Question

Give a possible set of values of the four quantum numbers for all the electrons in a boron atom and a nitrogen atom if each is in the ground state.

Short answer

For a boron atom in the ground state with 5 electrons, the possible sets of quantum numbers are: 1. \( (n=1, l=0, m_l=0, m_s=+1/2) \) 2. \( (n=1, l=0, m_l=0, m_s=-1/2) \) 3. \( (n=2, l=0, m_l=0, m_s=+1/2) \) 4. \( (n=2, l=0, m_l=0, m_s=-1/2) \) 5. \( (n=2, l=1, m_l=-1, m_s=+1/2) \) For a nitrogen atom in the ground state with 7 electrons, the possible sets of quantum numbers are: 1. \( (n=1, l=0, m_l=0, m_s=+1/2) \) 2. \( (n=1, l=0, m_l=0, m_s=-1/2) \) 3. \( (n=2, l=0, m_l=0, m_s=+1/2) \) 4. \( (n=2, l=0, m_l=0, m_s=-1/2) \) 5. \( (n=2, l=1, m_l=-1, m_s=+1/2) \) 6. \( (n=2, l=1, m_l=0, m_s=+1/2) \) 7. \( (n=2, l=1, m_l=+1, m_s=+1/2) \)

Step-by-step solution

Boron Quantum Numbers

1. n = 1, l = 0, m_l = 0, m_s = +1/2 (1s electron) 2. n = 1, l = 0, m_l = 0, m_s = -1/2 (1s electron) 3. n = 2, l = 0, m_l = 0, m_s = +1/2 (2s electron) 4. n = 2, l = 0, m_l = 0, m_s = -1/2 (2s electron) 5. n = 2, l = 1, m_l = -1, m_s = +1/2 (2p electron)

Nitrogen Quantum Numbers

1. n = 1, l = 0, m_l = 0, m_s = +1/2 (1s electron) 2. n = 1, l = 0, m_l = 0, m_s = -1/2 (1s electron) 3. n = 2, l = 0, m_l = 0, m_s = +1/2 (2s electron) 4. n = 2, l = 0, m_l = 0, m_s = -1/2 (2s electron) 5. n = 2, l = 1, m_l = -1, m_s = +1/2 (2p electron) 6. n = 2, l = 1, m_l = 0, m_s = +1/2 (2p electron) 7. n = 2, l = 1, m_l = +1, m_s = +1/2 (2p electron)

Key concepts

Electron Configuration

Electron configuration refers to the distribution of electrons in the orbitals of an atom. Understanding how electrons are arranged in an atom is key to comprehending chemical reactions and bonds. Electrons fill orbitals starting from the lowest energy level, following what is known as the Aufbau principle. The configuration is noted using a series of numbers, letters, and superscripts that represent the principal quantum number (n), the orbital type (s, p, d, f), and the number of electrons in those orbitals.

For example, the electron configuration of a boron atom, which has 5 electrons, is written as 1s² 2s² 2p¹. This notation indicates that there are two electrons in the 1s orbital, two electrons in the 2s orbital, and one electron in one of the 2p orbitals. The understanding of how to write out electron configurations is crucial because it allows us to predict the chemical properties of an atom.

Filling Order

The order in which subshells are filled is determined by their energy levels. An easy way to remember the order is to use the n+l rule, or by referring to a periodic table that often provides a visual aid to see the progression of filling orbitals.

Quantum Mechanical Model

The quantum mechanical model of the atom describes electrons in terms of a wave function, giving rise to electron clouds rather than fixed orbits. This model uses four quantum numbers to describe the unique state of an electron, similar to an address that indicates where an electron is likely to be found.

The first is the principal quantum number (n) which indicates the energy level and distance from the nucleus. The second is the angular momentum quantum number (l) which defines the shape of the orbital. The magnetic quantum number (m_l) designates the orientation of the orbital in space, and the spin quantum number (m_s) refers to the spin direction of the electron.

Probability and Uncertainty

One fundamental aspect of the quantum mechanical model is the concept of probability. This model implies that we can only predict the probability of finding an electron in a certain region around the nucleus, not the exact path it follows. This is a departure from the classical view of electrons orbiting the nucleus like planets around the sun.

Ground State Electron Configuration

The ground state electron configuration of an atom is the arrangement of electrons in the lowest possible energy levels or orbitals. This is the most stable electron configuration and the starting point for understanding an atom's reactivity and bonding capabilities.

For instance, the ground state electron configuration of the boron atom, with the quantum numbers provided in the exercise, would be 1s² 2s² 2p¹. For a nitrogen atom, the configuration would be 1s² 2s² 2p³, as indicated by the seven quantum number sets describing its seven electrons.

Hund's Rule

It is also essential to understand Hund's Rule when dealing with the ground state electron configuration. The rule states that electrons will fill degenerate orbitals (orbitals of the same energy level, like the three 2p orbitals) singly as far as possible before pairing up. This leads to a more stable arrangement due to electron repulsion and is observed in the step-by-step solution by the separate occupancy of the 2p orbitals in the case of the nitrogen atom.