Question

Which of the following orbital designations is (are) not correct? a. $1d$ b. $2d$ c. $4d$ d. $5f$

Short answer

The incorrect orbital designations are $1d$ and $2d$.

Step-by-step solution

Remember the rules for angular momentum quantum numbers

For the designated orbitals, recall that the angular momentum quantum number l has the following ranges depending on the principal quantum number (n): - For n = 1, l can only be 0 (s orbital) - For n = 2, l can be 0 (s orbital) or 1 (p orbital) - For n = 3, l can be 0 (s orbital), 1 (p orbital), or 2 (d orbital) - For n = 4, l can range from 0 (s orbital) to 3 (f orbital) - Higher values of n follow the pattern where l can range from 0 to (n-1).

Evaluate each option

We will now analyze each given option based on the rules from Step 1. a. For a 1d orbital, n = 1. However, as per the quantum number rules, when n=1, the only possible orbital type is an 's' orbital (l=0). Therefore, 1d is not a correct orbital designation. b. For a 2d orbital, n = 2. Here, the quantum number rules state that for n=2, only 's' (l=0) and 'p' (l=1) orbitals are possible. Thus, 2d is also not a correct orbital designation. c. For a 4d orbital, n = 4. In this case, for n=4, the possible orbital types are 's' (l=0), 'p' (l=1), 'd' (l=2), and 'f' (l=3). So, 4d is a correct orbital designation. d. For a 5f orbital, n = 5. Based on the rules for quantum numbers, when n=5, the possible orbital types are 's' (l=0), 'p' (l=1), 'd' (l=2), 'f' (l=3), and 'g' (l=4). Therefore, 5f is a correct orbital designation.

Present the final answer

Based on the analysis in Step 2, the orbital designations that are not correct are: a. 1d b. 2d

Key concepts

Orbital Designations

Orbitals are regions around an atom's nucleus where the probability of finding electrons is highest.
These orbitals are designated by a combination of numbers and letters, which indicate specific energy levels and shapes.
For example, an orbital designation like $3p$ tells us it's in the third energy level with a 'p' shape.

• The number represents the principal quantum number $n$, indicating the energy level.
• The letter indicates the type or shape of the orbital, governed by the angular momentum quantum number $l$.

Despite their importance, not all combinations are possible.
For example, in the exercise provided, a designation like $1d$ is not correct because the number of the energy level (1) does not support a 'd' orbital.
This is because each energy level can only support orbitals given by $l=0$ to $n-1$.
Learning correct designations helps predict electron arrangements in atoms, which are crucial for understanding chemical behavior.

Angular Momentum Quantum Number

The angular momentum quantum number $l$ defines the shape of an orbital.
It can have integer values from $0$ to $n-1$, where $n$ is the principal quantum number.
This means each energy level supports several types of orbitals.
For instance, at $n=3$, $l$ can be $0,1,$ or $2$, corresponding to 's', 'p', and 'd' orbitals respectively.

• An $l$ value of $0$ indicates an 's' orbital, spherical in shape.
• When $l=1$, it's a 'p' orbital, which is dumbbell-shaped.
• If $l=2$, you have a 'd' orbital, which is more complex in shape, often clover-like.

The rule $l=n-1$ is critical; it tells you the types of orbitals the quantum number can describe.
For example, in the $2d$ orbital designation seen in the exercise, $n=2$ and should only allow $l=0$ or $1$, meaning only 's' or 'p' orbitals are valid.
Therefore, $2d$ is an incorrect designation.

Principal Quantum Number

The principal quantum number $n$ is fundamental for understanding atomic orbitals.
It indicates an electron's energy level and its relative distance from the nucleus.
Higher $n$ values mean the electrons are further from the nucleus and have higher energy.

• $n=1$ is the first energy level and only supports 's' orbitals ($l=0$).
• With $n=2$, you can have 's' ($l=0$) and 'p' ($l=1$) orbitals.
• As $n$ increases further, more orbital types become available: 'd', 'f', 'g', etc.

The relationship between $n$ and $l$ is given by $l$ ranging from $0$ to $n-1$.
This simple but powerful relation helps us understand which orbitals exist at specific energy levels.
For example, in the exercise, $4d$ is a valid orbital because with $n=4$, $l=2$ (a 'd' orbital) is possible.
Recognizing the right combinations of $n$ and $l$ ensures accurate predictions of electron configurations, vital for chemistry.