Problem 61 - Chemistry and Chemical Reactivity

Question

Rank the following orbitals in the H atom in order of increasing energy: \(3 s, 2 s, 2 p, 4 s, 3 p, 1 s,\) and \(3 d\)

Step-by-Step Solution

Answer

Arrange as: 1s < 2s = 2p < 3s = 3p = 3d < 4s.

Step 1: Understand the Basics

In a hydrogen atom, the energy of an electron primarily depends on the principal quantum number (n). The higher the value of n, the higher the energy level. Thus, we need to first arrange the given orbitals based on their principal quantum number.

Step 2: List the Orbitals by Principal Quantum Number

The given orbitals are: 3s, 2s, 2p, 4s, 3p, 1s, and 3d. The principal quantum numbers are: - 1s (n=1) - 2s, 2p (n=2) - 3s, 3p, 3d (n=3) - 4s (n=4). Arrange them: 1s, 2s, 2p, 3s, 3p, 3d, 4s.

Step 3: Rank Orbitals Within Each Principal Quantum Number

For orbitals with the same principal quantum number, all sublevels (s, p, d, f) are degenerate in the hydrogen atom. Thus, within the same principal level, s, p, d, etc., have the same energy level. For example, 2s and 2p are of the same energy in hydrogen.

Step 4: Arrange Them from Lowest to Highest Energy

Now, we proceed to arrange the orbitals in increasing energy. Follow the order of increasing principal quantum number: 1. **1s** 2. **2s, 2p** (equal energy for same n) 3. **3s, 3p, 3d** (equal energy for same n) 4. **4s**

Step 5: Verify the Arrangement

In hydrogen, no subshell has energy splitting like multi-electron atoms ([n+l] rule does not apply similarly), so the order is primarily by principal quantum number. Double-checking confirms that 1s < 2s = 2p < 3s = 3p = 3d < 4s.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted as \(n\), is one of the fundamental concepts in quantum mechanics, specifically in the study of atomic orbitals. This number essentially describes the energy level of an electron in an atom. The higher the value of \(n\), the further the electron is from the nucleus, and consequently, the higher its energy.• **Relationship with orbitals:** The principal quantum number directly influences the size and energy of an orbital. For example, a 2s orbital with \(n=2\) is larger and has more energy than a 1s orbital with \(n=1\).• **Energy levels:** In a hydrogen atom, orbitals with the same \(n\) are said to be degenerate, meaning they have the same energy. For instance, the 2s and 2p orbitals both belong to \(n=2\) and thus have the same energy in hydrogen.Understanding the principal quantum number is key to determining the energy levels in the hydrogen atom and helps in predicting the arrangement of electrons across different orbitals.

Orbital Energy Levels

Orbital energy levels are crucial when it comes to understanding how electrons occupy different regions around the nucleus. In a hydrogen atom, these energy levels are defined by the principal quantum number \(n\), and there is a straightforward direct link between \(n\) and the energy level of the orbital.• **Energy and quantum number:** The energy of an orbital increases with an increase in the principal quantum number. Thus, a 3s orbital has more energy compared to a 2s or a 1s orbital.• **Hydrogen atom specifics:** In hydrogen, the energy level is solely determined by \(n\) because the other contributing quantum numbers (like \(l\), which discusses orbital shape) do not cause energy splitting as they do in multi-electron atoms.Knowing how to rank these energy levels helps chemists in predicting electron configurations and understanding chemical properties of hydrogen atoms.

Quantum Mechanics

Quantum mechanics provides the theoretical framework to understand the behavior of electrons within an atom. It's a field that explores the probability of the location of an electron rather than defining an exact position. • **Electron behavior:** In quantum mechanics, electrons occupy regions called orbitals, defined by different quantum numbers which give a probability distribution rather than a fixed path. • **Applications:** This field is not limited to hydrogen atoms, but extends to larger and more complex atoms, where it helps in deriving electron configurations and predicting atomic interactions. • **Key principles:** It involves concepts like quantization, where energy levels are divided into discrete values (quantized) rather than being continuous. Quantum mechanics revolutionizes our understanding of microscopic particles that classical physics couldn't explain.

Degenerate Orbitals

Degenerate orbitals refer to orbitals that have the same energy level. In the hydrogen atom, this is particularly evident within the same principal quantum number, where all orbitals, such as s, p, and d, have equal energy.• **Energy equivalence in hydrogen:** In orbitals like 2s and 2p (both having \(n=2\)), there is no energy difference in a hydrogen atom.• **Multi-electron atoms:** Unlike in hydrogen, degenerate orbitals in atoms with multiple electrons can experience energy splitting, where different sublevels within the same principal quantum number may vary in energy.• **Importance in chemistry:** Understanding degenerate orbitals is fundamental in predicting electron configurations and in explaining phenomena like atomic spectra and chemical bonding in hydrogen and beyond.