Problem 26 - Chemistry The Central Science

Question

(a) Which two of the following are isotopes of the same element: \(\frac{31}{16} \mathrm{X}, \frac{31}{15} \mathrm{X}, \frac{32}{16} \mathrm{X}\) ? (b) What is the identity of the element whose isotopes you have selected?

Step-by-Step Solution

Answer

(a) The two isotopes of the same element are \(\frac{31}{16} \mathrm{X}\) and \(\frac{32}{16} \mathrm{X}\), since they both have the same atomic number, Z = 16. (b) The element with atomic number 16 is Sulfur (S), so the isotopes are \(\frac{31}{16}\mathrm{S}\) and \(\frac{32}{16}\mathrm{S}\).

Step 1: Understanding the notation

The notation for an isotope is given as \(\frac{A}{Z}\mathrm{X}\), where A represents the mass number (number of protons and neutrons), Z represents the atomic number (number of protons), and X is the symbol of the element. To identify isotopes of the same element, we need to compare the atomic numbers (Z) of each given option.

Step 2: Compare atomic numbers

We have the following isotopes given in the exercise: 1. \(\frac{31}{16} \mathrm{X}\) 2. \(\frac{31}{15} \mathrm{X}\) 3. \(\frac{32}{16} \mathrm{X}\) Comparing the atomic numbers (Z), we can see that option 1 and option 3 have the same atomic number (Z = 16). So, these two are isotopes of the same element.

Step 3: Identify the element

Now that we know that \(\frac{31}{16} \mathrm{X}\) and \(\frac{32}{16} \mathrm{X}\) are isotopes of the same element, we need to determine the identity of this element. The atomic number (Z) determines the identity of the element. The atomic number (Z) for both these isotopes is 16. We know that an element with atomic number 16 is Sulfur (symbol S). So, the isotopes are: 1. \(\frac{31}{16}\mathrm{S}\) 2. \(\frac{32}{16}\mathrm{S}\)

Key Concepts

Atomic Number

The atomic number, denoted by \( Z \), is a fundamental concept when it comes to understanding the structure of an atom. It represents the number of protons in an atom's nucleus and is crucial because it defines the identity of an element.

For example, if the atomic number is 16, it means there are 16 protons in the nucleus of the atom. Every element on the periodic table has a unique atomic number, which distinguishes it from other elements.

• The atomic number is always written as a subscript to the left of the element symbol (e.g., \( _{16}\text{X} \) for any element \( X \) with an atomic number of 16).
• Because isotopes have the same element, they share the same atomic number.

Atomic numbers also help in arranging elements on the periodic table. The periodic table is organized in a way that elements are listed in increasing order of their atomic number. This systematic arrangement is what makes the periodic table such a powerful tool for scientists and students alike.

Mass Number

Mass number, indicated by \( A \), is another critical aspect of an atom's structure. It represents the total number of protons and neutrons in an atom's nucleus. Unlike the atomic number, the mass number can vary among isotopes of a single element.

For instance, considering isotopes \( ^{31}_{16}\text{S} \) and \( ^{32}_{16}\text{S} \):

• Both have the same atomic number (16 protons), signifying they are isotopes of the element sulfur.
• The two isotopes have different mass numbers (31 and 32, respectively); the difference arises because \(^32_{16}\text{S} \) has one more neutron than \(^31_{16}\text{S} \).

Understanding the mass number is essential for:

• Calculating the number of neutrons in an atom by subtracting the atomic number from the mass number \( (A - Z) \).
• Providing insight into the isotope's stability and radioactive properties.

Always remember that while the atomic number is constant for isotopes of the same element, the mass number varies depending on the number of neutrons.

Element Identification

Element identification heavily relies on the atomic number because it uniquely specifies an element. Even when dealing with isotopes, where the mass number might shift due to varying numbers of neutrons, the atomic number remains a steadfast identifier.

For example, if given isotopes with notation \( ^{31}_{16}\text{X} \) and \( ^{32}_{16}\text{X} \), you can decode the identity of the element by the atomic number 16. This directly points to sulfur (with the symbol \( \text{S} \)) since sulfur is the only element with this atomic number.

Points to consider when identifying elements:

• Use the atomic number as the primary clue in determining the elemental identity.
• The symbol given in conjunction with the atomic number reinforces the element’s identity but is secondary in isotopic contexts.
• Recognizing element symbols and their atomic numbers will improve your understanding of chemical equations, reactions, and isotopic variants.

Element identification isn't just important for labeling atoms on paper. It’s fundamental in fields like chemistry and physics where precise knowledge of atomic composition is essential.