Problem 53 - Chemistry: An Atoms-Focused Approach

Question

Complete the following nuclear equations describing the preparation of isotopes for nuclear medicine. a. \(^{197} \mathrm{Au}+? \rightarrow^{199} \mathrm{Hg}+\beta\) b. \(^{64} \mathrm{Ni}+^{1} \mathrm{H} \rightarrow^{64} \mathrm{Cu}+?\) c. \(^{63} \mathrm{Cu}+? \rightarrow^{66} \mathrm{Ga}+^{1} \mathrm{n}\) d. \(^{67} \mathrm{Zn}+^{1} \mathrm{n} \rightarrow^{67} \mathrm{Cu}+?\)

Step-by-Step Solution

Answer

a. The missing particle in the nuclear reaction \(^{197} \mathrm{Au}+? \rightarrow^{199} \mathrm{Hg}+\beta\) is \(^2_2 \mathrm{He}\). b. The missing particle in the nuclear reaction \(^{64} \mathrm{Ni}+^{1} \mathrm{H} \rightarrow^{64} \mathrm{Cu}+?\) is \(^1_0 \mathrm{n}\). c. The missing particle in the nuclear reaction \(^{63} \mathrm{Cu}+? \rightarrow^{66} \mathrm{Ga}+^{1} \mathrm{n}\) is \(^4_2 \mathrm{He}\). d. The missing particle in the nuclear reaction \(^{67} \mathrm{Zn}+^{1} \mathrm{n} \rightarrow^{67} \mathrm{Cu}+?\) is \(^1_1 \mathrm{H}\).

Step 1: Determine missing mass number

Let A be the mass number of the missing particle. Using the conservation of mass numbers, we have: \(197 + A = 199\) Solving for A, we get: \(A = 2\)

Step 2: Determine missing atomic number

Let Z be the atomic number of the missing particle (number of protons). Using the conservation of atomic numbers, we have: \(79 + Z = 80 - 1\) Solving for Z, we get: \(Z = 2\)

Step 3: Identify the missing particle

From the obtained mass and atomic numbers, we can identify the missing nuclear isotope: \(^2_2 \mathrm{He}\) Putting the missing particle in the equation, we get the complete nuclear equation for a: \(^{197} \mathrm{Au}+^2_2 \mathrm{He} \rightarrow^{199} \mathrm{Hg}+\beta\) #b. Balance mass numbers and atomic numbers# For b, we have the nuclear reaction: \(^{64} \mathrm{Ni}+^{1} \mathrm{H} \rightarrow^{64} \mathrm{Cu}+?\) We need to find the missing particle in this reaction.

Step 4: Determine missing mass number

Let A be the mass number of the missing particle. Using the conservation of mass numbers, we have: \(64 + 1 = 64 + A\) Solving for A, we get: \(A = 1\)

Step 5: Determine missing atomic number

Let Z be the atomic number of the missing particle (number of protons). Using the conservation of atomic numbers, we have: \(28 + 1 = 29 + Z\) Solving for Z, we get: \(Z = 0\)

Step 6: Identify the missing particle

From the obtained mass and atomic numbers, we can identify the missing nuclear isotope: \(^1_0 \mathrm{n}\) Putting the missing particle in the equation, we get the complete nuclear equation for b: \(^{64} \mathrm{Ni}+^{1} \mathrm{H} \rightarrow^{64} \mathrm{Cu}+^1_0 \mathrm{n}\) #c. Balance mass numbers and atomic numbers# For c, we have the nuclear reaction: \(^{63} \mathrm{Cu}+? \rightarrow^{66} \mathrm{Ga}+^{1} \mathrm{n}\). We need to find the missing particle in this reaction.

Step 7: Determine missing mass number

Let A be the mass number of the missing particle. Using the conservation of mass numbers, we have: \(63 + A = 66 + 1\) Solving for A, we get: \(A = 4\)

Step 8: Determine missing atomic number

Let Z be the atomic number of the missing particle (number of protons). Using the conservation of atomic numbers, we have: \(29 + Z = 31\) Solving for Z, we get: \(Z = 2\)

Step 9: Identify the missing particle

From the obtained mass and atomic numbers, we can identify the missing nuclear isotope: \(^4_2 \mathrm{He}\) Putting the missing particle in the equation, we get the complete nuclear equation for c: \(^{63} \mathrm{Cu}+^4_2 \mathrm{He} \rightarrow^{66} \mathrm{Ga}+^{1} \mathrm{n}\) #d. Balance mass numbers and atomic numbers# For d, we have the nuclear reaction: \(^{67} \mathrm{Zn}+^{1} \mathrm{n} \rightarrow^{67} \mathrm{Cu}+?\) We need to find the missing particle in this reaction.

