Question

Which of the following nuclides have magic numbers of both protons and neutrons: (a) helium-4, (b) oxygen-18, (c) calcium- 40 , (d) zinc-66, (e) lead-208?

Short answer

The nuclides with magic numbers of both protons and neutrons are: helium-4 (He-4), calcium-40 (Ca-40), and lead-208 (Pb-208).

Step-by-step solution

Identify the number of protons and neutrons in each nuclide

Using the notation 'XY', where 'X' represents the element and 'Y' represents the number of nucleons (protons and neutrons), we can identify the number of protons and neutrons for each nuclide. We do this using the atomic number (number of protons) of each element: (a) helium-4: atomic number (protons) = 2 (b) oxygen-18: atomic number (protons) = 8 (c) calcium-40: atomic number (protons) = 20 (d) zinc-66: atomic number (protons) = 30 (e) lead-208: atomic number (protons) = 82

Calculate the number of neutrons for each nuclide

Now that we have the atomic number for each element, we can calculate the number of neutrons by subtracting the atomic number from the total number of nucleons: (a) helium-4: neutrons = 4 - 2 = 2 (b) oxygen-18: neutrons = 18 - 8 = 10 (c) calcium-40: neutrons = 40 - 20 = 20 (d) zinc-66: neutrons = 66 - 30 = 36 (e) lead-208: neutrons = 208 - 82 = 126

Check if the number of protons and neutrons are magic numbers

Now we'll compare the number of protons and neutrons for each nuclide with the magic numbers (2, 8, 20, 28, 50, 82, and 126): (a) helium-4: protons (2) and neutrons (2) are both magic numbers. (b) oxygen-18: protons (8) are a magic number, but neutrons (10) are not. (c) calcium-40: protons (20) and neutrons (20) are both magic numbers. (d) zinc-66: neither protons (30) nor neutrons (36) are magic numbers. (e) lead-208: protons (82) and neutrons (126) are both magic numbers.

Identify which nuclides have magic numbers of both protons and neutrons

Based on our calculations in the previous steps, we can see that the following nuclides have magic numbers of both protons and neutrons: (a) helium-4 (c) calcium-40 (e) lead-208

Key concepts

Nuclides

In the world of nuclear physics, the term "nuclide" refers to a distinct species of an atom, identifiable through its specific number of protons and neutrons. Each nuclide represents a unique atom. For example, when you see something like "helium-4," this is a nuclide, indicating the element helium with a total of four nucleons (two protons and two neutrons).
Understanding nuclides is crucial because they play a vital role in the stability of an atom. The mix of protons and neutrons within the nucleus defines not just the identity of the nuclide, but also its properties. Some combinations of these particles are more stable than others, and this stability often involves what we call "magic numbers." These are special numbers of protons or neutrons that create very stable nuclides.

Protons and Neutrons

Protons and neutrons are subatomic particles found in the atom's nucleus, and they are collectively called nucleons. Protons carry a positive charge, while neutrons have no charge. Despite their opposing charges, they are close relatives by mass and size. Together, they define much about an element and its behavior.
For the nuclide helium-4, for instance, the two protons define helium's chemical nature, while the two neutrons contribute to its overall mass. Knowing the count of protons and neutrons allows scientists to delve into the heart of nuclear structure, particularly the concept of magic numbers, which you find frequently in stable nuclides.

Atomic Number

The term "atomic number" is fundamentally important in chemistry and physics. It represents the number of protons in the nucleus of an atom. Each element on the periodic table is distinguished by its unique atomic number. Thus, for helium-4, the atomic number is 2, due to its two protons.
Atomic numbers determine the identity of an element and can be used alongside the total nucleon count to find out how many neutrons are present: simply subtract the atomic number from the nucleon number (also known as the mass number). In exercises exploring nuclides and magic numbers, verifying atomic numbers helps quickly identify which elements and their stable isotopes fit these patterns.

Nucleons

Nucleons are the building blocks of the nucleus and consist of protons and neutrons. Each atom within an element is identified by the number of nucleons it contains. Nucleons are responsible for the majority of an atom's mass.
The sum of nucleons, or nucleon number, is essential in nuclear physics because it aids in differentiating between isotopes, as well as calculating the neutron-proton composition, especially in the search for magic numbers. For instance, in the case of Zinc-66, the total nucleon count tells us about its specific form and helps differentiate it from another form, like Zinc-64.

Isotopes

Isotopes are variations of elements that have the same number of protons but different numbers of neutrons. This means that while their atomic numbers are equal, their mass numbers are different. Identifying isotopes is crucial for understanding how elements behave under different conditions and how stable they can be.
Taking oxygen-18 as an example – this isotope has 8 protons, like all oxygen isotopes, but 10 neutrons instead of the usual 8 seen in oxygen-16. This difference makes oxygen-18 unique in its properties and uses. In examining nuclides, the recognition of isotopes helps in identifying those that possess magic numbers, as these stable isotopes demonstrate unique nuclear characteristics seen nowhere else.