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

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

The activity of a radioactive sample was measured over $12 \mathrm{h}$ with the net count rates shown in the accompanying table. (a) Plot the logarithm of the counting rate as a function of time. (b) Determine the decay constant and half-life of the radioactive nuclei in the sample. (c) What counting rate would you expect for the sample at $t=0 ?$ (d) Assuming the efficiency of the counting instrument is $10.0 \%$, calculate the number of radioactive atoms in the sample at $t=0.$ $$\begin{array}{cc}\text { Time (h) } & \text { Counting Rate (counts/min) } \\\hline 1.00 & 3100 \\2.00 & 2450 \\4.00 & 1480 \\6.00 & 910 \\8.00 & 545 \\10.0 & 330 \\12.0 & 200 \\ \hline\end{array}$$

The activity of a radioactive sample was measured over $12 \mathrm{h}$ with the net count rates shown in the accompanying table. (a) Plot the logarithm of the counting rate as a function of time. (b) Determine the decay constant and half-life of the radioactive nuclei in the sample. (c) What counting rate would you expect for the sample at $t=0 ?$ (d) Assuming the efficiency of the counting instrument is $10.0 \%$, calculate the number of radioactive atoms in the sample at $t=0.$
$$\begin{array}{cc}\text { Time (h) } & \text { Counting Rate (counts/min) } \\\hline 1.00 & 3100 \\2.00 & 2450 \\4.00 & 1480 \\6.00 & 910 \\8.00 & 545 \\10.0 & 330 \\12.0 & 200 \\
\hline\end{array}$$

Key Concepts

Half-Life

The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. Mathematically, it is related to the decay constant by the expression t?/? = ln(2)/?, and it provides a convenient measure of the rate of decay regardless of the initial quantity of material.

Activity and Number of Radioactive Atoms Relationship

The activity of a radioactive sample is directly related to the number of radioactive atoms and the decay constant through the relation A = ?N. This relationship allows one to calculate the total number of radioactive atoms from the measured activity, after accounting for instrument efficiency and other experimental factors.

Decay Constant

The decay constant (?) is a parameter that quantifies the probability per unit time that a nucleus will decay. It is the key factor in characterizing a radioactive sample’s rate of decay and is directly related to the half-life of the radioactive substance.

Counting Efficiency

Counting efficiency refers to the fraction of radioactive decays that are actually detected by a measurement instrument. It is a crucial concept in experimental nuclear physics because the observed counting rate must be corrected for this efficiency to accurately reflect the true activity and number of radioactive atoms present in a sample.

Exponential Decay Law

The exponential decay law mathematically models the decrease in activity of a radioactive sample. It states that the activity, or counting rate, at any time is given by an equation of the form A(t) = A? exp(-?t), where A? is the initial activity and ? (lambda) is the decay constant.

Radioactive Decay

This concept refers to the process by which an unstable atomic nucleus loses energy by radiation. In physics and chemistry, it is understood that the rate of decay is proportional to the number of atoms present, leading to a predictable exponential decrease in the number of undecayed nuclei over time.

Logarithmic Transformation for Linearization

Taking the natural logarithm of the exponential decay equation transforms it into a linear relationship (ln A(t) = ln A? - ?t). This transformation simplifies the process of determining the decay constant by allowing one to plot ln(activity) versus time and then perform linear regression to find the slope, which is equal to -?.

Transcript

00:01 Question 48 is based on the data shown in this table, which i got in the textbook.

00:05 It measures the activity of the sample over a period of 12 hours.

00:10 Part a, the question asks to plot the logarithm of the counting rate as a function of the time in hours.

00:15 So i did that in excel.

00:18 I went ahead and i posted here.

00:20 So here we see a relatively straight line.

00:24 Or x -axis, of course, time.

00:28 Sorry, our x -axis is of course time.

00:31 And on our y -axis, we have the natural logarithm of our.

00:34 Counting rate our activity.

00:39 So that means if we have something where we have our activity is our initial activity times our exponential decay by plotting the natural logarithm of this ratio, so r over r not, and then take the natural logarithm of both signs.

00:54 You see here we're plotting on our x -axis.

00:57 It's dependent on two terms, i guess just one, sorry, lambda the decay constant.

01:03 So that means because it's a straight line, the slope of this line would be equivalent to the decay constant, which is pretty useful, actually, and especially in part b, it asks us to determine this decay constant and half -life, where i did this already in excel, i use the graving function to determine what this would be.

01:23 So again, based on this function, that the slope, our decay constant is the slope, and of course, it's negative.

01:30 So the decay concept itself would not be negative, it's just the slope itself has be negative.

01:36 Because our x -axis is in units of hours, then the units for the key constant have to be one over hours.

01:43 Great, and then all it's due to solve for half -life is just take the natural logarithm of 2 and divide that by lambda...

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