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Technetium-99 (æ?r) has an excited state that decays by emission of a gamma ray. The half-life of the excited state is 360 min. What is the activity, in curies, of \(1.00 \mathrm{mg}\) of this excited isotope? Because we have the half-life \(\left(t_{1 / 2}\right)\) we can determine the decay constant since \(\lambda t_{1 / 2}=0.693 .\) The activity of a sample is \(\lambda N .\) In this case, $$ \lambda=\frac{0.693}{t_{1 / 2}}=\frac{0.693}{21600 \mathrm{~s}}=3.21 \times 10^{-5} \mathrm{~s}^{-1} $$ We also know that \(99.0 \mathrm{~kg}\) of Tc contains \(6.02 \times 10^{26}\) atoms. A mass \(m\) will therefore contain \([m /(99.0 \mathrm{~kg})]\left(6.02 \times 10^{26}\right)\) atoms. In our case, \(m=1.00 \times 10^{-6} \mathrm{~kg}\), and so $$ \begin{aligned} \text { Activity } &=\lambda N=\left(3.21 \times 10^{-5} \mathrm{~s}^{-1}\right)\left(\frac{1.00 \times 10^{-6} \mathrm{~kg}}{99.0 \mathrm{~kg}}\right)\left(6.02 \times 10^{26}\right) \\ &=1.95 \times 10^{14} \mathrm{~s}^{-1}=1.95 \times 10^{14} \mathrm{~Bq} \end{aligned} $$

Step by step solution

01

Determine the decay constant

First, calculate the decay constant \( \lambda \) using the half-life formula: \( \lambda t_{1/2} = 0.693 \). Given \( t_{1/2} = 360 \text{ mins} = 21600 \text{ s} \), thus \( \lambda = \frac{0.693}{21600 \text{ s}} = 3.21 \times 10^{-5} \text{ s}^{-1} \).

02

Calculate the number of atoms in the sample

We know \(99.0 \text{ kg}\) of Tc contains \(6.02 \times 10^{26}\) atoms. For a mass \(m = 1.00 \times 10^{-6} \text{ kg (1 mg)} \), the number of atoms \(N\) is \( \left(\frac{1.00 \times 10^{-6} \text{ kg}}{99.0 \text{ kg}}\right) \times 6.02 \times 10^{26} = 6.08 \times 10^{18}\).

03

Calculate the activity in becquerels

Use the formula for activity \(A = \lambda N\): \(A = (3.21 \times 10^{-5} \text{ s}^{-1}) \times 6.08 \times 10^{18} = 1.95 \times 10^{14} \text{ Bq}\).

04

Convert becquerels to curies

Finally, convert the activity from becquerels to curies. Since \(1\text{ Ci} = 3.7 \times 10^{10}\text{ Bq}\), the activity in curies is \(\frac{1.95 \times 10^{14} \text{ Bq}}{3.7 \times 10^{10} \text{ Bq/Ci}} = 5.27 \text{ Ci}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decay Constant Calculation

The decay constant, denoted as \( \lambda \), is a fundamental concept in radioactive decay. It represents the probability per unit time that a given nucleus will decay. Understanding \( \lambda \) is crucial, as it connects the half-life of a radioactive isotope with its decay kinetics.
To calculate the decay constant \( \lambda \) from the half-life \( t_{1/2} \), use the formula:

• \( \lambda = \frac{0.693}{t_{1/2}} \)

The factor 0.693 is derived from the natural logarithm of 2 (ln(2)), which relates to the exponential nature of decay.
In practical terms, if you know the half-life of an isotope, just divide 0.693 by the half-life (converted to the desired unit) to find \( \lambda \). For example, if Technetium-99's half-life is 360 minutes, convert that time to seconds for uniformity. Then, calculate \( \lambda \) by plugging into the formula to get \( \lambda = 3.21 \times 10^{-5} \text{ s}^{-1} \). This conversion ensures \( \lambda \) is in units \( \text{s}^{-1} \), which is standard for decay constants.

Half-Life

The half-life of a radioactive substance is the time it takes for half of the radioactive nuclei in a sample to decay. It is an essential aspect in nuclear physics, as it provides insight into the rate at which a isotope disintegrates.
Half-life is often used in a variety of applications such as radiometric dating, medicine, and nuclear power. It is unique to each isotope and remains constant over the isotope's lifetime. Understanding half-life helps quantify and predict the long-term behavior of radioactive substances.
Since half-life (\( t_{1/2} \)) is related inversely to the decay constant, they are used interchangeably to describe the rate of decay. For Technetium-99, a half-life of 360 minutes (or 6 hours) tells us that after 6 hours, only half of the initial amount of the isotope will remain radioactive.

Activity Calculation

The concept of "activity" in radioactive materials pertains to how many atoms in a sample decay per unit of time. It's often expressed in terms of becquerels (Bq) or curies (Ci), reflecting the intensity of the radioactive substance.
Activity (\( A \)) is calculated using the formula:

• \( A = \lambda N \)

where \( N \) is the number of radioactive atoms present and \( \lambda \) is the decay constant.
In the case of our example with Technetium-99, once you have determined \( \lambda = 3.21 \times 10^{-5} \text{ s}^{-1} \) and know \( N \) (or number of atoms), you can calculate activity. By plugging the values, you determine the sample's activity in becquerels, which can be further converted to curies using the conversion factor \( 1 \text{ Ci} = 3.7 \times 10^{10} \text{ Bq} \). This conversion is particularly useful in contexts where traditional units like curies are preferred over becquerels, such as in the medical field.