Question

Derive the ground-state term symbols for the following atoms: a. \(\mathrm{F}\) b. Na c. \(P\)

Short answer

The ground-state term symbols for the given atoms are: a. F: \(^3P_2\) b. Na: \(^2S_0\) c. P: \(^2P_2\)

Step-by-step solution

a. Ground-state term symbol for F

First, we need to find the electron configuration of F (fluorine), which has an atomic number of 9. Following the Aufbau principle, the configuration is: 1s² 2s² 2p⁵ Now, we should figure out the value of the total orbital angular momentum L and the total spin angular momentum S for the last occupied shell (i.e., the 2p shell): For the 2p shell with 5 electrons, which has 3 orbitals (m_l = -1, 0, +1), we have: L = |Σm_l| = |-1| + |0| + |+1| = 2 (for 3 electrons occupying different orbitals) m_l = +1 for the fourth electron m_l = -1 for the fifth electron L_total = L + Σ(|m_l|) = 2 + (+1) + (-1) = 2 To find the total S, we start calculating the spin of individual electrons: S = |Σm_s| = |(+1/2) + (-1/2)| = 0 (for the electrons in m_l = -1 and m_l = +1) S = +1/2 for the two remaining electrons S_total = S + Σs = 0 + (+1/2) + (+1/2) = 1 Then, we compute the term symbol as follows: Term symbol = 2S+1 L, where L is denoted by capital letters S, P, D, F, G... Term symbol = 2S+1 P_L = 2(1)+1 P₂ = \(^3P\)_2 Thus, the ground-state term symbol for F is \(^3P_2\).

b. Ground-state term symbol for Na

First, we need to find the electron configuration of Na (sodium), which has an atomic number of 11. Following the Aufbau principle, the configuration is: 1s² 2s² 2p⁶ 3s¹ Now, we should figure out the value of the total orbital angular momentum L and the total spin angular momentum S for the last occupied shell (i.e., the 3s shell): For the 3s shell with 1 electron: L = Σm_l = 0 S = Σm_s = +1/2 The term symbol will be computed as follows: Term symbol = 2S+1 L = 2(1/2)+1 S₀ = \(^2S\)_0 Thus, the ground-state term symbol for Na is \(^2S_0\).

c. Ground-state term symbol for P

First, we need to find the electron configuration of P (phosphorus), which has an atomic number of 15. Following the Aufbau principle, the configuration is: 1s² 2s² 2p⁶ 3s² 3p³ Now, we should figure out the value of the total orbital angular momentum L and the total spin angular momentum S for the last occupied shell (i.e., the 3p shell): For the 3p shell with 3 electrons, which has 3 orbitals (m_l = -1, 0, +1), we have: L = |Σm_l| = |-1| + |0| + |+1| = 2 S = |Σm_s| = |(+1/2) + (-1/2) + (+1/2)| = 1/2 Then, we compute the term symbol as follows: Term symbol = 2S+1 L = 2(1/2)+1 P₂ = \(^2P\)_2 Thus, the ground-state term symbol for P is \(^2P_2\).

Key concepts

Electron Configuration

Understanding electron configuration is key to unlocking many aspects of an atom's behavior. It's like a roadmap indicating how electrons are arranged in an atom. Electrons reside in energy levels or "shells," surrounding the atomic nucleus. Each shell is made up of subshells, named s, p, d, and f, based on their angular momentum. The sequence fills from a lower energy level to a higher one.
For instance, let’s look at Fluorine (F). It has 9 electrons; hence, its electron configuration follows the pattern 1s² 2s² 2p⁵. This tells us that the first shell has 2 electrons in the s subshell, the second shell also starts with 2 s electrons, and then there are 5 electrons in the p subshell of the second shell. Sodium (Na) and Phosphorus (P) follow similar rules: with configurations 1s² 2s² 2p⁶ 3s¹ for Na and 1s² 2s² 2p⁶ 3s² 3p³ for P, the arrangement of their electrons affects the properties and term symbols, which reveal details about their angular momentum.

Orbital Angular Momentum

The concept of orbital angular momentum, denoted as L, is tied to the movement of electrons in their respective orbitals. Each type of orbital (s, p, d, f) has its specific angular momentum values.
For example, in the case of p orbitals, each of those can be represented by magnetic quantum numbers (m_l = -1, 0, +1). When figuring out L, you sum up these magnetic quantum numbers for each electron in the particular shell, taking the absolute values.

• For fluorine (F), the 5 electrons in the 2p sub-shell contribute to its orbital angular momentum.
• The calculations involve adding the magnetic quantum numbers of these electrons, which results in a total orbital angular momentum of 2.

Understanding these numbers assists us in deriving term symbols, which are a powerful way to convey an atom's potential energetics and electron configuration in a succinct form.

Spin Angular Momentum

Every electron spins on its axis, similar to how Earth spins around its axis. This spinning creates spin angular momentum. The quantum mechanical feature of electrons means that only two "spin" states exist: "+1/2" and "-1/2".
When deriving term symbols, the summation of these spin "motions" (indicated by m_s) is essential. For certain configurations:

• In Sodium (Na), there’s one unpaired electron in the 3s shell with a spin of "+1/2".
• For Phosphorus (P), the spin calculations from its outermost electrons result in a total spin angular momentum of "1/2".

Spin angular momentum unites with orbital angular momentum to determine the term symbol, which helps classify atomic states according to their energy and behavior in a magnetic field.

Aufbau Principle

The Aufbau Principle is like a set of construction rules for building an atom's electron shell. In German, "Aufbau" means "build-up," which is quite fitting, as this principle explains how electrons fill up an atom’s orbitals in a step-wise manner. Electrons occupy the lowest energy orbitals available before moving to higher ones, hence providing a systematic chemistry approach.
Consider how it's applied for fluorine, sodium, and phosphorus:

• For F, electrons start populating from the 1s orbitals before moving sequentially to fill the 2s and 2p orbitals.
• Na, with an additional electron, utilizes this principle to extend electron placement into the 3rd shell.
• Similarly, P follows this orderly filling to derive a unique electron configuration, impacting its chemical properties.

The Aufbau Principle is pivotal in chemistry, guiding the electron configuration and influencing the physical and chemical properties of elements.