# Rate of Disintegration

Rate of disintegration

The rate of disintegration is independent of temperature, concentration etc. the number of atoms of radioactive element disintegrating in unit time, at any instance, is proportional to the number of atoms present.

Thus

$-\dfrac{dN}{dt} \propto N \\[3mm] \text{or} -\dfrac{dN}{dt} = \lambda N$          ……(1)

Where dN is the number of nuclei which decay during the time interval dt out of N nuclei present at time t, $\lambda$ is a proportionality constant and is called disintegration or decay constant. On integration eq. 1 and on rearrangement we get,

$\lambda = \dfrac{2.303}{t} \log \dfrac{N_0}{N}$             ………(2)

Where $N_0$ is number of nuclei initially present.

• Radioactive decay is always of Ist order.

If any time, $t_{1/2}, N = N_0 / 2$ then eq. 2 becomes

$t_{1/2} = \dfrac{0.693}{\lambda}$            ………..(3)

$t_1/2$ is known as half life period, i.e., the time required for disintegration of one half of the original amount of radioactive substance. It is the characteristic property like decay constant of the radioactive substance. Average life period $(\tau)$ is related with disintegration constant or half life period as given below:

$\tau \text{or} T = \dfrac{1}{\lambda} = 1.44 t_{1/2}$

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