The radio-isotopes O 14 and O 15 are β + emitters with half lives 71 and 124 seconds, respectively. If a sample just collected contains equal moles of both isotopes, in what time (in seconds), the nuc

Question

The radio-isotopes O 14 and O 15 are β + emitters with half lives 71 and 124 seconds, respectively. If a sample just collected contains equal moles of both isotopes, in what time (in seconds), the nucleidic ratio of O 14 and O 15 becomes 1:4?

TestHub · Accepted Answer

Step 1: The amount of remaining nuclei is given by $N = N_{0} e^{- λ t}$ , where $λ = \frac{l n ( 2 )}{T _{1/2}}$ .

Step 2: Setting up the ratio of the two isotopes: $\frac{N _{O^{14}}}{N _{O^{15}}} = \frac{e ^{- λ_{1} t}}{e ^{- λ_{2} t}} = e^{(λ_{2} - λ_{1}) t} = \frac{1}{4}$ .

Step 3: Taking the natural logarithm on both sides yields $(λ_{1} - λ_{2}) t = ln (4) = 2 ln (2)$ .

Step 4: Substituting $λ = \frac{l n ( 2 )}{T _{1/2}}$ simplifies the equation to: $(\frac{1}{71} - \frac{1}{124}) t = 2 ⟹ t = \frac{2 \times 71 \times 124}{124 - 71} \approx 332.23s$ .

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