Problem 1 - University Physics with Modern Physics

Question

How many protons and how many neutrons are there in a nucleus of the most common isotope of (a) silicon, \(_{14}^{28} \mathrm{Si} ;\) (b) rubidium, \(\frac{85}{37} \mathrm{Rb} ;\) (c) thallium, \(_{81}^{205} \mathrm{Tl} ?\)

Step-by-Step Solution

Answer

Silicon: 14 protons, 14 neutrons. Rubidium: 37 protons, 48 neutrons. Thallium: 81 protons, 124 neutrons.

Step 1: Understanding isotope notation

The isotope notation \( _Z^A \text{X} \) provides two pieces of information: \( Z \) is the atomic number, representing the number of protons, and \( A \) is the mass number, representing the sum of protons and neutrons in the nucleus.

Step 2: Determine protons in silicon (\( _{14}^{28} \text{Si} \))

For silicon (\( _{14}^{28} \text{Si} \)), the atomic number \( Z = 14 \). Therefore, there are 14 protons in the nucleus of silicon.

Step 3: Calculate neutrons in silicon

Subtract the number of protons from the mass number to find neutrons: \( 28 - 14 = 14 \). Therefore, there are 14 neutrons in silicon.

Step 4: Determine protons in rubidium (\( \frac{85}{37} \text{Rb} \))

For rubidium (\( \frac{85}{37} \text{Rb} \)), the atomic number \( Z = 37 \). Therefore, there are 37 protons in the nucleus of rubidium.

Step 5: Calculate neutrons in rubidium

Subtract the number of protons from the mass number to find neutrons: \( 85 - 37 = 48 \). Therefore, there are 48 neutrons in rubidium.

Step 6: Determine protons in thallium (\( _{81}^{205} \text{Tl} \))

For thallium (\( _{81}^{205} \text{Tl} \)), the atomic number \( Z = 81 \). Therefore, there are 81 protons in the nucleus of thallium.

Step 7: Calculate neutrons in thallium

Subtract the number of protons from the mass number to find neutrons: \( 205 - 81 = 124 \). Therefore, there are 124 neutrons in thallium.

Key Concepts

Atomic Number

The atomic number, symbolized by the letter \( Z \), is a fundamental property of an element. It defines the number of protons in the nucleus of an atom. This number is crucial because it determines the element's identity on the periodic table. For example, an element with an atomic number of 14 is always silicon.

To identify the atomic number in isotope notation, look for the subscript in the notation format \( _{Z}^{A}\text{X} \). The subscript \( Z \) gives the number of protons in the nucleus.

• Silicon's isotope notation is \( _{14}^{28} \text{Si} \), meaning it has 14 protons.
• Rubidium's notation \( \frac{85}{37} \text{Rb} \) reveals 37 protons.
• Thallium, noted as \( _{81}^{205}\text{Tl} \), contains 81 protons.

Each element has a unique atomic number, which ensures no two elements are the same. This number is also indicative of how an element interacts chemically, as electrons equal protons in a neutral atom.

Mass Number

The mass number, denoted by the letter \( A \), indicates the total number of protons and neutrons in an atom's nucleus. Unlike the atomic number, the mass number can vary among isotopes of the same element because the number of neutrons can differ.

To find the mass number in isotope notation, check the superscript \( A \) in the notation \( _{Z}^{A}\text{X} \). It sums the protons and neutrons for a particular isotope. Let's delve into examples:

• Silicon has a mass number of 28, suggesting 14 protons and 14 neutrons.
• Rubidium's mass number of 85 indicates it has 37 protons and 48 neutrons.
• Thallium, with a mass number of 205, includes 81 protons and 124 neutrons.

While the mass number is essential for understanding isotopes, it is not listed on the periodic table, highlighting the importance of isotope notation for conveying this information.

Neutrons in Nucleus

To find the number of neutrons in a nucleus, you subtract the atomic number \( Z \) from the mass number \( A \). Neutrons contribute significantly to an atom's mass but not to its charge, playing a key role in atomic stability.

Here's the simple calculation to find neutrons:

• Silicon has 28 (mass number) - 14 (atomic number) = 14 neutrons.
• Rubidium has 85 - 37 = 48 neutrons.
• Thallium has 205 - 81 = 124 neutrons.

Knowing the number of neutrons is crucial for recognizing isotopes of the same element, which differ only by neutron count. Neutrons are neutral, so they don't affect the charge balance but do impact nuclear stability and properties like isotope decay.