Problem 94 - Chemistry Matter and Change

Question

What is the maximum number of electrons that can be contained in an atom’s orbitals having the following principal quantum numbers? a. 3 c. 6 b. 4 d. 7

Step-by-Step Solution

Answer

The maximum number of electrons for each principal quantum number (n) is as follows: a. For n = 3, the maximum number of electrons is 18. b. For n = 4, the maximum number of electrons is 32. c. For n = 6, the maximum number of electrons is 72. d. For n = 7, the maximum number of electrons is 98.

Step 1: Apply the Formula for Maximum Number of Electrons

For each principal quantum number (n), apply the formula 2 * n^2 to find the maximum number of electrons. a. For n = 3, the maximum number of electrons is: 2 * (3^2) = 2 * 9 = 18 b. For n = 4, the maximum number of electrons is: 2 * (4^2) = 2 * 16 = 32 c. For n = 6, the maximum number of electrons is: 2 * (6^2) = 2 * 36 = 72 d. For n = 7, the maximum number of electrons is: 2 * (7^2) = 2 * 49 = 98

Step 2: List the Maximum Number of Electrons for Each Principal Quantum Number

Based on our calculations in Step 1, the maximum number of electrons for each principal quantum number is as follows: a. For n = 3, the maximum number of electrons is 18. b. For n = 4, the maximum number of electrons is 32. c. For n = 6, the maximum number of electrons is 72. d. For n = 7, the maximum number of electrons is 98.

Key Concepts

Principal Quantum Number

Quantum numbers are essential in describing the arrangement of electrons in an atom. The principal quantum number, often denoted as \( n \), is one of the four quantum numbers used to describe the unique quantum state of an electron.

• The principal quantum number determines the size of the electron orbital. The higher the value of \( n \), the larger the orbital.
• It also tells us the relative energy level of the electron. Electrons with higher \( n \) values have higher energy levels.
• For each principal quantum number \( n \), there are \( n^2 \) orbitals. This indicates how many possible orientations electron pairs can occupy.

Understanding \( n \) helps predict the possible orbits and energy levels electrons may reside in, which is crucial for comprehending more complex electron arrangements and chemical behavior.

Electron Configuration

Electron configuration refers to the specific arrangement of electrons in an atom's orbitals, which are defined by the atom's unique set of quantum numbers. This configuration helps to understand how elements interact to form chemical bonds, and how they exhibit specific chemical properties.

• Electrons fill orbitals starting from the lowest available energy state to the highest, according to the Aufbau principle.
• Every electron in an atom will be configured in increasing levels of \( n \), from 1, 2, 3, and so forth.
• The configuration is often represented using the notation style — for example, \( 1s^2 2s^2 2p^6 \) represents the filling of the "1s," "2s," and "2p" orbitals respectively, showing the arrangement in increasing energy levels.

This pattern and style of filling help to predict chemical reactivity and bonding capacity as well.

Maximum Electrons in Orbitals

The maximum number of electrons that can occupy an atom's energy level corresponds with the principal quantum number, and is calculated with the expression \( 2n^2 \). Here's how this formula works:

• For any principal quantum number \( n \), the equation gives the total maximum electrons that can be contained in that energy level.
• The calculation \( 2n^2 \) accounts for each set of orbitals' capacity to host two electrons each, with opposite spins.
• For instance, if \( n = 3 \), substituting in the formula: \( 2 \times (3^2) = 18 \), which corresponds to the cumulative capacity of orbitals such as 3s, 3p, and 3d.

This is a fundamental principle used to determine the occupancy limits of electron shells, which assists in understanding potential electron distributions and ionization energy differences across periods in the periodic table.