A sample consists of 1.00 ×10^6 radioactive nuclei with a half-life of 10.0 h. No other nuclei a

step 1 We have in this question a sample of a radioactive nucleiclite. We are given the initial number

A sample consists of 1.00 ×10^6 radioactive nuclei with a...

A sample consists of $1.00 \times 10^{6}$ radioactive nuclei with a half-life of 10.0 h. No other nuclei are present at time $l=0 .$ The stable daughter nuclei accumulate in the sample as time goes on. (a) Derive an equation giving the number of daughter nuclei $N_{d}$ as a function of time. (b) Sketch or describe a graph of the number of daughter nuclei as a function of time. What are the maximum and minimum numbers of daughter nuclei, and when do they occur? What are the maximum and minimum rates of change in the number of daughter nuclei, and when do they occur?

Key Concepts

Differentiation and Rate of Change Analysis

The rate of change of the number of daughter nuclei can be found by differentiating the accumulation function with respect to time. This analysis helps identify when the daughter production is at its fastest (or slows down) and provides insights into the kinetics of the transformation process.

Daughter Nucleus Accumulation

As radioactive parent nuclei decay, they transmute into daughter nuclei. The accumulation of these daughter products over time is directly linked to the decay of the parent nuclei. Understanding this process involves setting up differential equations that describe how the daughter nucleus population increases as the parent population decreases.

Graphical Analysis of Decay Processes

Graphing the number of daughter nuclei as a function of time helps visualize the accumulation process. Such graphs typically show a gradual increase reaching a maximum plateau, reflecting the asymptotic approach to a final value as the parent nuclei become depleted. This visualization aids in understanding the overall behavior of the system, including maximum and minimum values of the daughter nuclei count and their rates of change.

Radioactive Decay

This concept centers on the process by which unstable atomic nuclei lose energy by emitting radiation. The rate at which this decay occurs is typically described by an exponential decay law characterized by a decay constant. It provides the mathematical framework for predicting the number of undecayed nuclei over time.

Half-life and Decay Constant

Half-life is the time required for half of a given number of radioactive nuclei to decay. The decay constant, which is derived from the half-life, is used to quantify the probability per unit time of decay and appears in the exponential decay equation. Together, these concepts allow for the conversion between the discrete notion of half-life and the continuous mathematical model of decay.

Transcript

00:01 We have in this question a sample of a radioactive nucleiclite.

00:06 We are given the initial number is 10 to power 6 nucleates and we are given the half -life is 10 hours.

00:20 Well, what you want to find is the equation for the number of daughter nucleates over time.

00:26 So we start off with just all the nucleide being the parent nucleic.

00:34 Well we know that the amount of parent nucleide after a certain period of time, the parent after some time is given as the initial number multiplied by exponential negative lambda t.

00:56 Where lambda, the decay constant is just on 2.

01:01 Over t half right oh this is number of parents but what we want is the number of daughter nuclei right number of daughter nuclei is basically the number of parents that has decay over time so the number of parent that has decayed be taking the initial amount minus away the number there is left right number that's left and this will give us n n .0 times 1 minus e negative 2 t half times t and this would be equals to the number of daughter nucleic at time t since for every decade parent we get one daughter nuclei therefore we can actually simplify this a bit substituting n not to be 10 of our 6.

02:18 Right, and we can evaluate long 2 over t half.

02:24 We just take long 2 divided by 10 to be exponential negative 0 .0693 times t.

02:36 But take note that our t will need to be in terms of hours, like since we kept t half to be in terms of hours, like 10 hours.

02:47 So this t is in terms of hours.

02:55 Now we can sketch this curve quite simply.

02:58 This is an exponential decay curve...

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