## Radioactive Decay Formula in Chemistry

**Radioactive decay** is the spontaneous disintegration or emission of the atomic particles like alpha, beta, and gamma from the nucleus in the form of nuclear energy of radioactive matter or substances. Therefore, the radioactive decay simply determines by counting the number of alpha, beta, and gamma radiation in a given time and half-life is the period of time when half of the radioactive nucleus of the elements undergo disintegrates. Radioactivity is one important natural phenomenon obeying the first-order kinetics uses mainly in radiocarbon dating and age of matter or materials. The rate of these first-order radioactive decay depends only on the single power of the concentration of the radioactive isotopes. Therefore, dN/dt = -kN, where N = number of the atoms of the disintegrating radioactive element, dt = time over which the decay is measured, and k = rate constant.

### Radioactive Decay Constant

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 N_{0} = number of the atoms present at the time t = 0 and N = number of atoms present after the t time interval. The radioactive decay formula represented as, (dN/N) = -kt, and the integrating equation is written as N = N_{0}e^{-kt}.

### Half-life in radioactive decay

After a certain period of time, the value of (N_{0}/N ) becomes one-half of the radioactive elements. This period of decay is called the half-life of radioactive chemical material or substances.

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

The rate law for radioactive decay, ln(N/N_{0}) = -kt

But when t = t_{½}, N = N_{0}/2

∴ t_{½} = ln2/k

or, t_{½} = 0.693/k

This equation shows the relation between half-life and radioactive decay rate constants are independent of the amount of the radio-element present at a given time. Therefore, the half-life of polonium-213 isotope = 4.2 × 10^{-6} sec and bismuth-209 isotope = 3 × 10^{7} years.

### Average life Period of Elements

The average life period of an atom of the radioactive element tells us the average span of time after which the atom will disintegrate or decay. The length of time a radio-element atom can live before it disintegrates may have values from zero to infinity. This explains the gradual radioactive decay of the radioisotopes instead of the decay of all the atoms at the same time. Therefore, the total number of radioactive atoms (N_{0}) is composed of many small numbers of atoms like dN_{1}, dN_{2}, dN_{3}, etc, each with its own life span t_{1}, t_{2}, t_{3}, etc.

The average life of radio-element is reciprocal of its radioactive decay constant in nuclear chemistry. This radioactive decay result can also be derived in a very simple way of learning chemistry or physics. If radioactive isotopes have an average life, then the product of disintegration constant and average life must be unity. Therefore, from the half-life and average life formula, we get the following radioactive decay equation, t_{½} = 0.693/k = 0.693 t_{av}.

## Radioactive Carbon dating Formula

Radiocarbon dating or carbon dating in radioactivity is a method for determining the age of organic martial based on the accurate determination of the ratio of radioactive decay of carbon isotopes present in our environment. The radiocarbon dating method was developed by Willard Libby, the University of Chicago in 1940 and receive a Nobel prize in chemical science for his work in 1960. Carbon-14 produced in the atmosphere by the interaction of neutron particles with ordinary nitrogen or cosmic reaction.

_{7}N^{14} + _{+1}e^{0} → _{6}C^{14} +_{1}H^{1}

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 carbon-14 and carbon-12 drops from the steady-state ratio in the living matter.

_{6}C^{14} → _{7}N^{14} + _{-1}e^{0} (t_{½} = 5760 years)

By measuring this ratio and comparing radioactive decay with living plants, we can estimate when the plant died or the age of dead substances.

Problem: A piece of wood was found to have the carbon-14 and carbon-12 ratio of 0.7 times that in the living plant. Calculate the approximate period when the plant died (t_{₁/₂} = 5760 years).

Solution: We know that radioactive decay constant (k) = 0.693/t_{½} = 0.693/5760 years = 1.20 ×10^{-4} yr^{-1}. When we putting the value, radioactive decay equation time, = (2.303 × 0.155)/(1.20 × 10^{-4}) years = 2970 years.

### Age of rock deposits by Half-life

Knowledge of the radioactive decay model of certain radioactive isotope helps to determine the age of various rock deposits in nuclear chemistry. Let us consider uranium-containing rock formed many years ago. The radioactive uranium-235 started to decay and end with lead-207 to form the decay series or chain. The half-life of the intermediate members being small compared to that of uranium-235 (4.5 × 10^{9} years). Therefore, uranium started to decay many-many years ago must have been completely converted to the stable lead-207 during this extra-long period.

Hence remaining uranium-235 and the lead-207 present at zero time when the rock solidified. Therefore, both N_{0} and N are known. In radioactivity, the decay constant(k) is calculated from the knowledge of the half-life of uranium-235. Thus the age of the rock can be calculated.

Problem: A radioactive sample of uranium (t_{½} = 4.5 × 10^{9} 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 by the law of radioactive decay?

Solution: 11.9 gm of uranium-238 = 11.9/238 = 0.05 mole of uranium and 10.3 gm of lead-206 = 10.3/206 = 0.05 mole of lead-206. Therefore, the mole number of uranium-238 at zero time = (0.05+0.05) = 0.010 mole and decay constant

= 0.693/(4.5 × 10^{9} year) = 0.154 × 10^{-9} yr^{-1}. From the radioactive decay law in nuclear chemistry, time (t) = (2.303 × log2)/(0.154 × 10^{-9}) = 4.5 × 10^{9} year.

### Avogadro Number of Radium-226

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

∴ N = N_{0}/m_{Ra
}where N_{0} = Avogadro number

Mass number of radium = 226

k × N = k × (N_{0}/m_{Ra})

where k = 0.693/t_{½}

In nuclear chemistry, one gram of radioactive radium-226 isotope undergoes 3.7 × 10^{10} decay per second and half-life = 1590 year. Therefore, the Avogadro number of radium-226 calculated from the radioactive decay law = 6.0 × 10^{23}.