Problem 7 - Introductory Chemistry Atoms First

Question

Suppose \({ }_{20}^{40} \mathrm{Ca}\) is the product of a beta-emission nuclear reaction. What was the parent isotope?

Step-by-Step Solution

Answer

The parent isotope of \(_{20}^{40}\text{Ca}\) after a beta-emission nuclear reaction is \(_{19}^{40}\text{K}\) (potassium-40).

Step 1: Understanding Beta Emission

In a beta-emission nuclear reaction, a neutron in the nucleus is converted into a proton, and an electron (called beta particle) is emitted. This results in an increase in the atomic number (number of protons) by 1, while the mass number (sum of protons and neutrons) remains constant.

Step 2: Identify the mass number (A) and atomic number (Z) of the product isotope

From the given product isotope, we have: \(_{\text{Z}}^{\text{A}}\text{Element}\), which is \(\(_{20}^{40}\text{Ca}\)\). Here, the atomic number Z = 20 and the mass number A = 40.

Step 3: Determine the atomic number (Z) of the parent isotope

Since beta-emission causes an increase in the atomic number by 1, we can determine the atomic number of the parent isotope by subtracting 1 from the atomic number of the product isotope: Parent atomic number \(=\text{Z}-1=20-1=19\)

Step 4: Determine the mass number (A) of the parent isotope

The mass number of the parent isotope remains constant during beta-emission. Therefore, the mass number of the parent isotope is the same as the product isotope, which is A = 40.

Step 5: Identify the parent isotope

With an atomic number of 19 and a mass number of 40, the parent isotope is \(_{19}^{40}\text{K}\) (potassium-40).

Key Concepts

Nuclear Reaction

A nuclear reaction is a process that changes the composition, structure, or energy of an atomic nucleus. Unlike chemical reactions which involve electrons, nuclear reactions deal with the nucleus of atoms.
Nuclear reactions can be initiated by:

• Collisions with other particles
• Decay processes that happen naturally

During these reactions, elements can change into other elements. This happens when the nuclei of atoms accept or emit particles.
Understanding nuclear reactions is crucial in fields like nuclear energy, medicine, and astrophysics as they explain the transformation of elements and release of energy at an atomic level.

Parent Isotope

A parent isotope is an original unstable radioactive isotope that undergoes decay to form a more stable daughter isotope.
In the context of nuclear reactions, the parent isotope is the starting material which transforms over time.
Key characteristics of a parent isotope include:

• Instability, leading to decay over time
• Decaying through processes such as alpha, beta, or gamma decay
• Having a measurable half-life which indicates the time taken for half of the substance to decay

The identification of the parent isotope is essential in understanding the process of radioactive decay and predicting the product isotopes formed from such reactions.

Atomic Number

The atomic number, denoted as Z, is a fundamental property of an element and represents the number of protons in the nucleus of an atom. It is what defines the identity of an element.
For example, the atomic number of carbon is 6, meaning every carbon atom has 6 protons.
Important aspects of atomic number include:

• Determines the element's position on the periodic table
• Affects the chemical properties of the element
• Increased by one unit during beta decay, where a neutron converts into a proton

In nuclear reactions like beta emission, knowing the atomic number helps to identify changes that occur to the nucleus, such as transformations and new element formation.

Mass Number

The mass number, represented by A, is the total number of protons and neutrons in an atom's nucleus. It provides information about the specific isotope of an element.
The mass number is calculated as:
\[ A = Z + N \]where Z is the atomic number, and N is the number of neutrons.
Vital points regarding mass number include:

• It stays the same during beta decay because the total number of nucleons remains constant
• It differs between isotopes of an element due to variation in neutron number
• Does not dictate the chemical properties of the element

Understanding mass number is fundamental when tracing nuclear reactions and calculating the specific isotopes that are formed as reaction products.