Problem 57 - Chemistry: The Central Science

Question

Give the numerical values of \(n\) and \(l\) corresponding to each of the following orbital designations: (a) \(3 p\), (b) \(2 s\), (c) \(4 f\), (d) \(5 d\).

Step-by-Step Solution

Answer

(a) 3p: \(n = 3\), \(l = 1\) (b) 2s: \(n = 2\), \(l = 0\) (c) 4f: \(n = 4\), \(l = 3\) (d) 5d: \(n = 5\), \(l = 2\)

Step 1: (a) 3p orbital quantum numbers

The 3p orbital has the principal quantum number \(n = 3\). The letter 'p' corresponds to the angular momentum quantum number \(l = 1\). So for the 3p orbital, \(n = 3\) and \(l = 1\).

Step 2: (b) 2s orbital quantum numbers

The 2s orbital has the principal quantum number \(n = 2\). The letter 's' corresponds to the angular momentum quantum number \(l = 0\). So for the 2s orbital, \(n = 2\) and \(l = 0\).

Step 3: (c) 4f orbital quantum numbers

The 4f orbital has the principal quantum number \(n = 4\). The letter 'f' corresponds to the angular momentum quantum number \(l = 3\). So for the 4f orbital, \(n = 4\) and \(l = 3\).

Step 4: (d) 5d orbital quantum numbers

The 5d orbital has the principal quantum number \(n = 5\). The letter 'd' corresponds to the angular momentum quantum number \(l = 2\). So for the 5d orbital, \(n = 5\) and \(l = 2\). To summarize: (a) 3p orbital: \(n = 3\), \(l = 1\) (b) 2s orbital: \(n = 2\), \(l = 0\) (c) 4f orbital: \(n = 4\), \(l = 3\) (d) 5d orbital: \(n = 5\), \(l = 2\)

Key Concepts

Principal Quantum Number

The principal quantum number, symbolized as n, is a fundamental concept in quantum mechanics that denotes the size and energy level of an electron orbital. Think of it like the floor number in a hotel: the higher the floor number, the higher up you are. In the context of an atom, the higher the value of n, the further away the orbital is from the nucleus, and the higher the energy level of the electrons in that orbital.

For example, in our exercise, the value of n for a 3p orbital is 3. This indicates that the electrons in this orbital are in the third energy level of the atom. It's essential to understand that n starts at 1 and goes up from there, and the value of n can give you insights into the electron's properties like its potential energy and its distance from the nucleus.

Angular Momentum Quantum Number

The angular momentum quantum number, represented as l, tells us about the shape of the electron orbital. Imagine it as different floor plans for rooms on the same floor of our hotel analogy. Each floor plan corresponds to a different room shape. For example, the s orbital floor plan, with l = 0, is spherical, whereas the p, d, and f orbitals have progressively more complex shapes marking the areas where there is a high probability of finding an electron.

In quantum mechanics, l can have integer values ranging from 0 up to n - 1. So, for a principal quantum number n = 3, l can be 0, 1, or 2, corresponding to s, p, and d orbitals, respectively. Identifying the correct value of l helps in predicting the orbital shape of an electron and understanding the atom's electronic structure.

Electron Orbitals

Electron orbitals are regions within an atom where electrons are most likely to be found. Unlike the orbits of planets around the sun, orbitals are not circular paths but rather complex three-dimensional shapes around the nucleus, where there is a high probability of locating an electron. They are specified by three quantum numbers: the principal (n), angular momentum (l), and magnetic (m_l) quantum numbers.

Each type of orbital (s, p, d, and f) has a unique shape and orientation in space. The s orbitals are spherical, while p orbitals are dumbbell-shaped, d orbitals can be cloverleaf-shaped, and f orbitals are even more complex. Understanding the distinction between these orbitals is crucial for grasping the arrangement of electrons in an atom and predicting chemical properties and behaviors.

Quantum Mechanics in Chemistry

Quantum mechanics plays a pivotal role in chemistry as it explains the behavior of atoms and molecules at the microscopic level. According to quantum theory, energy, such as that of an electron in an atom, is quantized, meaning it can only take on specific discrete values. This is why we have distinct electron energy levels and orbitals within an atom – a concept that underpins the entire structure of the periodic table.

The application of quantum mechanics to chemistry, known as quantum chemistry, enables chemists to understand and predict the properties of molecules, such as stability and reactivity. It provides the tools to describe bonds between atoms, the shapes of molecules, and how they interact with light and other forms of energy. Essentially, quantum mechanics lays the groundwork for most of the theoretical explanations and predictions in chemical research and technology.