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Chapter 31: Problem 60

$$ \text { A sample of ore containing radioactive strontium }{ }_{38}^{90} \mathrm{Sr} \text { has an } $$ activity of \(6.0 \times 10^{5} \mathrm{Bq} .\) The atomic mass of strontium is \(89.908 \mathrm{u},\) and its half-life is 29.1 yr. How many grams of strontium are in the sample?

Short Answer

Expert verified
The sample contains approximately \(1.185 \times 10^{-7} \text{ grams of strontium.} \)

Step by step solution

01

Calculate the number of atoms using activity

The formula relating the number of radioactive atoms (\( N \)) to the activity (\( A \)) is \( A = \frac{N \ln(2)}{T_{1/2}} \), where \( T_{1/2} \) is the half-life in seconds. First, convert the half-life from years to seconds: \( 29.1 \text{ years} \times 365.25 \text{ days/year} \times 24 \text{ hours/day} \times 3600 \text{ s/hour} = 9.176 \times 10^{8} \text{ s} \). Then rearrange the formula to \( N = \frac{A \cdot T_{1/2}}{\ln(2)} \) and substitute values: \( N = \frac{6.0 \times 10^{5} \times 9.176 \times 10^{8}}{0.693} \approx 7.942 \times 10^{14} \text{ atoms.} \)
02

Convert the number of atoms to moles

To find the number of moles of strontium, use Avogadro's number (\( 6.022 \times 10^{23} \text{ atoms/mol} \)): \( \text{Moles of Sr} = \frac{N}{6.022 \times 10^{23}} = \frac{7.942 \times 10^{14}}{6.022 \times 10^{23}} \approx 1.319 \times 10^{-9} \text{ moles.} \)
03

Calculate the mass of strontium in grams

Finally, calculate the mass of strontium using its molar mass \( (89.908 \text{ g/mol}) \): \( \text{Mass of Sr} = 1.319 \times 10^{-9} \times 89.908 \approx 1.185 \times 10^{-7} \text{ grams.} \)

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Key Concepts

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

Strontium-90
Strontium-90 is a radioactive isotope of strontium, represented as
  • with the chemical symbol \( ^{90}\text{Sr} \) and
  • an atomic number of 38.
This isotope is known for its relatively long half-life of 29.1 years, meaning it takes this amount of time for half of a given quantity to decay. This property makes it particularly interesting and useful in various fields, including scientific research and nuclear medicine. Strontium-90 is a byproduct of nuclear reactions, such as the fission of uranium and plutonium in nuclear reactors. Due to its radioactive nature, it needs to be handled with care, as it poses health risks through exposure. Strontium-90 is also used in medicine, particularly in radiation therapy, where its beta radiation can target and destroy cancer cells.
Half-life Calculation
The half-life calculation is a fundamental concept in understanding radioactive decay. The half-life (
  • designated as \( T_{1/2} \)
  • defines the time it takes for half of the radioactive nuclei in a sample to decay.
This measure is critical in determining the rate at which radioactive substances decrease in activity over time. To calculate the number of atoms remaining at any given time, or how many atoms originally existed based on current activity, we use the half-life formula involving activity \( A \), Avogadro's Number, and the decay constant \( \ln(2)/T_{1/2} \). This formula allows scientists and engineers to determine the decay period and plan for safe disposal or usage of radioactive materials. By knowing the half-life of Strontium-90, which is 29.1 years, we can understand how long it will remain active and manage its applications or removal accordingly.
Atomic Mass
The atomic mass of an element is the weighted average mass of the atoms in a naturally occurring sample of the element, measured in atomic mass units (u). In the case of Strontium-90:
  • the atomic mass is given as 89.908 u,
  • which indicates a very close approximation to the isotope's actual weight.
Atomic mass is crucial when calculating the number of moles from a given mass of an element, or conversely, calculating the mass from a given number of moles. This principle was applied in the example exercise, where the number of moles of Strontium-90 was determined from the number of atoms. The calculated moles were then multiplied by the atomic mass to find the sample's total mass in grams. Understanding atomic mass thus plays a significant role in chemical reactions, nuclear physics, and any calculations involving quantifying matter at the molecular or atomic level.

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Most popular questions from this chapter

$$ \text { Radon }{ }_{86}^{220} \mathrm{Rn} \text { produces a daughter nucleus that is radioactive. The } $$ daughter, in turn, produces its own radioactive daughter, and so on. This process continues until lead \({ }_{\mathrm{} 82}^{208} \mathrm{Pb}\) is reached. What are the total number \(N_{a}\) of \(\alpha\) particles and the total number \(N_{\beta}\) of \(\beta^{-}\) particles that are generated in this series of radioactive decays?

A copper penny has a mass of \(3.0 \mathrm{g}\). Determine the energy \((\mathrm{in}\) MeV) that would be required to break all the copper nuclei into their constituent protons and neutrons. Ignore the energy that binds the electrons to the nucleus and the energy that binds one atom to another in the structure of the metal. For simplicity, assume that all the copper nuclei are \({ }_{29}^{63} \mathrm{Cu}\) (atomic mass \(=62.939598 \mathrm{u})\).

Radioactive Dating of a Mystery Object. In a science fiction movie, a strange object is discovered buried deep in the ice of western Antarctica. It appears to be a radioisotope thermoelectric device from a spacecraft, the function of which is to convert thermal energy released in the decay of its radioactive contents into electrical energy. The radioactive material is identified as americium-241 ( \({ }_{95}^{241} \mathrm{Am}\) ), which has a half-life of 432 years. (a) If \({ }_{95}^{24}\) Am undergoes alpha decay, what is its daughter nucleus? (b) What is the activity you would expect for \(1.00 \mathrm{g}\) of \({ }_{95}^{241} \mathrm{Am}\) (in Bq)? (c) In the movie, \(1.00 \mathrm{g}\) of material is extracted from the device, and the activity is measured to be \(4.00 \times 10^{10}\) Bq. Assuming the device was initially loaded with \(100 \%^{241} \mathrm{Am}\) how old is the device? Express your answer in years.

$$ \text { A one-gram sample of thorium }{ }_{90}^{228} \text { Th contains } 2.64 \times $$$10^{21}\( atoms and undergoes \)\alpha\( decay with a half-life of 1.913 yr \)\left(1.677 \times 10^{4} \mathrm{h}\right)\( Each disintegration releases an energy of 5.52 MeV \)\left(8.83 \times 10^{-13} \mathrm{J}\right) .\( Assume that all of the energy is used to heat a \)3.8-\mathrm{kg}\( sample of water. Concepts: (i) How much heat \)Q\( is needed to raise the temperature of a mass \)m\( of water by \)\Delta T\( degrees? (ii) The energy released by each disintegration is \)E\(. What is the total energy \)E_{\text {tural }}\( released by a number \)n\( of disintegrations? (iii) What is the number \)n\( of disintegrations that occur during a time \)t ?\( Calculations: Find the change in temperature of the \)3.8-\mathrm{kg}$ sample of water that occurs in one hour.

$$ \text { Osmium }{ }_{76 }^{191} \text { Os (atomic mass }=190.960920 \mathrm{u} \text { ) is converted into } $$ iridium \({ }_{77}^{191} \) Ir (atomic mass \(=190.960584 \mathrm{u}\) ) via \(\beta\) -decay. What is the energy (in MeV) released in this process?
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