Question

Arrange the following atoms in order of increasing effective nuclear charge experienced by the electrons in the \(n=3\) electron shell: \(\mathrm{K}, \mathrm{Mg}, \mathrm{P}, \mathrm{Rh} , \mathrm{Ti}.\)

Short answer

The atoms arranged in the order of increasing effective nuclear charge experienced by the electrons in the n=3 electron shell are: Mg, P, K, Ti, Rh.

Step-by-step solution

Determine the atomic numbers

To find the effective nuclear charge, we first need the atomic numbers (Z) of the given elements. The atomic numbers can be found using a periodic table. The atomic numbers of the given elements are: K: 19 Mg: 12 P: 15 Rh: 45 Ti: 22

Calculate the number of inner electrons (shielding electrons)

We next determine the number of electrons in the inner shells, which shield the outer electrons from the full positive charge of the nucleus. For the n=3 electron shell, there will be two inner shells (n=1 and n=2), with a total of 10 electrons (2 electrons in n=1 and 8 electrons in n=2).

Calculate the effective nuclear charges (Z_eff)

The effective nuclear charge (Z_eff) for electrons in the n=3 shell can be calculated using the following formula: \(Z_\text{eff} = Z - S\) Where Z is the atomic number and S is the number of shielding electrons. For our elements, the Z_eff values are: K: 19 - 10 = 9 Mg: 12 - 10 = 2 P: 15 - 10 = 5 Rh: 45 - 10 = 35 Ti: 22 - 10 = 12

Arrange the elements in the order of increasing Z_eff

Now, we will arrange the elements in the order of increasing effective nuclear charge experienced by the electrons in the n=3 electron shell: Mg (2) < P (5) < K (9) < Ti (12) < Rh (35) So, the given atoms arranged in the order of increasing effective nuclear charge are: Mg, P, K, Ti, Rh.

Key concepts

Atomic Number

The atomic number, often denoted by the symbol \( Z \), is a fundamental property of an element. It tells us the number of protons found in the nucleus of an atom. The atomic number is crucial because it defines the identity of the element. For example, if an atom has an atomic number of 19, it is potassium (K).

Each element has a unique atomic number. This number isn't just important for dictating the element's identity, but it also tells us about the number of electrons in a neutral atom. Electrons are negatively charged particles, and in a neutral atom, the number of electrons equals the number of protons which is the atomic number.

In the periodic table, elements are arranged in ascending order based on their atomic numbers. This organization helps us predict the properties of elements and understand their similarities and differences.

Electrons

Electrons are tiny, negatively charged particles that orbit the nucleus of an atom. The arrangement of electrons in an atom is crucial in determining the chemical behavior of the atomic species.

Electrons are arranged in various energy levels or shells around the nucleus. These shells are denoted by the principal quantum number \( n \). For example, in a potassium atom, electrons are arranged in shells n=1, n=2, and n=3.

The outer shell, especially if it is not completely filled, plays a major role in chemical reactions. In our case, the electrons in the n=3 shell are particularly important because they determine the effective nuclear charge that these electrons experience. The electrons closer to the nucleus partially shield the outer electrons from the full positive charge of the nucleus, which affects how strongly these outer electrons are held.

Shielding

Shielding is an essential concept in understanding atomic structure and the behavior of electrons in an atom. It refers to the phenomenon where inner-shell electrons partially block or shield the outer-shell electrons from the complete nuclear charge.

The extent of shielding informs us about the effective nuclear charge \( Z_\text{eff} \), which is the net positive charge experienced by an electron in the valence shell. The more electrons there are in inner shells, the greater the shielding effect they will provide.

For instance, in the case of potassium (K), there are 10 inner electrons providing shielding for the electrons in the n=3 shell. This reduces the full nuclear charge (19 for K) to the effective charge felt by the outer electrons, calculated as \( Z_\text{eff} = Z - S \), where \( S \) is the number of inner electrons providing shielding.

Periodic Table

The periodic table is a systematic arrangement of chemical elements according to their atomic number, electron configuration, and recurring chemical properties. It is a vital tool in chemistry, as it summarizes the known information about the chemical behavior of the elements.

The structure of the periodic table allows us to determine trends in atomic size, electronegativity, ionization energy, and effective nuclear charge, among others. Elements in the same column, known as groups, usually have similar properties. On the other hand, elements in the same row, or period, exhibit gradually changing properties as you move across.

For our purposes, the periodic table allows us to quickly find the atomic numbers of elements, which is the first step in calculating the effective nuclear charge. By understanding the position of an element in the periodic table, we can infer a lot about its electronic structure and its chemical reactivity.