Question

Which orbital in each of the following pairs has the higher energy? (a) \(2 s\) or \(3 s\) (b) \(2 p_{x}\) or \(3 p_{x}\) (c) \(2 p_{x}\) or \(2 p_{y}\) (d) \(4 p_{\mathrm{y}}\) or \(4 p_{\mathrm{z}}\)

Short answer

(a) 3s; (b) 3p_x; (c) Same energy; (d) Same energy.

Step-by-step solution

Understanding Orbital Energy Levels

In atomic orbitals, energy levels increase as the principal quantum number, denoted by \( n \), increases. Therefore, an orbital with a higher principal quantum number will typically have more energy. This rule applies generally when comparing orbitals of the same type, such as \( s \), \( p \), or \( d \) orbitals.

Comparing s Orbitals

For part (a), we compare \(2s\) and \(3s\) orbitals. Between these, \(3s\) has a higher principal quantum number than \(2s\). Thus, \(3s\) orbital has more energy than \(2s\).

Comparing p_x Orbitals

Part (b) asks us to compare \(2p_x\) and \(3p_x\) orbitals. Here, \(3p_x\) has a higher principal quantum number than \(2p_x\), indicating that \(3p_x\) has more energy.

Comparing p_x and p_y Orbitals

For part (c), we compare orbitals \(2p_x\) and \(2p_y\). Both orbitals belong to the same principal energy level and subshell, so they possess equal energy. This means neither orbital has more energy than the other.

Comparing p_y and p_z in Same Level

In part (d), we compare orbitals \(4p_y\) and \(4p_z\). Since both belong to the same principal energy level and type of subshell, they have the same energy. Hence, neither \(4p_y\) nor \(4p_z\) has higher energy than the other.

Key concepts

Principal Quantum Number

The principal quantum number, represented by the letter \( n \), is fundamental in determining the energy levels of atomic orbitals. In simple terms, it indicates the relative size and energy of an orbital. As \( n \) increases, the orbital becomes larger and the electron it contains is further from the nucleus. This increase in distance from the nucleus typically results in a higher energy level. For example, orbitals with \( n = 3 \) are at a higher energy level than those with \( n = 2 \).
Ultimately, the principal quantum number plays a crucial role in understanding the energy hierarchy amongst the various atomic orbitals. It helps in predicting electron configurations and is fundamental to the entire quantum mechanical model of the atom.

Atomic Orbitals

Atomic orbitals are regions in an atom where electrons are most likely to be found. Each orbital can hold a maximum of two electrons, and their arrangement represents different shapes and energy levels within an atom.
The concept of atomic orbitals is key to understanding the structure and behavior of atoms. Orbitals are described by quantum numbers which provide critical insight into their energy, shape, and orientation in space. The primary quantum numbers include the principal quantum number \( n \), the azimuthal quantum number \( l \), and the magnetic quantum number \( m \).
In essence, atomic orbitals form the basis for understanding chemical bonding and properties, as they describe how electrons are arranged and distributed in an atom.

s Orbitals

s orbitals are spherical in shape and are the simplest type of atomic orbitals. They correspond to the azimuthal quantum number \( l = 0 \). One of the key features of s orbitals is that they have the same shape regardless of their energy level or principal quantum number \( n \). However, their size and energy level do increase with a higher \( n \).
The important point to remember is that within the same principal energy level, the s orbital is always the lowest energy orbital due to its closeness to the nucleus. This characteristic plays a major role in electron configurations, where s orbitals are filled before other types, like p or d orbitals, in the electron shell.

p Orbitals

p orbitals are more complex in shape compared to s orbitals. They resemble a dumbbell and are aligned along particular axes within three-dimensional space. The azimuthal quantum number \( l \) for p orbitals is 1, and they occur in groups of three: \( p_x \), \( p_y \), and \( p_z \). These designations indicate the orientation of the orbitals along the respective x, y, and z axes in space.
As with s orbitals, p orbitals increase in size and energy as the principal quantum number \( n \) increases. Although p orbitals have the same energy within the same principal energy level (for instance, \( 2p_x \), \( 2p_y \), and \( 2p_z \)), they are generally higher in energy compared to s orbitals in the same principal shell. This increased energy is due to their shape and the electron density being further from the nucleus. Understanding p orbitals is integral in predicting molecular shapes and chemical reactivity.