Step 10: Determine missing mass number

Let A be the mass number of the missing particle. Using the conservation of mass numbers, we have: \(67 + 1 = 67 + A\) Solving for A, we get: \(A = 1\)

Step 11: Determine missing atomic number

Let Z be the atomic number of the missing particle (number of protons). Using the conservation of atomic numbers, we have: \(30 + 0 = 29 + Z\) Solving for Z, we get: \(Z = 1\)

Step 12: Identify the missing particle

From the obtained mass and atomic numbers, we can identify the missing nuclear isotope: \(^1_1 \mathrm{H}\) Putting the missing particle in the equation, we get the complete nuclear equation for d: \(^{67} \mathrm{Zn}+^{1} \mathrm{n} \rightarrow^{67} \mathrm{Cu}+^1_1 \mathrm{H}\)

Key Concepts

Understanding Isotopes

Isotopes are variations of elements that contain the same number of protons but different numbers of neutrons. This means that although they have the same atomic number, their mass numbers differ. Isotopes are significant in various scientific fields because they help scientists understand and predict the behavior of elements in different environments.
Here are some key points to grasp about isotopes:

• **Atomic Structure:** Isotopes possess the same position in the periodic table since they share an atomic number. However, the number of neutrons influences their physical properties and mass number.
• **Stability:** Some isotopes are stable while others are radioactive. Radioactive isotopes decay over time, releasing radiation that can be used in numerous applications.
• **Applications:** In nuclear medicine, isotopes serve critical roles due to their radioactive properties. They can be used for imaging, treatment, and diagnosis of various conditions.

It's crucial to understand isotopes fully because they form the backbone of activities in nuclear chemistry and nuclear physics.

Role of Nuclear Medicine

Nuclear medicine is a specialized branch of medicine that utilizes radioactive isotopes for diagnosis, imaging, and treatment of diseases. It combines elements of chemistry, physics, biology, and medicine to provide insights that other imaging methods cannot.
Here's how nuclear medicine functions in a nutshell:

• **Diagnostic Imaging:** Radioactive isotopes are introduced into the body, where they emit gamma rays. The emission is captured by instruments to create medical images. This makes it easier to see how organs and tissues are functioning.
• **Therapeutic Applications:** Certain treatments exploit the radioactive properties of isotopes to target and destroy diseased cells, while minimizing damage to healthy tissue.
• **Safety and Precision:** The use of isotopes in nuclear medicine is highly regulated to ensure safety and efficacy. This also helps in delivering targeted treatment, reducing side effects.

Understanding the interface of isotopes and nuclear medicine can significantly enhance comprehension of their diagnostic and therapeutic potentials.

Exploring Mass Number

The mass number is a fundamental concept in understanding atomic structure. It refers to the total count of protons and neutrons within an atom's nucleus. Unlike the atomic number, which is consistent among isotopes of a given element, the mass number can vary.
Significant aspects of the mass number include:

• **Calculation:** An element’s mass number is calculated by adding the total number of protons and neutrons. For example, for the isotope ^{197}_{79}Au, the mass number is 197, indicating a composition of 79 protons and 118 neutrons.
• **Mass Number vs. Atomic Mass:** It is pivotal to distinguish between the mass number and atomic mass, the latter of which reflects an element’s average mass accounting for its isotopes.
• **Relevance in Reactions:** In nuclear equations, balancing mass numbers ensures the conservation of mass, signifying stability in nuclear reactions and transformations.

Understanding mass numbers allows students to make accurate predictions regarding the properties and behaviors of isotopic substances.

Unveiling Atomic Number

The atomic number is one of the most crucial identifiers of an element. It is defined by the number of protons found in the nucleus of an atom and determines the element's position in the periodic table. Here are some important considerations about the atomic number:

• **Identification Role:** Each element has a unique atomic number, making it essential for identifying and differentiating elements.
• **Stability and Reactivity:** The atomic number influences an element’s chemical properties and reactivity, crucial for understanding interactions in chemistry.
• **Periodic Table Placement:** Consistent across isotopes, the atomic number organises elements in the periodic table, providing insight into electronic configurations and chemical behavior.

Appreciating the atomic number's role lays foundational knowledge pivotal for further exploration of elements and their interactions in physical and chemical processes.