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  • Consider an initially pure sample of <img alt="{ }^A \mathrm{X}" src="/latex-image/?%7B%20%7D%5EA%20%5Cmathrm%7BX%7D" /

Consider an initially pure \mathrm{M\, gm} sample of { }^A \mathrm{X}, an isotope that has a half life of T hour, what is it's initial decay rate \left(N_A=\right.\text { Avogrado No.) }

Option: 1

\frac{\mathrm{MN}_{\mathrm{A}}}{\mathrm{T}}


Option: 2

\frac{0.693 \mathrm{MN}_{\mathrm{A}}}{\mathrm{T}}


Option: 3

\frac{0.693 \mathrm{MN}_{\mathrm{A}}}{\mathrm{AT}}


Option: 4

\frac{2.303 \mathrm{MN}_{\mathrm{A}}}{\mathrm{AT}}


Answers (1)

best_answer

\mathrm{N}=\mathrm{N}_0 \mathrm{e}^{-\lambda \mathrm{t}} \Rightarrow\left|\frac{\mathrm{dN}}{\mathrm{dt}}\right|=\mathrm{N}_0 \lambda \mathrm{e}^{-\lambda \mathrm{t}}

\text { Initially at } t=0,\left|\frac{\mathrm{dN}}{\mathrm{dt}}\right|_{\mathrm{t}=0}=\mathrm{N}_0 \lambda

\text { where } N_0=\text { Initial number of undecayed atoms }

=\frac{\text { Mass of the sample }}{\text { Mass of a single atom of } X}=\frac{M}{A / N_A}=\frac{M_A}{A}

\therefore\left|\frac{\mathrm{dN}}{\mathrm{dt}}\right|_{\mathrm{t}=0}=\frac{\mathrm{MN}_{\mathrm{A}} \lambda}{\mathrm{A}}=\frac{0.693 \mathrm{MN}_{\mathrm{A}}}{\mathrm{AT}}

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shivangi.bhatnagar

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