What mass of $14 \mathrm { C }$, with a half-life of $5730 \mathrm { y }$, do you need to provide a decay rate of $280.0 \mathrm {~Bq}$ ? The mass of ${ } ^ { 14 } \mathrm { C }$ is $14.00 \mathrm { u }$, and $1 \mathrm { u } = 1.6605 \times 10 ^ { - 27 } \mathrm {~kg}$.

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The decay rate (activity, A) of a radioactive substance is given by the formula $A = \lambda N$, where $\lambda$ is the decay constant and $N$ is the number of radioactive atoms. The decay constant $\lambda$ can be calculated from the half-life $T_{1/2}$ using the formula $\lambda = \frac{\ln(2)}{T_{1/2}}$. Given the half-life of $^{14}C$ is 5730 years, we first convert it to seconds ($T_{1/2} = 5730 \times 365.25 \times 24 \times 3600 \, \text{s}$) and then calculate $\lambda$. Knowing $\lambda$, we can solve for $N$ using the given decay rate $A = 280.0 \, \text{Bq}$. Finally, we find the mass $m$ by multiplying $N$ by the mass of one $^{14}C$ atom ($14.00 \, \text{u}$ converted to kg using $1 \, \text{u} = 1.6605 \times 10^{-27} \, \text{kg}$). The calculations lead to the mass of $^{14}C$ required to provide the given decay rate, which matches the option indicating $1.70 \times 10^{-12} \, \text{kg}$.

What Mass Of 14C14 \mathrm { C }14C , with a Half-Life Of

What Mass Of $14 \mathrm { C }$ , with a Half-Life Of