Draw the 3pz, 3dxy, 3dx2-y2, and 3dz2 atomic orbitals in the correct coordinate system and give n and l quantum numbers

Draw the 3pz, 3dxy, 3dx2-y2, and 3dz2 atomic orbitals in the correct coordinate system and give n and l quantum numbers. Label any nodes.

The Correct Answer and Explanation is:

Correct Answer:

Orbitals and Quantum Numbers:

1. 3pz orbital
   • Principal quantum number: n = 3
   • Angular momentum quantum number: l = 1
   • Shape: Dumbbell aligned along the z-axis
   • One angular node (a nodal plane at the xy-plane)
   • Radial node: 1 radial node
2. 3dxy orbital
   • n = 3
   • l = 2
   • Shape: Cloverleaf with lobes between x- and y-axes
   • Two angular nodes (along the x and y axes)
   • Radial node: 1 radial node
3. 3dx²−y² orbital
   • n = 3
   • l = 2
   • Shape: Cloverleaf with lobes along the x and y axes
   • Angular nodes: Planes between the axes (e.g. x = y and x = −y)
   • Radial node: 1 radial node
4. 3dz² orbital
   • n = 3
   • l = 2
   • Shape: Dumbbell along z-axis with a donut-shaped ring in the xy-plane
   • Angular node: Cone-like regions between lobes
   • Radial node: 1 radial node

Sketch (Use Standard x, y, z 3D axes):

[Not able to draw in plain text, but here’s a description to guide the sketch]

• The 3pz orbital has two lobes above and below the xy-plane along the z-axis, separated by a planar node at z = 0.
• The 3dxy orbital has four lobes between the x and y axes (at 45°, like NE, NW, SE, SW directions).
• The 3dx²−y² orbital has four lobes aligned with the x and y axes (positive and negative directions), shaped like a four-leaf clover.
• The 3dz² orbital has two lobes along the z-axis with a torus-shaped ring in the xy-plane, around the nucleus.

Explanation

Atomic orbitals describe regions in space where there is a high probability of finding an electron. These orbitals are defined by quantum numbers: the principal quantum number (n) determines the energy level and size, while the angular momentum quantum number (l) defines the orbital’s shape.

For the 3pz orbital, n = 3 and l = 1. The “p” orbitals are dumbbell-shaped, and the “z” subscript shows orientation along the z-axis. The 3pz orbital has one angular node, a planar region at the xy-plane where the probability of finding the electron is zero. In addition, because n = 3, it has one radial node where the probability also drops to zero at a certain radius.

The 3d orbitals have l = 2, and with n = 3, they also have one radial node. The 3dxy orbital lies in the xy-plane with lobes between the x and y axes, and nodes along both axes. The 3dx²−y² orbital has lobes aligned directly along the x and y axes, and its angular nodes lie between these directions. The 3dz² orbital is distinct: it consists of two elongated lobes along the z-axis with a donut-shaped ring in the xy-plane, making it the most symmetric among d orbitals.

Nodes are key features. Angular nodes are planes or cones where electron probability is zero due to the shape of the orbital, and radial nodes occur at specific distances from the nucleus due to the wavefunction’s radial part. Understanding these orbitals is essential for grasping chemical bonding and electron configurations.