According to current regulations, the maximum permissible dose of strontium-90 in the body of an adult is 1 mCi (1 * 10^-3 Ci). Using the relationship rate = kN, calculate the number of atoms of strontium-90 to which this dose corresponds. To what mass of strontium-90 does this correspond? The half-life for strontium-90 is 28.8 years.

Verified step by step guidance

Step 1: Understand the relationship rate = kN, where 'rate' is the activity in curies (Ci), 'k' is the decay constant, and 'N' is the number of radioactive atoms. The activity given is 1 mCi, which is 1 * 10^{-3} Ci.

Step 2: Calculate the decay constant 'k' using the half-life formula: k = \frac{0.693}{t_{1/2}}, where t_{1/2} is the half-life of strontium-90, which is 28.8 years. Convert the half-life into seconds for consistency in units.

Step 3: Substitute the values of 'rate' and 'k' into the equation rate = kN to solve for 'N', the number of atoms of strontium-90.

Step 4: Use Avogadro's number (6.022 * 10^{23} atoms/mol) to convert the number of atoms 'N' to moles of strontium-90.

Step 5: Calculate the mass of strontium-90 by multiplying the moles obtained in Step 4 by the molar mass of strontium-90, which is approximately 89.9077 g/mol.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This decay occurs at a predictable rate characterized by the half-life, which is the time required for half of the radioactive atoms in a sample to decay. Understanding this concept is crucial for calculating the number of remaining atoms after a certain period and for determining the activity of a radioactive substance.

Activity and Dose

The activity of a radioactive substance, measured in curies (Ci), indicates the rate at which decay occurs, defined as the number of decays per second. The dose refers to the amount of radiation absorbed by the body, and regulations often specify maximum permissible doses to ensure safety. In this context, converting the dose from millicuries to the number of atoms involves understanding the relationship between activity, decay constant, and the number of radioactive atoms present.

Mass and Molar Relationships

To find the mass of a substance from the number of atoms, one must use the molar mass, which is the mass of one mole of a substance expressed in grams. The relationship between the number of atoms, moles, and mass is given by Avogadro's number (approximately 6.022 x 10^23 atoms/mole). This concept is essential for converting the calculated number of strontium-90 atoms into a corresponding mass, allowing for practical applications in health and safety regulations.

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Textbook Question

Nuclear scientists have synthesized approximately 1600 nuclei not known in nature. More might be discovered with heavy-ion bombardment using high-energy particle accelerators. Complete and balance the following reactions, which involve heavy-ion bombardments:

(a) ⁶₃Li + ⁵⁶₂₈Ni → ?

(b) ⁴⁰₂₀Ca + ²⁴⁸₉₆Cm → ¹⁴⁷₆₂Sm + ?

(d) ⁴⁰₂₀Ca + ²³⁸₉₂U → ⁷⁰₃₀Zn + 4 ¹₀n + 2 ?

Textbook Question

In 2010, a team of scientists from Russia and the United States reported creation of the first atom of element 117, which is named tennessine, and whose symbol is Ts. The synthesis involved the collision of a target of ²⁴⁹₉₇Bk with accelerated ions of an isotope which we will denote Q. The product atom, which we will call Z, immediately releases neutrons and forms ²⁹⁴₁₁₇Ts: ²⁴⁹₉₇Bk + Q → Z → ²⁹⁴₁₁₇Ts + 3 ¹₀n (a) What are the identities of isotopes Q and Z? (c) Collision of ions of isotope Q with a target was also used to produce the first atoms of livermorium, Lv. The initial product of this collision was ²⁹⁶₁₁₆Lv. What was the target isotope with which Q collided in this experiment?

Textbook Question

In 2010, a team of scientists from Russia and the United States reported creation of the first atom of element 117, which is named tennessine, and whose symbol is Ts. The synthesis involved the collision of a target of ²⁴⁹₉₇Bk with accelerated ions of an isotope which we will denote Q. The product atom, which we will call Z, immediately releases neutrons and forms ²⁹⁴₁₁₇Ts: ²⁴⁹₉₇Bk + Q → Z → ²⁹⁴₁₁₇Ts + 3 ¹₀n (b) Isotope Q is unusual in that it is very long-lived (its half-life is on the order of 1019 yr) in spite of having an unfavorable neutron-to-proton ratio (Figure 21.1). Can you propose a reason for its unusual stability?

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Textbook Question

Each of the following transmutations produces a radionuclide used in positron emission tomography (PET).

(a) In equations (i) and (ii), identify the species signified as 'X.'

(i) ¹⁴N(p,α)X

(ii) ¹⁸O(p,X)¹⁸F

(iii) ¹⁴N(d,n)¹⁵O

Textbook Question

The nuclear masses of 7Be, 9Be, and 10Be are 7.0147, 9.0100, and 10.0113 amu, respectively. Which of these nuclei has the largest binding energy per nucleon?