Problem 9 - Chemistry The Central Science

Question

(a) Evaluate the expressions \(2 \times 1,2 \times(1+3)\) \(2 \times(1+3+5),\) and \(2 \times(1+3+5+7)\) . (b) How do the atomic numbers of the noble gases relate to the numbers from part (a)? (c) What topic discussed in Chapter 6 is the source of the number \(" 2 "\) in the expressions in part (a)?

Step-by-Step Solution

Answer

The evaluated expressions from part (a) are 2, 8, 18, and 32. These numbers are closely related to the atomic numbers of the noble gases, specifically Helium (2), Neon (10), Argon (18), and Krypton (36). The connection between these values may be related to the maximum number of electrons in specific energy levels, contributing to the stability of noble gases. However, without more context on Chapter 6, it's difficult to determine the exact significance of the number 2 in these expressions.

Step 1: Evaluate the expressions

For part (a), we need to evaluate the following expressions: \( 2 \times 1 \), \( 2 \times (1 + 3) \), \( 2 \times (1 + 3 + 5) \), and \( 2 \times (1 + 3 + 5 + 7) \) Let's start evaluating each expression one by one: 1. \( 2 \times 1 = 2 \) 2. \( 2 \times (1 + 3) = 2 \times 4 = 8 \) 3. \( 2 \times (1 + 3 + 5) = 2 \times 9 = 18 \) 4. \( 2 \times (1 + 3 + 5 + 7) = 2 \times 16 = 32 \) So, the evaluated expressions are 2, 8, 18, and 32.

Step 2: Relate the expressions to the atomic numbers of noble gases

In part (b), we're supposed to find a connection between the values we got in the previous step (2, 8, 18, 32) and the atomic numbers of the noble gases. Let's list down the atomic numbers of the first few noble gases: 1. Helium (He) - 2 2. Neon (Ne) - 10 3. Argon (Ar) - 18 4. Krypton (Kr) - 36 We can see that there is a connection between the expressions' values and the atomic numbers of the noble gases. The atomic numbers of the noble gases are close to the evaluated expressions, except for Neon (10) and Krypton (36), which differ by 2 units each.

Step 3: Identify the source of the number "2" in the expressions

In part (c), we have to discuss the significance of the number "2" in the expressions and how it relates to a topic in Chapter 6 (which hasn't been specified in the given context). Nevertheless, the number "2" has different significances in chemistry. For example, it could refer to the maximum number of electrons in the first energy level (1s orbital) or the stability of noble gases due to having completely filled energy levels. Without more information on the specific context of Chapter 6, we cannot pinpoint the exact significance of the number 2.

Key Concepts

Atomic Numbers

Atomic numbers play a crucial role in identifying elements, including noble gases. The atomic number is defined as the number of protons present in an atom's nucleus. This number not only determines the identity of the element but also its position in the periodic table.
The noble gases are a group of elements with distinctive properties, known for being inert or unreactive due to their full valence electron shells. The atomic numbers of noble gases range as follows:

• Helium (He): 2
• Neon (Ne): 10
• Argon (Ar): 18
• Krypton (Kr): 36

The atomic numbers reveal that their electron configurations achieve stability, making them chemically non-reactive. These elements have complete octets in their valence shells (except Helium, which has a full 1s shell), correlating with their inert nature. Understanding atomic numbers helps in predicting the chemical behavior and reactivity of elements.

Electron Configuration

Electron configuration describes how electrons are distributed among an atom's orbitals. It is a systematic way to note the arrangement of electrons resulting from their placement in various energy levels and sublevels.
Noble gases are known for their stable electron configurations, which contribute to their lack of chemical reactivity. These configurations achieve the lowest energy state of an atom. For example:

• Helium (He): 1s²
• Neon (Ne): 1s² 2s² 2p⁶
• Argon (Ar): 1s² 2s² 2p⁶ 3s² 3p⁶
• Krypton (Kr): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p⁶

Each noble gas achieves a full valence shell, making them particularly stable. This configuration provides insight into why noble gases do not typically form chemical bonds. The full outer electron shells signify energy levels are maximized in terms of stability, making these elements highly inert.

Energy Levels

Energy levels, also known as electron shells, are an integral part of atomic structure that influences the behavior of electrons in an atom. Each energy level can hold a certain number of electrons; this is primarily what determines how atoms interact chemically.
The noble gases exemplify elements with fully occupied energy levels. This full occupation results in almost negligible reactivity because there is no tendency to gain or lose electrons. In an atom:

• The first energy level can hold up to 2 electrons (1s orbital)
• The second energy level can hold up to 8 electrons (2s and 2p orbitals)
• Subsequent energy levels fill in a more complex order due to the presence of d and f orbitals

The concept of filled energy levels also elucidates the nature of the expressions given in the exercise, where stability is linked with a full complement of electrons in the orbitals. Understanding the configuration of energy levels can demystify why noble gases possess their distinctive unreactive properties.