Question

Explain the following variations in atomic or ionic radii: (a) \(1^{-}>1>1^{+},(b) C a^{2+}>M g^{2+}>B e^{2+}\) (c) \(\mathrm{Fe}>\mathrm{Fe}^{2+}>\mathrm{Fe}^{3+}\).

Short answer

In summary, the variations in atomic or ionic radii can be explained as follows: (a) \(1^{-} > 1 > 1^{+}\): The anion (negative ion) has increased shielding due to the additional electron, making it larger than the neutral atom. The cation (positive ion), on the other hand, has decreased shielding and an increased effective nuclear charge, making it smaller than the neutral atom. (b) \(Ca^{2+} > Mg^{2+} > Be^{2+}\): The atomic radii are influenced by the number of electron shells and effective nuclear charge. As we move from \(Be^{2+}\) to \(Ca^{2+}\), the ions have additional electron shells, making their atomic radii larger despite the increased effective nuclear charge. (c) \(Fe > Fe^{2+} > Fe^{3+}\): The atomic radii decrease as we move from the neutral iron atom to the iron cations due to the increased effective nuclear charge and reduced shielding effects from the lost electrons.

Step-by-step solution

Case (a): \(1^{-} > 1 > 1^{+}\)

In this case, we are comparing an anion (negative ion) with the same atom in its neutral state, and a cation (positive ion) with the same atom in its neutral state. Let's analyze the factors that influence the atomic radii. 1. Effective nuclear charge: The more positive the nucleus is, the stronger the pull it has on the electrons, resulting in a smaller atomic radius. 2. Shielding: The presence of more electrons results in increased shielding, meaning the electrons in the outermost shell experience less of the nucleus's positive charge since it is being 'blocked' by the inner electrons. Comparing the three situations: - \(1^{-}\): The anion has one extra electron than the neutral atom, which results in increased shielding and reduces the effective nuclear charge on the outer electron. Therefore, the anion is larger than the neutral atom. - \(1 > 1^{+}\): The cation has one less electron than the neutral atom, resulting in decreased shielding and an increased effective nuclear charge on the outer electron. Therefore, the cation is smaller than the neutral atom.

Case (b): \(Ca^{2+} > Mg^{2+} > Be^{2+}\)

In this case, we are comparing three cations with the same charge but with different atomic numbers. Let's analyze the trends in atomic radii. 1. Number of electron shells: The more electron shells an ion has, the larger its atomic radius will be due to more electrons' presence. 2. Effective nuclear charge: The higher the atomic number, the more protons are in the nucleus, and the stronger the force attracting electrons, resulting in a smaller radius. Comparing the three ions: - \(Ca^{2+}\) has the highest atomic number (20) and loses two electrons, resulting in an atomic number of 18. (Electronic Configuration: \(1s^2 2s^2 2p^6 3s^2 3p^6\)) - \(Mg^{2+}\) has an atomic number of 12 and also loses two electrons, resulting in an atomic number of 10. (Electronic Configuration: \(1s^2 2s^2 2p^6\)) - \(Be^{2+}\) has the lowest atomic number (4) and also loses two electrons, resulting in an atomic number of 2. (Electronic Configuration: \(1s^2\)) As we move from \(Be^{2+}\) to \(Ca^{2+}\), the ions are gaining additional electron shells. The increase in shells outweighs the effect of increased effective nuclear charge, making the atomic radii larger.

Case (c): \(Fe > Fe^{2+} > Fe^{3+}\)

In this case, we are comparing a neutral iron atom with iron cations carrying a 2+ and 3+ charge. Let's analyze the trends in atomic radii: 1. Effective nuclear charge: The more positive the nucleus is, the stronger the pull it has on the electrons, resulting in a smaller atomic radius. 2. Shielding: The removal of electrons may reduce the shielding effect and the outer electrons will experience a stronger effective nuclear charge. Comparing the three situations: - \(Fe > Fe^{2+}\): In this comparison, the \(Fe^{2+}\) cation has lost two electrons from its outer shell. This loss reduces the shielding effect and increases the effective nuclear charge, resulting in a smaller atomic radius for the cation. - \(Fe^{2+} > Fe^{3+}\): In this comparison, the \(Fe^{3+}\) cation has lost an additional electron from the \(Fe^{2+}\) cation. This further loss decreases the shielding effect and increases the effective nuclear charge, resulting in a smaller atomic radius for the \(Fe^{3+}\) cation. Overall, the atomic radii decrease as we move from the neutral iron atom to the iron cations due to the increased effective nuclear charge and reduced shielding effects.

