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Chapter 13: Problem 25

A sample of radioactive material is said to be carrier-free when no stable isotopes of the radioactive element are present. Calculate the mass of strontium in a carrierfree \(5-\mathrm{mCi}\) sample of \({ }^{90} \mathrm{Sr}\) whose half-life is \(28.8 \mathrm{yr}\).

Short Answer

Expert verified
The mass of strontium in the given carrier-free \(5-\mathrm{mCi}\) sample of \({ }^{90} \mathrm{Sr}\) is \(2.77 x 10^{-14} g\)

Step by step solution

01

Calculation of Number of Atoms in Sample

The radioactivity (given in mCi) of a sample can be used to calculate the number of atoms in the sample. Convert radioactivity in mCi to Ci. This is done because the conversion factor between Curie and atoms is based on 1 Ci: \(1 mCi = 10^{-3} Ci\). So, the radioactivity of the sample = \(5x10^{-3} Ci\). Then apply the conversion factor which states that 1 Ci of radioactivity equals \(3.7x10^{10} decays/s\). The radioactivity of the sample in decays per second = \(5x10^{-3} Ci x 3.7x10^{10} decays/s.Ci = 1.85x10^{8} decays/s\). As the decay rate equals the number of radioactive atoms present, number of atoms of Sr-90 in the sample = \(1.85x10^{8} atoms\)
02

Conversion of Number of Atoms to Mass

Now, we can convert the number of atoms into mass using the atomic weight of Sr-90. The atomic weight is 90 g/mol. Using the Avogadro's number (used to convert moles to atoms) which is \(6.02x10^{23} atoms/mol\), we calculate the mass of the sample. Mass of the sample = \( (1.85x10^{8} atoms / 6.02x10^{23} atoms/mol) x 90 g/mol = 2.77 x 10^{-14} g\)

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Key Concepts

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

Carrier-free Isotopes
Carrier-free isotopes refer to samples of a radioactive element that do not contain any stable isotopes of that element. This term is important in ensuring the purity of a radioactive sample, often used in scientific research or medical applications. In a carrier-free sample, all of the isotopes present are radioactive. This absence of stable isotopes means the sample's behavior is entirely dictated by the radioactive isotopes present.
This purity is crucial for precise experimental results.
Having a carrier-free sample allows for unambiguous measurements of the isotope's properties without interference from stable isotopes. Here, the strontium sample is ensured to be fully composed of \(^{90}\text{Sr}\) atoms, allowing accurate calculations of its radioactivity and mass.
Half-life Calculation
Half-life is a key concept in understanding radioactive decay, defined as the time it takes for half of a radioactive sample to decay.
For \(^{90}\text{Sr}\), we know that its half-life is 28.8 years. This constant gives us a measure of how long the isotope remains active. Half-life is used to calculate how many atoms from a given sample decay over a certain period.
Knowing an isotope's half-life is crucial in different scenarios:
  • In waste management, planning how to store spent materials safely
  • In medicine, calculating dosages for treatments.

The step-by-step solution uses radioactivity to find out how many \(^{90}\text{Sr}\) atoms exist, applying this concept.
Strontium-90
Strontium-90 is a radioactive isotope of strontium, prominently known for its role in the aftermath of nuclear reactions and explosions. It is a beta emitter and has a half-life of 28.8 years. This considerable half-life signals long-term persistence in the environment, which is significant for both safety assessment and radiological studies.
Strontium-90 is used in various fields, notably:
  • Medicine, for radiation therapy in specific types of cancer
  • Industry, for applications such as radiological tracing and testing.

Its decay into \(^{90}\text{Y}\) is a significant source of beta radiation. In nuclear science, studying \(^{90}\text{Sr}\) provides vital insights into nuclear fission processes, environmental contamination, and radioactive decay sequences. The exercise highlights the calculation of \(^{90}\text{Sr}\) mass in a carrier-free sample, utilizing its radioactive properties and half-life.

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Most popular questions from this chapter

The concept of radioactive half-life was described in Section 13.4, and Equation \(13.11\) gives the relationship between \(T_{1 / 2}\) and \(\lambda\). Another parameter that is often useful in the description of radioactive processes is the mean life, \(\tau\). Although the half-life of a radioactive isotope is accurately known, it is not possible to predict the time when any individual atom will decay. The mean life is a measure of the average length of existence of all the atoms in a particular sample. Show that \(\tau=1 / \lambda\). (Hint: Remember that \(\tau\) is essentially an average value, and use the fact that the number of atoms that decay between \(t\) and \(t+d t\) is equal to \(d N\). Furthermore, note that these \(d N\) atoms have a finite time of existence, \(t\) )

Consider the hydrogen atom to be a sphere of radius equal to the Bohr radius, \(a_{0}\), given by Equation \(3.29\) in Chapter 3, and calculate the approximate value of the ratio of the nuclear mass density to the atomic mass density.

A laboratory stock solution is prepared with an initial activity due to \({ }^{24} \mathrm{Na}\) of \(2.5 \mathrm{mCi} / \mathrm{mL}\), and \(10 \mathrm{~mL}\) of the stock solution is diluted (at \(t_{0}=0\) ) to a working solution with a total volume of \(250 \mathrm{~mL}\). After \(48 \mathrm{~h}\), a \(5-\mathrm{mL}\) sample of the working solution is monitored with a counter. What is the measured activity? (Note: \(1 \mathrm{~mL}=\) 1 milliliter, and the half-life of \({ }^{24} \mathrm{Na}\) is \(15.0 \mathrm{~h}\).)

When a material of interest is irradiated by neutrons, radioactive atoms are produced continually and some decay according to their given half-lives. (a) If radioactive atoms are produced at a constant rate \(R\) and their decay is governed by the conventional radioactive decay law, show that the number of radioactive atoms accumulated after an irradiation time \(t\) is $$ N=\frac{R}{\lambda}\left(1-e^{-\lambda t}\right) $$ (b) What is the maximum number of radioactive atoms that can be produced?

(a) Find the radius of the 12 C nucleus. (b) Find the force of repulsion between a proton at the surface of a \({ }^{12}{6}\) C nucleus and the remaining five protons. (c) How much work (in MeV) must be done to overcome this electrostatic repulsion and put the last proton into the nucleus? \((\mathrm{d})\) Repeat \((\mathrm{a}),(\mathrm{b})\), and (c) for \({ }_{92}^{238} \mathrm{U}\).
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