Step 1: Understanding Ionization Enthalpy
Ionization enthalpy refers to the energy required to remove an electron from a neutral atom in its gaseous state. Lower ionization enthalpy indicates that the element can lose electrons more easily. The elements A and B have the lowest ionization enthalpy values in their group. This suggests that they are relatively easier to ionize compared to other members of their group.
Step 2: Analyzing the Nature of Their Ions
Element A: Since A has a low ionization enthalpy, it will lose electrons easily and thus, the A\(^{2+}\) ion will readily accept electrons (reduced). Hence, element A behaves as a
reducing agent
Element B: On the other hand, element B has a slightly higher ionization enthalpy compared to A. The B\(^{4+}\) ion tends to attract electrons, which makes it a strong
oxidizing agent
Thus, the nature of their ions is reducing (for A\(^{2+}\)) and oxidizing (for B\(^{4+}\)).
The question asks about the nature of ions A2+ and B4+ from group 14 elements based on their ionization enthalpy. To answer this, we need to understand the relationship between ionization enthalpy, group properties, and the behavior of ions as reducing or oxidizing agents.
Step 1: Understand the Context
Group 14 elements are carbon (C), silicon (Si), germanium (Ge), tin (Sn), and lead (Pb). The ionization enthalpy values provided (708 and 715 kJ mol-1) are the lowest among their group, indicating that A and B are the heavier elements at the bottom of the group. Typically, among these elements, tin (Sn) and lead (Pb) have the lowest ionization enthalpy.
Step 2: Identifying Elements and Their Ions
Given the ionization enthalpies, it is reasonable to assume:
We need to evaluate the nature of Sn2+ and Pb4+ ions.
Step 3: Evaluating the Ions as Reducing or Oxidizing Agents
The reducing or oxidizing nature of an ion depends on its ability to donate or accept electrons:
Step 4: Conclusion
The ions Sn2+ and Pb4+ align as reducing and oxidizing agents, respectively. Thus, the correct answer is:
Reducing and Oxidizing
This choice is justified by the tendency of Sn2+ to oxidize itself (act as a reducing agent) and Pb4+ to reduce itself (act as an oxidizing agent). Such behavior is typical for these ions based on their position in the periodic table and the stability of their oxidation states.
Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ (\sin x \cos y)(f(2x + 2y) - f(2x - 2y)) = (\cos x \sin y)(f(2x + 2y) + f(2x - 2y)), \] for all \( x, y \in \mathbb{R}. \)
If \( f'(0) = \frac{1}{2} \), then the value of \( 24f''\left( \frac{5\pi}{3} \right) \) is: