When a radioactive substance decays, the fraction of atoms present at time t is f(t)=e^-k t, w

When a radioactive substance decays, the fraction of atoms...

Key Concepts

Exponential Decay

Exponential decay describes a process where the quantity of a substance decreases at a rate proportional to its current amount. In the context of radioactive decay, it is modeled by the function f(t) = exp(–kt), where the remaining fraction of material decreases exponentially with time.

Decay Constant

The decay constant, denoted by k, is a positive parameter that indicates the probability per unit time that a radioactive atom decays. It is central to characterizing the rate of radioactive decay and is linked to the half-life of the substance through the equation k = ln(2)/Half-life.

Integration by Parts

Integration by Parts is a fundamental technique in calculus used to transform the integral of a product of functions into a potentially simpler form. In this problem, it is used to convert the integral expression for the average lifetime, originally given by –??? t f'(t) dt, into the more manageable form ??? f(t) dt.

Half-Life

The half-life of a radioactive substance is the time required for half of the original atoms to decay. It is an important parameter that quantifies the decay process and is inversely related to the decay constant via the relation k = ln(2)/Half-life.

Average Lifetime

The average lifetime, or mean life, of a radioactive atom is the expected time until decay. Through integration by parts, it is shown that this average lifetime can be expressed as A = ??? f(t) dt, and for an exponential decay process, it simplifies to A = 1/k. This relationship connects the decay constant directly to the average decay time.

Transcript

00:01 Okay, we have a radioactive substance and its decay constant is the letter k, and k is positive.

00:09 And then f of t is the fraction of the atoms left after time t, and that's equal to e to the minus kt.

00:17 Then the average life of an atom is equal to a, and it's this improper integral zero to infinity, t, f prime of t, dt.

00:27 So we're going to show that a is equal to the integral of f of t, and then we're going to find a using that f of t above there.

00:35 And then the last question is, what's the average life of radon whose half -life is 3 .825 days? okay.

00:43 So first we're going to start with a equals minus zero to infinity, t, f prime of t, dt.

00:54 And we're going to say that as minus the limit as r goes to infinity.

00:59 0 to r, t, f prime of t, dt.

01:06 All right, we're going to integrate that by parts.

01:08 So we're going to let u, b, t, dv, be f prime of t, dt, so du would be dt, integral of f prime of t, dt, that be f of t.

01:26 So it is equal to minus the limit as r goes to infinity.

01:34 Uv so t f of t minus the integral v d u from zero to r so this equal to minus the limit as r goes to infinity r f of r minus minus the integral zero to r f of t d t minus zero f of zero d t minus zero f of zero minus the integral 0 to 0 f of t d t.

02:30 Okay, so we get minus.

02:33 Okay, r is going to infinity.

02:35 So we have infinity times f of infinity.

02:38 Can i remember, f is how much is left after time t? well, after infinity, it should have decayed to zero.

02:49 So the limit as r goes to infinity of f of r is zero.

02:56 All right.

02:56 So and then we have minus this integral.

03:05 So we have minus the limit as r goes to infinity, minus integral, zero to r, f of t, dt, minus zero times f of zero, that's zero, minus the integral from zero to zero, f of t, dt, and that's zero.

03:29 Oh, going crickety.

03:34 So we get a equals minus minus, that's.

03:36 Plus 0 to infinity f of t d t, which is what we're supposed to get.

03:44 Okay, so yay for us so far.

03:46 Now it says plug in what f of t is.

03:49 It is e to the minus kt.

03:52 So a is zero to infinity, e to the minus kt, dt...