Boundary Surface Diagram

A boundary surface diagram is one of the good diagrammatic representations of the shape of atomic orbitals. It is a resultant solution of the Schrödinger wave equation.

As we all know that the momentum and exact position of an electron cannot be determined (as per the Heisenberg uncertainty principle), so we calculate the probability density of finding the electron in a specific region. 

A boundary surface diagram can be explained either as a boundary surface or as a contoured surface that is drawn in a space for an orbital, where the value of probability density |ψ|2 is constant.

Note: The constant probability density of the boundary surface diagram is considered an acceptable and good approximation of orbital shape if the boundary surface encloses the volume or region with a probability density having more than 90%. It means that the boundary surface enclosing a constant probability density of 50% (for suppose) won't be considered good.

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Features of the Boundary Surface Diagram

• Size of the Surface Diagram

The boundary surface diagram of the orbital increases either in volume or size with an increase in the principal quantum number (n).

• Shape of the Surface Diagram

The orbital boundary surface diagram is independent of the principal quantum number.

For example, the boundary surface diagram of an s orbital is spherical, and so, it will remain spherical for 1s, 2s, 3s, 4s, or for any other general ns. It should be noted that the shape does not depend on the principle quantum number (n).

• Nodes in the Surface Diagram

Nodes are the regions having very low probability density, which goes to zero, typically. There exist (n-1) nodes in the s-orbital’s boundary surface diagram with the principal quantum number 'n.' Such nodes are also noticed in the surface diagram of p, d, f orbitals.

Shapes of the Orbitals

Probability Density

• ψ provides us the wave amplitude. The value of ψ contains no physical significance.

• Wheres, |Ψ|2 provides us the region, where the probability of finding an electron is maximum. It is known as a probability density.

Nodal Surfaces

The region where this function of probability density reduces to zero is known as simply nodes or nodal surfaces.

There are two types of Nodes, which are given below:

1. Angular nodes and, 

2. Nodal Planes

Angular nodes or nodal planes take place when the probability density wave function for the electron is given as zero along the directions specified by a specific angle, where the number of angular nodes is l.

Therefore we can say that

• The number of angular nodes is l,

• The number of radial nodes is n-l-1,

• Total number of nodes = No.of radial nodes + No.of angular nodes = n – 1.

Nodal Region or Radial Nodes

Radial nodes or the nodal region takes place when the probability density of the wave function for an electron is zero on a spherical surface of the radius. Thus, the number of radial nodes is = n – l – 1

Shapes of Boundary Surface Diagrams

It is a surface in space, where the density of probability is constant for a given orbital. This provides a good representation of the orbital shape. Also, this shape encloses the region or volume, where the probability of finding electrons is high.

Shape of S-Orbital

• All the s-orbitals will be Spherical in shape.

• The size of s orbital increases with an increase in n, which means 4s > 3s > 2s > 1s, and the electron is located further away to the point of the nucleus as the principal quantum number increases.

• The probability of finding out the electron at a given distance is equal in all directions.

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Shape of P-orbitals

• It holds 3 possible orientations.

• Each p orbital contains 2 sections, which are known as lobes that are on either side of the plane, passing via the nucleus.

• The probability density function can be given as zero on the plane, where the 2 lobes touch each other.

• The shape, energy, and size of the 3 orbitals are similar, whereas just the orientation is different.

• They are provided with the designations 2px, 2py, 2pz.

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Shape of D-Orbitals

• It contains 5 orientations.

• The shapes of the first 4D orbitals are the same as each other, whereas that of the 5th one is different from the others. But, all the 5 3d orbitals are equivalent in energy.

• The 5 d-orbitals are designated as dxy, dyz, dxz, dx²-y², dz².

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Difference between boundary Surface and Orbital Diagrams

• A boundary is a surface or line marking the extent of some feature. Hence, it is not similar to an orbital.

• An orbital can be given as a volume around an atomic nucleus, where an electron can be found with a given permitted energy state. 

• Even though the textbook illustrations depict the orbitals as zones having clear boundaries, it is NOT accurate. 

• In fact, there are no boundaries defined for the orbitals, in the sense of demarcating a volume within which the electron will always be found.