Chapter 21.8, Problem 3PPB

Interpretation Introduction

Interpretation: The time required to take for the radioactivity due to radon to fall to a level considered acceptable by the EPA if the starting activity is $\text{2}\text{.25}{\text{×10}}^{\text{3}}\text{dps/L}$ should be determined.

Concept Introduction:

• An unstable nucleus emits radiation spontaneously to become stable nuclei by losing energy. This process of emission of radiation by unstable nuclei is known as radioactive decay.
• Radioactive decay is in the first order kinetics. Rate of radioactive decay at a time t is,
• $\begin{array}{l}Rateofdecayatatime\text{R = k N}\\ \text{k-}\text{First}\text{order}\text{rate}{\text{constant and its unit is t}}^{-1}\\ \text{N-}\text{number}\text{of}\text{radioactive}\text{nuclei}\text{present}\text{at}\text{time}\text{'t}\end{array}$

  Suppose the number of radioactive nuclei at time zero is ${\text{N}}_{\text{0}}$ and at a time t is ${\text{N}}_{\text{t}}$,

                                   $\text{ln}\frac{{\text{N}}_{\text{t}}}{{\text{N}}_{\text{0}}}\text{=}\text{-kt}$

• Half-life of radioactive decay is the time required for a radioactive sample to decay to one half of the atomic nucleus.

                    Half-life of the radiation,  $t_{1/2}=\frac{0.693}{k}$

• The level considered acceptable by EPA is 4 picocuries.

$\text{1}\text{picocurie}\text{=}\text{3}\text{.70}\text{×}{\text{10}}^{\text{-2}}\text{disintegration}\text{per}\text{sec}$

To determine: Rate constant k for the given reaction.