A certain sample of a radioactive material decays at a rate...

A certain sample of a radioactive material decays at a rate of 548 per second at $t=0$. At $t=48$ minutes, the counting rate has fallen to 213 per second. (a) What is the half-life of the radioactivity? (b) What is its decay constant? (c) What will be the decay rate at $t=125$ minutes?

Key Concepts

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. In the context of radioactive decay, the number of undecayed nuclei (or the observed decay rate) decreases exponentially over time, meaning that the amount of material diminishes rapidly at first and then more slowly as time progresses.

Half-Life

Half-life is the time required for a quantity undergoing exponential decay to reduce to half of its initial value. In radioactive decay, it is a fundamental concept that quantifies how quickly a radioactive substance loses its activity, allowing for comparisons between different isotopes or decay processes.

Decay Constant

The decay constant is a parameter that characterizes the rate at which a substance decays exponentially. It is defined as the probability per unit time that a given nucleus will decay and is related to the half-life by a logarithmic relationship. This constant is crucial for formulating and solving equations involving exponential decay.

Logarithmic Methods

Logarithmic methods are used to solve equations involving exponential decay. They allow one to isolate time or the decay constant in the exponential decay equation by applying logarithms, making it possible to calculate parameters such as half-life or the remaining quantity at a specified time.

Transcript

00:01 In this problem, we're going to use the concepts of activity, the half -life, and the decay constant, along with their equation representations, to figure out the half -life and the decay constant, as well as the activity at some time, t, of this radionucleide, given the activity at t -equal, zero and the activity at some other time.

00:26 And so we can first realize what we were given.

00:30 The activity at the time t equals zero was 548 decays per second, and that decays per second is the unit of beccarell.

00:37 So this is 548 becorels.

00:40 We're also given the activity at the time t equals 48 minutes, which is 213 becorels.

00:47 And we can use this to find first to half -life.

00:50 If we, well, we can first write down the activity at any time t.

00:55 This is given as the activity at the time t equals 0 times e to the negative lambda t.

01:00 That's how it's typically written.

01:03 But we know that lambda is equal to the natural log of 2 over t 1 half based on the typical definition of the half life.

01:14 If you rearrange that just a little bit.

01:16 And if you plug that into this equation, this will be a sub 0, e to the negative natural log of 2 times t.

01:23 Over t one half and now we can solve for t one half and if we do that we will divide both sides by a sub zero and take the natural log so we'll have the natural log of a of t over a sub zero is equal to the negative natural log of two times t over t one half because the natural log of e to the x is just x so this right size we'll just become the exponent.

01:57 So now we can just solve for t one half if we multiply t one half on both sides and divide by this on both sides.

02:07 So we'll get that t one half is equal to the negative natural log of two times t over the natural log of a of t over a 0, which is relatively complicated at first.

02:23 But this allows us to calculate the half -life at any time.

02:25 Time t if we know the activity at that time.

02:29 And we were given that at time 48 minutes, we were given the activity...

Please give Ace some feedback