Indocr air pollution. Uranium is naturally present in rock...

Indocr air pollution. Uranium is naturally present in rock and soil. At one step in its series of radioactive decays, $^{233} \mathrm{U}$ produces the chemically incrt gas radon- $222,$ with a half-life of 3.82 days. The radon sceps out of the ground to mix into the atmosphere, typically making open air radioactive with activity 0.3 $\mathrm{pCi} / \mathrm{L}$ . In homes, 222 $\mathrm{Rn}$ can be a serious pollutant, accumulating to reach much higher activities in enclosed spaces. If the radon radioactivity bexceeds $4 \mathrm{pCi} / \mathrm{L},$ the Environmental Protection Agency suggests taking action to reduce it, such as by reducing infiltration of air from the ground. (a) Convert the activity 4 $\mathrm{pCi} / \mathrm{L}$ to units of becquerels per cubic meter. (b) How many 22 $\mathrm{Rn}$ atoms are in one cubic meter of air displaying this activity? (c) What fraction of the mass of the air does the radon constitute?

Key Concepts

Half-Life and the Decay Constant

The half-life of a radioactive isotope is the time required for half of the radioactive atoms to decay and is related to the decay constant through the equation ? = ln(2)/t1/2. Understanding this relationship is essential for calculating decay rates and predicting the behavior of radioactive substances over time. It allows for the translation of a half-life, which is often experimentally measured, into the decay constant needed for further quantitative analysis.

Mass Fraction Calculation

Calculating the mass fraction involves determining how much of a given component, such as radioactive radon, contributes to the total mass of a mixture like air. This is done by comparing the mass of the radioactive atoms to the mass of the system (in this case, air). This calculation is vital in environmental assessments to evaluate the potential impact of pollutants in terms of their concentration relative to the overall composition of the air.

Radioactivity Unit Conversion

This concept involves converting between different units of radioactivity and volume measurements. For example, activity is commonly given in picocuries (pCi) while the SI unit is the becquerel (Bq), with 1 Bq equaling one decay per second. Additionally, converting concentrations from per liter to per cubic meter requires an understanding that 1 cubic meter equals 1000 liters. These conversion skills are fundamental when comparing measurements or applying regulatory thresholds established in different units.

Radioactive Decay Law

The radioactive decay law, expressed as A = ?N, is a key principle in understanding how the activity (A) of a radioactive substance relates directly to the number (N) of undecayed atoms and the decay constant (?). This law provides the framework for determining how many radioactive atoms are present in a sample and how the activity changes over time. It is crucial for tasks that include calculating the number of atoms from measured activity.

Transcript

00:01 In the first part of this question we just want to convert from four picol curies per litre into back rows per meter cube.

00:18 This is the activity of radon gas in one liter of air.

00:23 I want to find what is the activity in terms of back row for in terms of meter cube, right? for each meter cube.

00:31 So of course the easiest one, one will be from pico and this 10 to power minus 12 kui's now to convert from kui's to back row we multiply by 3 .7 times 10 power of 10 and then we want to convert from liters to meter cube right so there is total of 1 ,000 liters per meter cube right therefore this will have to multiply by 1 ,000 and we should get 148 macros per meter cube.

01:28 Now to find what is the number of radon atoms that is in this one cubic meter that will give us this activity.

01:40 So activity is given as lambda times n, right? n is the number of atoms on nucleite.

01:49 And lambda is the decay constant.

01:54 Right, so what we need here, it would the decay constant since we already have the activity and it goes to and the decay constant is given as long two over the half life.

02:13 We are given that the half life is in terms of 3 .82 days we just have to convert that into seconds.

02:26 Our activity is we calculated above 148 backrose.

02:33 Multiply the by the t half.

02:41 3 .82 days multiplied by 8 .6 ,400.

02:46 This is a number of seconds per day.

02:50 Divided by long 2.

02:54 And we should get 7 .05 times 10 power of 7 .7 atoms...