Define the term decay constant of a radioactive nucleus.
Two nuclei P, Q have equal number of atoms at t = 0. Their half lives are 3 hours and 9 hours respectively. Compare their rates of disintegration, after 18 hours from the start.

Solution

Decay constant of a radioactive element is the reciprocal of the time during which the number of atoms left in the sample reduces to

\frac{1}{e}

times the number of atoms in the original sample.

Given, two nucleii P and Q.
P and Q have equal number of atoms at t=0.
Half life of P = 3 hours.
Half life of Q = 9 hours.

and, t = 18 hours

Number of half lives of P in 18 hours =

\frac{t}{T_{\frac{1}{2}}}

\frac{18}{3} = 6

Number of nuclei P left undecayed after 6 half lives is,

N_{1} = N {(\frac{1}{2})}^{6}

Number of half lives of Q in 18 hours=

\frac{t}{T_{\frac{1}{2}}}

\frac{18}{9} = 2

Number of nuclei of Q left undecayed after 2 half lives is,

N_{2} = N {(\frac{1}{2})}^{2}

Ratio of their decay rate is,

\frac{R_{1}}{R_{2}} = \frac{λ_{1} N_{1}}{λ_{2} N_{2}} = \frac{T_{2} N_{1}}{T_{1} N_{2}}

[∵ R = λ N and T = \frac{0.693}{λ}]

∴

\frac{R_{1}}{R_{2}} = \frac{9}{3} \times \frac{N {(\frac{1}{2})}^{6}}{N {(\frac{1}{2})}^{2}} = \frac{3}{16}

Hence ratio of disintegration is,

R_{1} : R_{2} = 3 : 16

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Define the term decay constant of a radioactive nucleus.
Two nuclei P, Q have equal number of atoms at t = 0. Their half lives are 3 hours and 9 hours respectively. Compare their rates of disintegration, after 18 hours from the start.

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Wave Optics

Define the term decay constant of a radioactive nucleus.Two nuclei P, Q have equal number of atoms at t = 0. Their half lives are 3 hours and 9 hours respectively. Compare their rates of disintegration, after 18 hours from the start.

Some More Questions From Wave Optics Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Define the term decay constant of a radioactive nucleus.
Two nuclei P, Q have equal number of atoms at t = 0. Their half lives are 3 hours and 9 hours respectively. Compare their rates of disintegration, after 18 hours from the start.