La désintégration d'un nucléide radioactif est une...

La désintégration d'un nucléide radioactif est une transformation qui obéit à une vitesse d'ordre 1, et qui est souvent prise comme exemple dans une étude cinétique. a) Quelle est l'activité exprimée en curies, d'une source radioactive constituée par $500 \mathrm{mg}$ de strontium-90? b) Quelle fraction de cette activité subsiste-t-il un an plus tard? c) Au bout de combien de temps cette activité est-elle réduite de $10 \%$ ? (les données nécessaires se trouvent dans l'Annexe $\mathrm{O}$ )

La désintégration d'un nucléide radioactif est une transformation qui obéit à une vitesse d'ordre 1, et qui est souvent prise comme exemple dans une étude cinétique.
a) Quelle est l'activité exprimée en curies, d'une source radioactive constituée par $500 \mathrm{mg}$ de strontium-90?
b) Quelle fraction de cette activité subsiste-t-il un an plus tard?
c) Au bout de combien de temps cette activité est-elle réduite de $10 \%$ ?
(les données nécessaires se trouvent dans l'Annexe $\mathrm{O}$ )

Key Concepts

Unit Conversion

Unit conversion is a technique used to convert a measurement expressed in one set of units to another set. In the context of radioactivity, this often includes converting between units such as curies and becquerels, which is essential for comparing and analyzing data from different sources or experiments.

Half-Life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. It is an important measure of the stability and longevity of a radioactive material and is directly related to the decay constant through the relation t½ = ln(2)/?.

Decay Constant

The decay constant (?) characterizes the probability per unit time that a nucleus will decay. It is linked to the half-life of a substance and is a key parameter in determining the rate of exponential decay.

Radioactive Activity

The activity of a radioactive sample is defined as the number of nuclear disintegrations per unit time. It quantifies how 'active' or 'intense' a radioactive source is and is typically expressed in units such as becquerels or curies.

First-Order Kinetics

In first-order kinetics, the rate of decay is directly proportional to the quantity of the substance that remains. This implies that the time it takes for a given amount of material to decay is constant relative to its current amount, leading to an exponential decrease over time.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. It is a random process at the level of single atoms, in that it is impossible to predict when a particular atom will decay, but given a large number of atoms, the decay rate can be described statistically.

Exponential Decay Law

The exponential decay law mathematically models the decrease in the number of radioactive nuclei (or its activity) over time. It is represented by the equation A(t) = A? exp(-?t), where A(t) is the activity at time t, A? is the initial activity, and ? is the decay constant.

Transcript

00:01 In this scenario, we're given an element strontium, and we're asked, we're also told that the function, a of t, is equal to a0, which is the initial amount of the strontium, times e to the negative 0 .0044t power is equal to the amount present.

00:19 So a of t is equal, is the amount present during the time t years of strontium.

00:26 And we're also given that the scientist has 500 grams of the strontium.

00:31 And for part a, we're asked to find the decay rate.

00:36 So for that, we can just look at the function we're given.

00:39 And we have to realize that when it's written in this notation, the decay is this formula over here, this number.

00:46 So we can take the absolute value of negative 0 .0044.

00:51 And this is actually a percentage point.

00:53 So we want to convert it back from decimal to percentage.

00:58 We want to move the decimal towards the right two spaces, and we get that the decay rate is equal to 0 .44%.

01:07 So the answer to part a is 0 .44.

01:11 And as for part b, we're asked how much trantium 90 is left after 10 years.

01:18 So t is equal to 10.

01:20 And we can just write the equation here.

01:24 So original amount as the problem said is 500 grams.

01:27 So 500 times e to the negative.

01:31 And since we're multiplying this number by 10, we get 0 .044.

01:36 And since e is a number, we can just calculate it as it is.

01:40 So 500 times e to the negative 0 .044 is equal to 478 .4.

01:52 About 400 so 478 4777 grams after 10 years so once we have this figure we asked part c what when will there be 400 grams left so now we know what a of t is equal to so we just fill in the values we know 400 is equal to again we're also we're still given 500 grams in the initial amounts and e times negative 0 .0044t.

02:33 So we're trying to find t in the scenario.

02:36 We can divide both sides by 500.

02:39 So we got 400 divided by 500 is equal to four fits or 0 .8 .0 .8 is equal to e to the negative 0 .0044t power...

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