Key concepts

Effective Nuclear Charge

The concept of effective nuclear charge ( Z_{eff} ) is crucial when considering atomic and ionic sizes. This charge refers to the net positive charge experienced by electrons in an atom. It is a balance between the total positive charge of the nucleus and the electron repulsion caused by electrons in lower energy levels. - Z_{eff} is important because it affects the size of an atom or ion. The greater the effective nuclear charge, the more tightly electrons are pulled in towards the nucleus, leading to a smaller atomic radius. - When atoms form cations by losing electrons, effective nuclear charge felt by remaining electrons increases, resulting in a smaller radius. - Conversely, when atoms become anions by gaining electrons, the effective nuclear charge per electron decreases, leading to a larger atomic radius. For example, in the set 1^{-}, 1, 1^{+} , the anion 1^{-} has more electrons resulting in a higher shielding and therefore, lower Z_{eff} , making the radius larger. In contrast, 1^{+} being a cation has fewer electrons, a higher Z_{eff} , and a smaller atomic radius.

Shielding Effect

The shielding effect plays a significant role in determining the size of atoms and ions. This phenomenon occurs when inner-shell electrons 'shield' or block the outer-shell electrons from experiencing the full force of the positive charge from the nucleus. - More electrons means more shielding, which can lead to outer electrons experiencing less nuclear attraction. This often results in a larger atomic or ionic size as seen in anions. - Fewer electrons lead to less shielding, allowing outer electrons to feel a greater pull from the nucleus, thus resulting in a smaller atomic or ionic size, typical of cations. In situations where an atom loses an electron to form a cation, such as Fe^{2+} and Fe^{3+} , there is less electron shielding, making the outer electrons more tightly attracted to the nucleus. Consequently, this results in a smaller atomic radius as more electrons are removed.

Cations and Anions

The formation of cations and anions involves the loss or gain of electrons, respectively. These processes have a direct impact on the size of an atom or ion. - **Cations** are positively charged ions formed by the loss of one or more electrons. Since they involve fewer electrons, the nuclear attraction becomes stronger, often resulting in a smaller radius compared to the neutral atom. - **Anions** are negatively charged ions formed by the gain of electrons, which increases the electron count, enhancing inner electron repulsion and often resulting in a larger radius. When comparing sizes, Fe^{2+} and Fe^{3+} cations are smaller than neutral Fe. This is because cations have lost electrons, resulting in reduced shielding and increased effective nuclear charge. Conversely, 1^{-} is larger than its neutral form due to extra electrons increasing the shielding effect.

Electron Configuration

Electron configuration provides insight into how electrons are distributed around an atom's nucleus in various energy levels or shells. This configuration is key to predicting the chemical and physical behaviors of atoms and ions. - Each added electron occupies the lowest available energy level following the Aufbau principle, leading to changes in the atom's size and charge distribution. - Shell completion can increase stability, such as the noble gases, while incomplete shells are more reactive. When elements like Ca^{2+}, Mg^{2+}, and Be^{2+} form, they lose electrons to achieve stable electron configurations: Ca^{2+} retains electrons in lower energy levels compared to its original state. The loss of electrons and the resulting electron configuration lead to changes in ionic radii. This is visible as Ca^{2+} sustains a larger ionic radius due to possessing outer shell electrons when compared to Be^{2+} , which has a very contracted electron configuration. This shrinking factor is due to the fewer electron shells present post electron-loss.