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# The radioactivity of a sample is $${R}_{1}$$ at a time $${T}_{1}$$ and $${R}_{2}$$ at a time $${T}_{2}$$. If the half-life of the specimen is $$T$$, the number of atoms that have disintegrated in the time $$({T}_{2}-{T}_{1})$$ is proportional to

A
$${R}_{1}{T}_{1}-{R}_{2}{T}_{2}$$
B
$${R}_{1}-{R}_{2}$$
C
$$({R}_{1}-{R}_{2})T$$
D
$$\cfrac{({R}_{1}-{R}_{2})}{T}$$
Solution
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#### Correct option is D. $$({R}_{1}-{R}_{2})T$$$${ R }_{ 1 }={ N }_{ 1 }\lambda$$$${ R }_{ 2 }={ N }_{ 2 }{ \lambda }$$$$R_1-R_2=(N_1-N_2)\lambda$$Also$$T=\cfrac { \log _{ e }{ 2 } }{ \lambda } \Rightarrow \lambda =\cfrac { \log _{ e }{ 2 } }{ T }$$$${ R }_{ 1 }-{ R }_{ 2 }=\left( { N }_{ 1 }-{ N }_{ 2 } \right) \lambda$$$$R_1-R_2=\left( { N }_{ 1 }-{ N }_{ 2 } \right) \cfrac { \log _{ e }{ 2 } }{ T }$$$$\Rightarrow \left( { N }_{ 1 }-{ N }_{ 2 } \right) \propto \left( { R }_{ 1 }-{ R }_{ 2 } \right) T$$

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