Question

Arrange the following atoms in order of increasing effective nuclear charge experienced by the electrons in the \(n=3\) electron shell: \(K, M g, P, R h\), and \(T\). Explain the basis for your order.

Short answer

The order of increasing effective nuclear charge experienced by the electrons in the n=3 electron shell for the given elements is: Mg, P, K, Ti, and Rh. The basis for this order is the increase in the number of protons and the variation in shielding effect as we move across the periodic table. Mg has the least effective nuclear charge because of its low atomic number and the least shielding electrons in the n=3 shell, while Rh has the highest due to its higher atomic number and lesser shielding effect on n=3 electrons.

Step-by-step solution

Find the atomic numbers of the given elements

First, we will look up the periodic table and find the atomic numbers for each element. The atomic number is the number of protons in the nucleus of an atom and determines the element's identity. Here are the atomic numbers for the given elements: K (Potassium): 19 Mg (Magnesium): 12 P (Phosphorus): 15 Rh (Rhodium): 45 Ti (Titanium): 22

Write electron configurations for the given elements

Using the atomic numbers, we will write the electron configurations for each element. This will help us understand their electron arrangement and shielding effect on the n = 3 shell. The electron configurations are: K: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹ Mg: 1s² 2s² 2p⁶ 3s² P: 1s² 2s² 2p⁶ 3s² 3p³ Rh: 1s² 2s² 2p⁶ 3s² 3p⁶ 3d¹⁰ 4s² 4p⁶ 4d⁸ 5s¹ Ti: 1s² 2s² 2p⁶ 3s² 3p⁶ 3d² 4s²

Calculate the effective nuclear charge for the n=3 shell

The effective nuclear charge (Zeff) represents the net positive charge experienced by an electron in an atom, taking into account the shielding and repulsion caused by other electrons. For an electron in n=3 shell, we will consider both the shielding by inner electrons and the presence of other n=3 electrons. The shielding factor is approximately equal to the number of inner electrons (n < 3); however, for the sake of simplicity, we will use the atomic numbers (Z) which are indicative of the attraction with the nucleus. The Zeff can be estimated as: K: Z = 19 Mg: Z = 12 P: Z = 15 Rh: Z = 45 Ti: Z = 22

Arrange the elements in increasing order of effective nuclear charge experienced by n = 3 electrons

Using the estimated Zeff for the electrons in the n=3 shell, we can now arrange the elements in increasing order: Mg (12) < P (15) < K (19) < Ti (22) < Rh (45)

Explain the basis for the order

The effective nuclear charge increases as we move across a period in the periodic table because the number of protons in the nucleus increases, leading to a greater attraction between the nucleus and the electrons in the same shell. On the other hand, elements with more shielding electrons cause the nuclear charge to be less experienced by the valence electrons. This is why Mg has the lowest Zeff (12) as it has the least number of protons and the least shielding electrons in the n = 3 shell, while Rh has the highest Zeff (45) as it has the most protons and the least shielding effect on the electrons in the n=3 shell. Therefore, the order of increasing effective nuclear charge experienced by the electrons in the n=3 electron shell for the given elements is: Mg, P, K, Ti, and Rh.

Key concepts

Electron Configuration

Electron configuration refers to the arrangement of electrons in an atom's electron shells and subshells. Understanding this concept is essential as it provides insight into the behavior and properties of an element. Every element has a unique electron configuration, reflecting the number of electrons that fill the atomic orbitals around its nucleus. This configuration can be predicted using the Aufbau principle, Hund's rule, and the Pauli exclusion principle.

In the given exercise, knowing the electron configurations helps determine the effective nuclear charge, as they illustrate how electrons distribute themselves among different energy levels. For example, potassium (K) has an electron configuration of \(1s^2 2s^2 2p^6 3s^2 3p^6 4s^1\), giving insights into its electron arrangement and how these electrons are shielded.

• Aufbau Principle: Electrons fill the lowest energy orbitals first.
• Hund's Rule: Electrons fill degenerate orbitals singly before pairing up.
• Pauli Exclusion Principle: No two electrons can have the same set of quantum numbers.

Periodic Table

The periodic table is a comprehensive chart that organizes elements based on their atomic number, electron configurations, and recurring chemical properties. It serves as a roadmap for understanding how atomic structure correlates with an element's behavior and characteristics. The table is structured into periods (horizontal rows) and groups (vertical columns), with elements exhibiting increasing atomic numbers from left to right across a period.

Understanding the periodic trends is key for predicting how the effective nuclear charge changes across different elements. For instance, as one moves across a period, the number of protons in the nucleus increases which generally results in an increase in the effective nuclear charge experienced by the outer electrons. This exercise employs this knowledge to arrange the atoms based on their effective nuclear charge.

• Periods: Indicate the number of electron shells.
• Groups: Elements within a group generally share similar chemical properties.

Shielding Effect

The shielding effect occurs when inner electrons partially block the attraction between the nucleus and the outer electrons. Inner electrons repel outer electrons, decreasing the net magnetic pull exerted by the protons in the nucleus upon the valence electrons. This phenomenon is crucial for calculating the effective nuclear charge that electrons experience.

The more inner electron shells an atom has, the greater the shielding effect, thereby reducing the nuclear charge felt by the outer electrons. In the context of the given elements, magnesium (Mg) has a smaller shielding effect than rhodium (Rh) because they contain fewer inner electrons, leading to different effective nuclear charges despite their configurations.

• Inner Electrons: Block nuclear pull on outer electrons.
• Valence Electrons: Experience a decreased force due to shielding.

Atomic Number

The atomic number of an element is the total count of protons in the nucleus of an atom. It is a fundamental property that determines an element's identity and its position on the periodic table. Elements are ordered in increasing atomic number as this represents an orderly increment in the number of protons.

Atomic number is particularly significant when assessing effective nuclear charge because it represents the strength of the nuclear charge itself. The higher the atomic number, the greater the number of protons in the nucleus, and thus, the stronger the potential attraction to surrounding electrons, even as the shielding effect is considered. In our exercise, elements are compared based on their atomic numbers to estimate effective nuclear charge, arranging them accordingly.

• Proton Count: Determines an element's properties and placement.
• Increases: As you move across a period in the periodic table.