A radioactive sample has equal numbers of ^15 O and ^19 O nuclei. Use the half-lives found in

A radioactive sample has equal numbers of ^15 O and ^19 O...

Key Concepts

Half-Life

Half-life is the time required for a quantity of radioactive nuclei to reduce to one-half of its initial amount. It is a fundamental concept in nuclear physics that describes the characteristic rate at which a radioactive substance decays, independent of its initial quantity.

Exponential Decay

Exponential decay describes the process by which the number of radioactive nuclei decreases over time in a way that is proportional to the current number. This process is mathematically modeled by an exponential function, reflecting the consistent, time-independent probability of decay for each nucleus.

Decay Constant

The decay constant is a parameter that quantifies the probability per unit time that a nucleus will decay. It directly relates to the half-life of a radioactive substance through the equation ? = ln(2)/T?/?, linking the probability model with the observable decay rate.

Radioisotopic Ratio Analysis

Radioisotopic ratio analysis involves comparing the decay behavior of two or more radioactive isotopes. By examining the ratios of their remaining quantities over time, one can infer differences in their decay rates and underlying nuclear properties, a technique that is essential in fields such as nuclear chemistry and radiometric dating.

Percent Decay Calculation

Percent decay calculation determines the fraction of a radioactive substance that has decayed over a given period, expressed as a percentage. This calculation is often used to quantify the extent of decay and assess the progress of a radioactive process relative to the original amount present.

Transcript

00:01 Hello, and in this question here we're going to be looking at the radioactyl decay of two different isotopes of oxygen.

00:08 These isotopes are oxygen 15 and oxygen 19.

00:13 So the number of nuclei after a given time t, so we're going to represent this with capital n, we're going to use a subscript 15 to refer to the number of oxygen 15 nuclei, and we'll use a subscript 19 to refer to the number of oxygen 19.

00:32 Nuclei.

00:33 But anyway, the number of oxygen 15 nuclei as a function of time is equal to the initial number of oxygen 15 nuclei multiplied by e to the minus lambda 15 times t, where lambda with subscript 15 is the decay constant of oxygen 15 and that is equal to, so lambda 15 is equal to, so lambda 15, is equal to the natural log of 2 divided by the half -life of oxygen 15.

01:09 Similarly, the number of nuclei of oxygen 19 as a function of time is equal to the initial number of oxygen 19 multiplied by e to the power of minus lambda, subscript 19, multiplied by time, where the decay constant is defined in the same way for oxygen 19 as it is for oxygen 15.

01:33 We're also told that initially that the initial number of oxygen 15 nuclei equals the initial number of oxygen 19 nuclei, and for ease of notation, we're just going to call this n -zero.

01:49 So because these two substances have different half -lives, they're going to decay at different rates.

01:56 And we're going to want to find out, when the number of oxygen 50, there's twice as many oxygen 15 nuclei as there is oxygen 19 nuclei.

02:10 So that means we're looking for when n, so the number of oxygen 15 nuclei is equal to twice the number of oxygen 19 nuclei.

02:21 We want to find out the time when this condition is true.

02:24 So, filling in these two expressions up here into our relation, well, first let's just say 2 is equal to the number of nuclei of oxygen 15 divided by the number of nuclear of oxygen 19.

02:43 I'm filling in what the number of nuclei is, where we get n0 multiplied by e to the minus lambda 15 times time, all divided by n0.

02:55 E to the minus 19.

02:59 Sorry, e to the minus lambda, subscript 19 multiplied by time.

03:06 Okay, well, we can cancel our n zeros to get two is equal to e to the, well, we're just going to say plus lambda subscript 19 minus lambda subscript 15 multiplied by time.

03:28 We now need to rearrange this equation for time.

03:31 To do this, we take the natural log of both sides of this equation.

03:37 To guess that time is equal to 1 divided by the decay constant of oxygen 19 minus the decay constant of oxygen 15, all multiplied by the natural log of 2.

03:51 So if we can determine the decay constants, we can find out the time for when this equation is, for when there is twice as many oxygen 50 nuclei as oxygen 19 nuclei.

04:04 Well, the decay constant of oxygen 50 is equal to the natural log divided by a natural log of 2 divided by the half -life of oxygen 15...

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