoms disintegrating in unit time (that is k) and average life must be unity.

tav × k = 1

or, tav = 1/k

∴ t½ = 0.693/k = 0.693 tav

Half-life and carbon dating

Radiocarbon dating is a method for determining the age of organic martial based on the accurate determination of the ratio of isotopes of carbon.

The radiocarbon dating method was developed by Willard Libby, the University of Chicago in 1940 and receive a Nobel prize in chemistry for his work in 1960.

Carbon-14 produced in the atmosphere by the interaction of neutron with ordinary nitrogen or cosmic reaction.

7N14 + +1e06C14 +1H1

Carbon reacts with atmospheric oxygen to form carbon dioxide and carbon dioxide taken by plants in photosynthesis and animals by eating plants.

But when the animal or plant dies, it stops exchanging carbon with its environment since there no fresh intake of stratospheric carbon dioxide and the dead matter out of equilibrium with the atmosphere.

The radiocarbon-14 continues to decay so that thereafter a number of years only a fraction of carbon-14 left on the died matter.

Therefore the ratio of the 14C/12C drops from the steady-state ratio in the living matter.

6C147N14 + -1e0 (t½ = 5760 years)

By measuring this ratio and comparing it with the ratio in living plants one can estimate when the plant died.

Problem
A piece of wood was found to have a 14C/12C ratio of 0.7 times that in the living plant. Calculate the approximate period when the plant died(t₁/₂ = 5760 years).

Solution
We know that radioactive decay constant

k = 0.693/t½ = 0.693/5760 years
= 1.20 ×10⁻⁴ yr⁻¹

log\frac{N_{0}}{N}=log\left (\frac{1.00}{0.70} \right )=0.155

\therefore 2.303log\frac{N_{0}}{N}=kt

or, t=\frac{2.303\, log\frac{N_{0}}{N}}{k}

Putting the value, above equation

t = (2.303 × 0.155)/(1.20 × 10-4) years
= 2970 years

Age of rock deposits by half-life

Knowledge of the rate of decay of certain radioactive isotope helps to determine the age of various rock deposits.

Let us consider uranium-containing rock formed many years ago. The uranium started to decay giving rise to the uranium – 235 to lead -207 series.

The half-life of the intermediate members being small compared to that of uranium -235 (4.5 × 109 years).

Uranium atoms that started decaying many-many years ago must have been completely converted to the stable lead-207 during this extra-long period.

Thus uranium-235 remaining and the lead-207 formed must together account for the uranium 235 present at zero time when the rock solidified. Thus both N0 and N are known.

k is known from the knowledge of the half-life of decay of uranium -235. Therefore the age of the rock calculated.

Problem
A sample of uranium (t½ = 4.5 × 109 years) ore found to contain 11.9 gm of uranium-235 and 10.3 gm of lead-207. How to calculate the age of the ore?

Solution

11.9 gm of uranium-238 = 11.9/238
= 0.05 mole of uranium

10.3 gm of lead-206 = 10.3/206
= 0.05 mole of lead -206

Mole number of uranium -238 at zero time
= (0.05+0.05)
= 0.010 mole

∴ Radioactive decay constant
= 0.693/(4.5 × 109 year)
= 0.154 × 10⁻⁹ yr-1

From the radiactive decay law

t = (2.303 × log2)/(0.154 × 10-9)
= 4.5 × 109 year

Avogadro number of radium – 226

Let 1 gm of radium – 226 contains N number of atoms.

∴ N = N0/mRa

where N0 = Avogadro number
mass number of radium = 226

k × N = k × (N0/mRa)
where k = 0.693/t½

k\times N=\frac{0.693\times N_{0}}{t_{\frac{1}{2}}\times m_{Ra}}

\therefore N_{0}=\frac{k\times N\times t_{\frac{1}{2}}\times m}{0.693}

One gram of radium – 226 undergoes 3.7 × 1010 disintegrations per second and half-life = 1590 year.

Thus the Avogadro number of radium – 226

N_{0}=\frac{k\times N\times t_{\frac{1}{2}}\times m}{0.693}

=\frac{3.7\times 10^{10}\times 1590\times 365\times 24\times 60\times 226}{0.693}

Thus the Avogadro number of radium-226
= 6.0 × 1023