Problem 39 - Chemistry and Chemical Reactivity

Question

State which of the following orbitals cannot exist according to the quantum theory: \(2 s, 2 d, 3 p, 3 f, 4 f,\) and \(5s\). Briefly explain your answers.

Step-by-Step Solution

Answer

2d and 3f do not exist according to quantum theory.

Step 1: Understanding Quantum Numbers

First, understand that orbitals are defined by quantum numbers. For any given principal quantum number, \(n\), the azimuthal or angular momentum quantum number, \(l\), must be an integer from 0 to \(n-1\). The possible \(l\) values then determine the type of orbital: \(s, p, d, f, ...\)

Step 2: Analyze Each Orbital

Examine each orbital given in the question:1. **2s**: For \(n = 2\), \(l\) can be 0 or 1. \(l = 0\) corresponds to an \(s\)-orbital. Therefore, 2s is possible.2. **2d**: For \(n = 2\), \(l\) can only be 0 or 1. \(d\)-orbitals correspond to \(l = 2\), so 2d does not exist.3. **3p**: For \(n = 3\), \(l\) can be 0, 1, or 2. \(l = 1\) corresponds to \(p\)-orbitals. Therefore, 3p is possible.4. **3f**: For \(n = 3\), \(l\) can be 0, 1, or 2. \(f\)-orbitals correspond to \(l = 3\), so 3f does not exist.5. **4f**: For \(n = 4\), \(l\) can be 0, 1, 2, or 3. \(l = 3\) corresponds to \(f\)-orbitals. Therefore, 4f is possible.6. **5s**: For \(n = 5\), \(l\) can be 0, 1, 2, 3, or 4. \(l = 0\) corresponds to \(s\)-orbitals. Therefore, 5s is possible.

Step 3: Conclusion

After analyzing each orbital:- 2d and 3f orbitals cannot exist because their angular momentum quantum number \(l\) is higher than the maximum allowed for their given principal quantum numbers \(n\), i.e., \(l < n\).

Key Concepts

Orbitals

In the world of quantum mechanics, an orbital is a region around an atom's nucleus where there is a high probability of finding an electron. These orbitals are not to be confused with actual paths of electrons; instead, they are more like clouds where electrons are likely to be found.
An important feature of orbitals is that they come in different shapes and sizes depending on the energy level and type. Different orbitals have specific labels like s, p, d, and f, which correspond to their shapes. The arrangement of electrons in these orbitals dictates an atom's chemical properties.
You can think about orbitals as rooms in a multi-story hotel (the atom), where each room can hold certain amounts of guests (electrons), and each floor (energy level/principal quantum number) has a different number of rooms (orbitals).

Principal Quantum Number

The principal quantum number, symbolized by \(n\), plays a key role in defining the size and energy level of an orbital. It can be any positive integer starting from 1. In our hotel analogy, the principal quantum number would represent different floors of the building. Each floor offers its distinct view and space.
The value of \(n\) determines the energy level of the electron in the atom. Higher values of \(n\) mean electrons are farther from the nucleus and have more energy. Importantly, as \(n\) increases, more types of orbitals (s, p, d, f) become permissible for that energy level.

• For \(n = 1\), only s-orbitals are possible.
• For \(n = 2\), s and p-orbitals are possible.
• For \(n = 3\), s, p, and d-orbitals are possible, and so on.

Angular Momentum Quantum Number

The angular momentum quantum number, denoted by \(l\), specifies the shape of an orbital and is crucial in determining which orbitals exist at each principal energy level. Each \(l\) value corresponds to a specific orbital shape: \(l = 0\) for s-orbitals, \(l = 1\) for p-orbitals, \(l = 2\) for d-orbitals, and \(l = 3\) for f-orbitals.
The range of \(l\) values for a given principal quantum number \(n\) is from 0 to \(n - 1\). For instance, if \(n = 3\), \(l\) can be 0, 1, or 2. This means, on this energy level, only s, p, and d-orbitals can exist. An \(f\)-orbital, which requires \(l = 3\) would not be possible until \(n\) becomes at least 4.

s, p, d, f Orbitals

Understanding s, p, d, and f orbitals is fundamental in quantum chemistry.

• s-orbitals: These are spherical in shape and exist at every energy level starting from \(n = 1\). They are simple and hold up to 2 electrons.
• p-orbitals: Shaped like dumbbells, these orbitals can hold up to 6 electrons and are available from \(n = 2\) onwards.
• d-orbitals: More complex in shape, these can contain up to 10 electrons and are found in energy levels starting from \(n = 3\).
• f-orbitals: With even more intricate shapes, these orbitals can accommodate up to 14 electrons, appearing from energy levels from \(n = 4\) upwards.

Recognizing these different types of orbitals helps understand how electrons occupy an atom's space and how they influence chemical bonding and properties.