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What is Half Life, Decay Constant and Their Relation?

The process of emitting radiation spontaneously from an unstable atom is the answer to the question that what is radioactivity. The nucleus of that atom wants to give up some energy in order to change into a more stable configuration. When a nucleus has too many neutrons, it emits a negative beta particle, which converts one of the neutrons to a proton. When a nucleus has too many protons, it emits a positron by converting a proton into a neutron. A nucleus with too much energy will emit a gamma ray, which will waste a lot of energy without affecting any of the particles in the nucleus. A nucleus with too much mass can emit an alpha particle, discarding four heavy particles in the process. What is the concept of a radioactivity and decay of atoms? What is half life? Decay constant and its relation with half life. All these questions are being answered here.

An atomic nucleus’ instability does not cause it to emit radiation right away. Instead, the chance of an atom disintegration is constant, there are random draws to determine that which atom will emit radiation and disintegrate to a more stable state.

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What is half life of a radioactive element?

What is Half Life and Decay Constant and Their Relationship?

The half-life is the amount of the time that it takes for half of the atoms in a given mass, to decay i.e. release radiation and transition to a more stable state. The half-lives of various atoms range from less than a second to billions of years. Half of the atoms in a mass of uranium 238 would spontaneously disintegrate in around 4.5 billion years, but half of the atoms in a mass of plutonium 239 would spontaneously disintegrate in only 24,000 years. The half-life of iodine 131, which is widely used in medicine, is just 8 days.

Cobalt-60 is a radioactive isotope of cobalt that is used in radiotherapy. It has a half-life of 5.26 years. After that time, a sample that originally contained 8 g of cobalt-60 would only contain 4 g of cobalt-60 and emit half as radiation. The sample will only produce 2 g of cobalt-60 after another 5.26-year interval. Since the unstable cobalt-60 nuclei decay into stable nickel-60 nuclei, which remain with the still undecayed cobalt, neither the volume nor the mass of the original sample decreases noticeably.

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Half life graph of cobalt is given as

What is Half Life and Decay Constant and Their Relationship?

This is the decay curve of cobalt. Half life of cobalt 60 is about 5.3 years.

The half-lives of different unstable atomic nuclei and the manner in which they decay are characteristics. The mechanisms of alpha and beta decay are normally slower than gamma decay. Beta decay has half-lives of up to one hundredth of a second, while alpha decay has half-lives of up to one millionth of a second. Although a wide range of half-lives for gamma emission has been published, gamma decay half-lives may be too short to calculate (around 10-14 seconds).

What is Half Life and Decay Constant and Their Relationship?

What is decay constant?

The proportionality between the size of a population of radioactive atoms and the rate at which the population declines due to radioactive decay is referred to as the decay constant. If N is the size of a population of radioactive atoms at a given time t and dN is the value by which the population shrinks over time dt, then

dN ∝ – N dt

dN = – λ N dt

where λ is called decay constant. If the value of λ is greater for some element then more of its atoms will decay in a particular interval, however, if value of λ is less, then small number of atoms will decay. Equation can be simplified as

λ = – (dN / N) / dt

dN / N is the fraction of decaying atoms. Thus decay constant is equal to the fraction of decaying atoms per unit time. Unit of decay constant is s-1. In above equation negative sign indicates that number of atoms is decreasing. This equation is a relationship between half life and decay constant.

The estimate of any radioactive element can be made by its half life or its decay constant. Below is the half life formula which relates it with decay constant.

λ T1/2 = 0.693

This is half life equation, so if one variable from this equation is known then other can be found quite easily.

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