Consider a 1.00-kg sample of natural uranium composed...

Consider a $1.00-\mathrm{kg}$ sample of natural uranium composed primarily of $288 \mathrm{U},$ a smaller amount $(0.720 \% \text { by }$ mass) of $^{235} \mathrm{U},$ and a trace $(0.005 \%)$ of $^{234} \mathrm{U},$ which has a half-life of $2.44 \times 10^{5}$ yr. (a) Find the activity in curies due to each of the isotopes. (b) What fraction of the total activity is due to each isotope? (c) Explain whether the activity of this sample is dangerous.

Key Concepts

Radiological Hazard

Assessing radiological hazard involves considering both the amount of radioactivity and the nature of the emitted radiation (such as type and energy). The danger posed by a radioactive substance does not depend solely on its activity level but also on factors like the isotope’s half-life, the type of radiation, and its behavior in biological systems. Proper evaluation considers these factors to determine actual risk to health and safety.

Isotopic Abundance and Specific Activity

Isotopic abundance refers to the relative amounts of different isotopes in a sample, while specific activity describes the radioactivity per unit mass of a substance. Even when an isotope is present in trace amounts, if it has a much shorter half-life, its specific activity can be significantly higher, thus contributing disproportionately to the overall activity.

Units of Activity (Curie and Becquerel)

Activity is commonly expressed in units such as the Curie and the Becquerel. The Curie is an older unit defined as 3.7 x 10^10 disintegrations per second, while the Becquerel is the SI unit and represents one decay per second. Converting between these units is essential for comparing and understanding radioactivity levels.

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This process follows statistical rules, and its progression over time can be quantified by techniques that allow prediction of decay on a macroscopic scale, despite its inherently random nature at the individual nucleus level.

Decay Constant

The decay constant is a proportionality factor that defines the probability per unit time that a single nucleus will decay. It is directly related to the half-life through the equation ? = ln(2)/t_(1/2), and it is fundamental in calculating the activity of a radioactive sample.

Half-life

The half-life of a radioactive isotope is the time required for half of the nuclei in a sample to decay. It is a key parameter for describing the rate of decay and is inversely related to the decay constant. A shorter half-life means a faster rate of decay and typically higher activity per atom.

Activity

Activity is the measure of the number of decays or disintegrations per unit time in a radioactive sample. It is calculated by multiplying the decay constant by the number of radioactive nuclei present, providing a quantitative assessment of the sample's radioactivity.

Transcript

00:01 Hello friends this is the problem based on activity of reductive material activity of reductive material is defined as lambda into n here lambda is decay constant of the sample and is number of reductive nuclei present and decay constant is defined as 0 .693 upon half life so on this concept this problem is based we have to use this concept here it is given sample of 1 kg of uranium in which 0 .72 % is urinium, 0 .05%, 0 .05 % uranium 234 and remaining uranium 238.

01:18 We have to find activity of each isotope be.

01:33 Fraction of each.

01:37 Each isotope with total c -part explain whether the activity of the sample is dangerous or not.

02:12 See here, it is also given half -life of uranium is 2 .44 into 10 to the power 5 here.

02:29 Let us start solving it.

02:32 Mass of uranium 238 in the sample is 0992 .928 in the sample is 0 .992.

02:45 Kg so number of uranium 238 nuclei in the sample mass of uranium 235 upon atomic weight into avogadro number 0 .99286 .023 into 10 to the power 23 upon 0 .23.

03:11 So it is to be 2 .51 into 10 to the power 24 nuclei now, activity of uranium 238 is lambda into an decay constant is 0 .693 upon half life, which is given for uranium 238, 4 .47 10 to the power 9.

03:58 This is half life of uranium 238.

04:02 It is to be used in second.

04:04 So in one year there are 3 .16 into 10 to the power seconds.

04:13 You will get the answer season per second.

04:17 So it is to be converted into curie.

04:26 So you will get 33 into 10 to the power minus 6 curie.

04:33 Or activity of uranium 238 is 33 microcury.

04:52 Now for uranium 235 mass of uranium 235 is 0 .0072 kg.

05:07 So number of uranium 235 nuclei present in the given sample 0 .0072 upon 0 .235 into 6 .023, 10 to the power 23...