Consider a hypothetical element that forms a solid with bands as shown in the Figure. A. Suppose that the isolated atom has configuration 1s22s2. If its equilibrium separation is r0= a, is the solid a conductor or an insulator? What if r0= b? Answer both questions for the case that the atomic configuration is B. 1s22s22p1 and C. 1s22s22p6. ## Consider a Hypothetical Element and Band Formation**Problem 13.34:** Consider a hypothetical element that forms a solid with energy bands as represented in Fig. 13.42:- **(a)** Suppose the isolated atom has the electron configuration \(1s^22s^2\). If its equilibrium separation is \(r_0 = a\), is the solid a conductor or an insulator? What if \(r_0 = b\)?- Answer both questions for the cases when the atomic configuration is: - **(b)** \(1s^22s^22p^1\) - **(c)** \(1s^22s^22p^6\)### Explanation of Fig. 13.42:- The graph shows the energy bands formed by the overlapping atomic orbitals \(1s\), \(2s\), \(2p\), and \(3s\) over a separation distance \(r\).- **Horizontal Axis (r):** Represents the distance between atoms in the solid.- **Vertical Bands:** - **1s Band:** This is depicted as a narrow band at the bottom, indicating lower energy levels. - **2s and 2p Bands:** Wider bands that overlap as the distance \(r\) decreases, indicating possible conductivity when overlaps occur. - **3s Band:** The widest and most spread band, suggesting higher energy availability and greater overlap potential at smaller distances.- **Dashed Lines \(a\) and \(b\):** These represent specific values of the equilibrium separation \(r_0\): - **At \(r_0 = a\):** Assess overlap to determine if the solid is a conductor (bands overlap) or an insulator (bands do not overlap). - **At \(r_0 = b\):** Re-evaluate overlap for conductor or insulator properties.This setup helps in analyzing electronic properties by observing band overlaps and identifying potential conductive behavior in the hypothetical solid.

When an atomic state is filled to its capacity, it can be proved that there are effectively no free electrons that participate in the conduction of current. An empty or partially filled state can have a non-zero number of free electrons. Hence, the outermost state of a single atomic element that is completely filled corresponds to the valance band of its solid. And, the lowest empty or partially filled state of an atomic element corresponds to the conduction band of its solid.A. For the 1s2 2s2 electronic configuration, the 2s state is completely filled and the solid’s valance band corresponds to the 2s state. The solid’s conduction band corresponds to the next state, that is, the 2p state. For r0 = a, there is a gap between the valance and the conduction bands, which an electron cannot overcome without external excitation. Hence, the solid would be an insulator. For r0 = b, the valance and the conduction bands overlap. So an electron can freely move from the valance band to the conduction band. Hence, the solid would be a conductor.B. For the 1s2 2s2 2p1 electronic configuration, the 2s state is completely filled and the solid’s valance band corresponds to the 2s state. The solid’s conduction band corresponds to the next state, that is, the partially filled 2p state. For r0 = a, there is a gap between the valance and the conduction bands, which an electron cannot overcome without external excitation. Hence, the solid would be an insulator. For r0 = b, the valance and the conduction bands overlap. So an electron can freely move from the valance band to the conduction band. Hence, the solid would be a conductor.C. For the 1s2 2s2 2p6 electronic configuration, the 2p state is completely filled and the solid’s valance band corresponds to the 2p state. The solid’s conduction band corresponds to the next state, that is, the 3s state. For r0 = a, or r0 = b, there is a gap between the valance and the conduction bands, which an electron cannot overcome without external excitation. Hence, the solid would be an insulator in both cases.

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Advanced Physics

•• Consider a hypothetical element that forms a solid with bands as shown in Fig. 13.42. (a) Suppose that the isolated atom has configuration 1s²2s². If its equilibrium separation is ro = a, is the solid a con- ductor or an insulator? What if ro = b? Answer both questions for the case that the atomic configuration is (b) 1s²2s²2p' and (c) 1s²2s²2p°. 3s 2p 2s 1s b. a

•• Consider a hypothetical element that forms a solid with bands as shown in Fig. 13.42. (a) Suppose that the isolated atom has configuration 1s²2s². If its equilibrium separation is ro = a, is the solid a con- ductor or an insulator? What if ro = b? Answer both questions for the case that the atomic configuration is (b) 1s²2s²2p' and (c) 1s²2s²2p°. 3s 2p 2s 1s b. a