(II) Calculate the mass of a sample of pure (40)/(19) K , with an initial decay rate of 2.0 ×10^

step 1 Okay, so this problem gives us the, that we're using potassium. It has an initial decay rate of

(II) Calculate the mass of a sample of pure (40)/(19) K ,...

Key Concepts

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This process occurs randomly for each nucleus, but statistically follows predictable patterns that can be described using mathematical models such as the exponential decay law.

Half-Life

The half-life of a radioactive isotope is the time required for half of the original number of radioactive nuclei to decay. It is a characteristic property of each radioactive substance and is used to calculate the decay constant and predict how the sample's activity decreases over time.

Decay Constant

The decay constant is a parameter that quantifies the probability per unit time that a nucleus will decay. It is directly related to the half-life by the relation ? = ln(2)/half-life, and it is a key factor in determining the disintegration rate or activity of a radioactive sample.

Activity

Activity is the rate at which a radioactive sample undergoes decay, typically measured in disintegrations per second (Becquerels). It reflects the number of nuclear decays occurring per second and is mathematically related to the number of radioactive atoms present and the decay constant.

Atomic Mass and Stoichiometry

Atomic mass and stoichiometry involve converting between the number of atoms in a sample and its mass. This is accomplished using quantities such as Avogadro’s number and the molar mass of the element, allowing one to calculate the mass of a sample based on the number of radioactive nuclei deduced from the observed activity.

Transcript

00:01 Okay, so this problem gives us the, that we're using potassium.

00:07 It has an initial decay rate of that, and then we have its half -life, and then our goal is to get the mass of the sample.

00:16 And so i'm going to go ahead and write out these givens.

00:20 So r is equal to 2 .05 times 10 to the 5, oops, not 2 .05, just plain 2 .05.

00:30 2 .0, 10 to the 5.

00:36 Inverse seconds.

00:39 And then i also want to write down the molar mass of potassium that isotope.

00:46 So we'll use that later, 39 .964 grams per mole.

00:58 And yeah, so now i think i'm ready to go.

01:01 So basically, the initial activity rate, i hope this is, the book somewhere, but it's just the initial number times lambda, and you get that from taking a derivative of n equals n -e -e -to -the -minus t -lam, and evaluating it time equals zero.

01:19 And so therefore, n -not is equal to r -not divided by lambda.

01:26 And then we know lambda is the l -n of two times divided by the half -life.

01:37 So times divided by t one -half.

01:39 Oh yeah, and i'll go ahead and put t -1 -half up here.

01:43 And when i look that up, it was 1 .265 times 10 to the nine years.

02:05 Okay.

02:07 So now to continue this calculation, so we just want to plug in our r not, which is this.

02:15 And then t1 .5, and you also want to convert that to years.

02:20 So i guess i'll just go ahead and write down what i put in my calculator.

02:25 So 2 .0.

02:26 Oh, wow.

02:29 Actually, i'm going to just copy and paste what i put into my calculator to get the first part.