Question

From memory, sketch the shape of the boundary surface for each of these atomic orbitals: (a) \(2 p_{z}\) (b) \(4 s\) (c) \(3 d_{x y}\)

Short answer

2pz: Dumbbell shape along z-axis; 4s: Spherical shape; 3dxy: Four-lobed, cloverleaf shape in xy plane.

Step-by-step solution

Understanding Atomic Orbitals

Atomic orbitals are regions in space around the nucleus where there is a high probability of finding an electron. Each type of orbital has a characteristic shape and is designated by a set of quantum numbers (principal, azimuthal, and magnetic). The shape and orientation can be derived from these quantum numbers.

Shape of 2pz Orbital

The 2pz orbital is a type of p orbital which has a dumbbell shape. The 'z' subscript indicates that the lobes of this orbital are aligned along the z-axis. A key feature of p orbitals is that they have one nodal plane passing through the nucleus where the probability density of finding an electron is zero.

Shape of 4s Orbital

The 4s orbital belongs to the s orbitals which have a spherical shape centered around the nucleus. As it progresses from 1s to 4s, the spherical shape expands, and the 4s orbital consists of several nodes (areas where the probability of finding an electron is zero). However, visually it would appear as an expanded spherical region in comparison to lower s orbitals.

Shape of 3dxy Orbital

The 3dxy orbital is one of the five d orbitals and is classified as a 'cloverleaf' shape. This particular orbital is oriented between the x and y axes, making its shape appear like a set of four lobes lying in the xy plane. Similar to other d orbitals, it has two nodal planes where electron probability is zero.

Key concepts

Quantum Numbers

Quantum numbers are fundamental to understanding atomic structure and the behavior of electrons in an atom. They provide a unique address or "coordinate system" for each electron. There are four types of quantum numbers:

• Principal Quantum Number ( n ): It indicates the main energy level or shell in which the electron resides. Values are positive integers (1, 2, 3,...). Larger numbers mean electrons are farther from the nucleus.
• Azimuthal Quantum Number ( l ): It describes the shape of the orbital, with possible values from 0 to (n-1). Each value corresponds to a different subshell (s, p, d, f).
• Magnetic Quantum Number ( m_l ): It specifies the orientation of the orbital in space, with possible values ranging from -l to +l.
• Spin Quantum Number ( m_s ): It describes the spin of the electron, having possible values of +1/2 or -1/2.

Understanding these numbers helps predict where electrons are likely to be found and their energy states. They form the basis for atomic orbital shapes.

Orbital Shapes

The shape of atomic orbitals is a fundamental concept in quantum chemistry, determining how electrons occupy space around the nucleus. Orbital shapes are primarily defined by quantum numbers.
Each type of orbital has a distinct shape:

• s orbitals: Spherical in shape. As you move to higher energy levels like 4s, the size increases, appearing as larger spheres.
• p orbitals: Characterized by a dumbbell shape. Each orientation of a p orbital, such as 2pz, 2px, and 2py, has two lobes with a nodal plane through the nucleus.
• d orbitals: Often described as "cloverleaf" shapes. They have more complex lobes, with dxy, dxz, and others having nodal planes based on their orientation and shape.

These shapes are crucial for understanding chemical bonding and the distribution of electrons, which impacts the atom's interactions.

Nodal Planes

Nodal planes are regions within atomic orbitals where the probability of finding an electron is zero. They are an essential feature of orbital shapes and arise due to wave-like properties of electrons.

• p Orbitals: Each p orbital has one nodal plane. For instance, in a 2pz orbital, the nodal plane is perpendicular to the z-axis.
• d Orbitals: These orbitals generally have two nodal planes. For example, the 3dxy orbital has nodal planes passing through the x and y axes.
• Higher Energy Levels: As orbitals move to higher principal quantum numbers (e.g., transitioning from 1s to 2s or 2p to 3p), more nodal planes may appear, which can change the shape and complexity of the electron distribution within those orbitals.

Understanding nodal planes is important for visualizing how electrons manage space in different types of atomic orbitals and their impact on chemical properties.