The activity of a radioactive sample was measured over 12 h , with the net count rates shown in

The activity of a radioactive sample was measured over 12 h...

Key Concepts

Half-life

Half-life is defined as the time required for half of the radioactive nuclei in a sample to decay. It is a key characteristic of a radioactive isotope and provides a convenient measure for comparing the decay rates of different substances. The concept of half-life is widely used in various fields, including nuclear physics, archaeology, and medicine, to estimate time periods and understand the kinetics of radioactive decay.

Counting Statistics

Counting statistics plays a vital role in the measurement of radioactive decay, as the detection of individual decay events is inherently random. Statistical analyses, often based on the Poisson distribution, are used to interpret the variability in measured net count rates. This approach allows for the estimation of uncertainties in the observed data, ensuring that the interpretation of activity measurements accounts for the probabilistic nature of radioactive decay.

Exponential Decay Law

The exponential decay law mathematically describes how the number of undecayed nuclei in a radioactive sample decreases exponentially over time. It is expressed by the equation N(t) = N0 exp(-?t), where N0 is the initial number of nuclei, ? is the decay constant, and t represents time. This law forms the foundation for analyzing decay data and estimating parameters such as the decay constant and half-life.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. This process is fundamentally random, with each nucleus possessing a fixed probability of decaying in a given time interval, leading to the characteristic behavior seen in nuclear physics and various applications such as dating techniques and nuclear medicine.

Activity

Activity refers to the rate at which a sample of radioactive material decays, typically measured as the number of disintegrations per unit time. This measurement is crucial in quantifying the strength or intensity of a radioactive source and is used to determine both safety precautions and the sample's utility in various scientific and industrial applications.

Transcript

00:01 Right, this question we are given a table of different count rates as well as the time.

00:12 So what we want to do over here is to plot a graph of long of the count rates over the time.

00:25 So we will take long count rates as our y -axis and our x -axis would be time.

00:41 And why are we doing this? because count rates is basically telling us the activity at that point in time.

00:49 And we know that activity is equals to the initial activity multiplied by exponential negative lambda.

01:01 Now if we take the natural log on both sides, we will get lon of a -0 -e -negative landa -t, which is equals to lone of a -0 plus lone of e negative lambda t.

01:25 Now lone of a -0 would be a constant plus negative lambda t, right, or just minus lambda t.

01:34 Now if we plot lone of a against t, t, what we will get is negative lambda will be our gradient of this graph, right, it will be a linear graph.

01:49 This is the gradient and long of a knot will be our y intercept.

02:01 And therefore, by plotting this graph, you will be able to find lambda as well as the initial activity.

02:14 Now let us plot the different points.

02:18 Right, so i'm going to plot for you, but what you should do is you take the table and you're going to take the lawn of all the count rates and then this will be your y points, plot them into x points and we should get a curve of something like this.

02:34 And i've drawn the blue line, which is the best fit line across the entire data points.

02:42 Right, so this is the best fit line.

02:43 Now to find what is the decay constant lambda, we will need to find what is our slope.

03:03 Alright, our gradient over here we can calculate by taking two different points.

03:10 So i'm going to take two points, one over here, and the other point i'm going to take is over here.

03:22 These points are exactly on the grid, so it is more complex.

03:26 Convenient, right? so this point over here is 8 .1, whereas the point over here on the right, so it is not 8, 1, it should be 1 8, right, x, y, and the point over here is 9, 6.

03:52 So the slope or the gradient, and we're looking at is just a negative 8 minus 6...

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