# Law of Radioactive Disintegration | Half Life

## Law of Radioactive Disintegration

According to
the law of * radioactive disintegration* the rate of spontaneous
disintegration of a radioactive element is proportional to the number of nuclei present at that time.

Mathematically, it can be written as

`\frac{dN}{dt}\infty N................(1)`

where, N is the number of atoms present at time t. Removing proportionality sign, we get

`\frac{dN}{dt}=-\lambda N............(2)`

where, Î» is a constant of proportionality and is known as * decay constant* of the element. Negative sign indicates that as t increase N decreases.

`\frac{dN}N=-\lambda dt.............(3)`

Integrating both sides, we have

`\int\frac{dN}N=-\lambda\int dt`

`\log_e(N)=-\lambda t+C.............(4)`

where C is constant of integration and is evaluated by the fact that at t = 0, number of atoms of the radioactive element is N_{0}. Using
this condition, we get

`C=\log_e(N_0).................(5)`

Substituting this value of C in Eq. (5), we get

`\log_e(N)=-\lambda t+\log_e(N_0)`

`\log_e(N)-\log_e(N_0)=-\lambda t`

Thus,

`N=N_0e^{-\lambda t}.................(6)`

Exponential decay curve |

## Units of Activity

The SI unit of activity is named after * Becquerel*. Active of a radioactive sample is said to be 1 becquerel if rate of decay is 1 nucleus per second.

Therefore,

becquerel = 1 Bq = 1 decay/sec

An older unit **Curie (Ci)** is still commonly used to measure activity.

Therefore,

1 curie = 1 Ci = 3.7 × `10^{10}` decay/sec = 3.7 × `10^{10}` Bq

Yet, another unit of activity of radioactive sample is ** rutherford (Rd)** and is defined as :

1 rutherford = 1 Rd = `10^{10}` decay/sec = `10^{10}` Bq

## Half-life

* Half-life* is the time at which the activity of sample has been reduced to one-half of its initial value.

If N_{0} is the initial number of radioactive nuclei
present, then in one half-life this number will reduce to N_{0}/2.
Therefore,

`N=\frac{N_0}2`

substituting N in Eq. (6), we get

`\frac{N_0}2=N_0e^{-\lambda t_{1/2}}`

`\1/2=e^{-\lambda t_{1/2}}`

`e^{\lambda t_{1/2}}=2`

Taking natural log on both sides,

`\lambda t_{1/2}=\log_e2`

`t_{1/2}=\frac{2.303\log_{10}\2}\lambda`

`[ \log_ex=2.303\log_{10}x\ ]`

`t_{1/2}=\frac{0.693}\lambda`

Therefore, * half-life* of a radioactive material depends upon the disintegration constant. The larger the value of disintegration constant, smaller is the half-life. Half-life of a radioactive material is its intrinsic property it cannot be altered by any physical or chemical means.

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