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Chapter 18: Problem 54

How does band theory explain the high electrical conductivity of mercury?

Short Answer

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Question: Explain the high electrical conductivity of mercury using band theory. Answer: The high electrical conductivity of mercury can be explained by the Band theory considering its electronic configuration ([Xe] 4f14 5d10 6s2), the position of its Fermi level within the partially filled s-band, and the high density of available states for conduction. The electrons in the s-band can easily participate in the conduction process, leading to a high electrical conductivity for mercury.

Step by step solution

01

Electronic configuration of mercury

Mercury has an atomic number of 80, so its electronic configuration is [Xe] 4f14 5d10 6s2. It has a completely filled 5d and 4f orbitals and 2 electrons in its outermost 6s orbital.
02

Energy band diagram of mercury

In a solid, the energy levels of isolated atoms form bands due to overlapping wavefunctions of nearby atoms (called a band structure). Since mercury has a filled 5d orbital and a partially filled 6s orbital, its band structure would have a full d-band and a partially full s-band.
03

Fermi level and conduction

The Fermi level represents the highest occupied electron energy level in a material at absolute zero temperature. In mercury, the Fermi level lies within the s-band because it has a partially filled 6s orbital. When a voltage is applied across a conductor, electrons near the Fermi level get excited to higher energy levels.
04

High conductivity of mercury

Since the Fermi level in mercury lies within the s-band, it has a high density of available electronic states for conduction. The electrons in the s-band can easily participate in the conduction process. Due to this, mercury exhibits high electrical conductivity. In conclusion, the high electrical conductivity of mercury can be explained by the Band theory considering its electronic configuration, the position of its Fermi level, and its partially filled s-band, allowing for a high density of available states for conduction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electronic Configuration
Understanding mercury's electronic configuration is essential to grasping its conductive properties.
Mercury, having an atomic number of 80, is configured as \([Xe] 4f^{14} 5d^{10} 6s^2\).
This notation shows a complete filling of both the 4f and 5d orbitals, while only the 6s orbital holds two electrons, making it partially filled.
This configuration influences how electrons in mercury interact and behave in a solid state.
  • Filled orbitals: 4f and 5d are fully occupied, meaning they don't participate much in conductivity.
  • Partially filled orbital: The 6s orbital, which impacts mercury's conductive abilities.
The electrons in the outermost shell, especially the 6s electrons, are the key players in conducting electricity.
Energy Band Diagram
The energy band diagram provides a visual understanding of electron behavior in solids.
When atoms form a solid, their atomic orbitals overlap, creating bands of energy levels.
In mercury, due to its electronic configuration, a filled d-band and a partially filled s-band emerge.
  • d-band: Filled with electrons, generally stable and less conductive.
  • s-band: Partially filled, making it more dynamic and conductive.
The structure of these bands is vital since it determines whether electrons can move freely, contributing to mercury's conductive properties.
Fermi Level
The Fermi level is a crucial concept for understanding electrical conductivity.
It indicates the highest energy level occupied by electrons at absolute zero.
In mercury, the Fermi level falls within the partially filled s-band.
This means that electrons near this level can easily gain energy and move to higher levels, facilitating electrical conduction.
  • Location within s-band: Allows easy electron transition, aiding conductivity.
  • Temperature effects: Even at room temperature, electrons can be excited.
The proximity of the Fermi level to available states is a primary reason for mercury's high conductivity.
Electrical Conductivity
Mercury's ability to conduct electricity is a practical outcome of its band structure.
Due to the placement of the Fermi level in the s-band, electrons have numerous available states to transition into.
This means they can readily flow under an electric field, which is the essence of electrical conductivity.
  • High density of states: Ensures many electrons can participate in conduction.
  • Low resistance to electron movement: Facilitates current flow.
The detailed arrangement of electrons and energy bands in mercury allows it to exhibit significant electrical conductivity, making it a unique and efficient conductor among metals.

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Most popular questions from this chapter

Ice is a molecular solid. However, theory predicts that, under high pressure, ice (solid \(\mathrm{H}_{2} \mathrm{O}\) ) becomes an ionic compound composed of \(\mathrm{H}^{+}\) and \(\mathrm{O}^{2-}\) ions. The proposed unit cell for ice under these conditions is a bec unit cell of oxygen ions with hydrogen ions in holes. a. How many \(\mathrm{H}^{+}\) and \(\mathrm{O}^{2-}\) ions are in each unit cell? b. Draw a Lewis structure for "ionic" ice.

Surfaces Plasma nitriding is a process for embedding nitrogen atoms in the surfaces of metals that hardens the surfaces and makes them more corrosion resistant. Do the nitrogen atoms in the nitrided surface of a sample of cubic closest-packed iron fit in the octahedral holes of the crystal lattice? (Assume that the atomic radii of \(\mathrm{N}\) and \(\mathrm{Fe}\) are \(75\) and \(126 \mathrm{pm}\), respectively.)

Minnesotaite \(\left[\mathrm{Fe}_{3} \mathrm{Si}_{4} \mathrm{O}_{10}(\mathrm{OH})_{2}\right]\) is a silicate mineral with a layered structure similar to that of kaolinite. The distance between the layers in minnesotaite is \(1940 \pm 10 \mathrm{pm}\) What is the smallest angle of diffraction of X-rays with \(\lambda=154\) pm from this solid?

Vanadium and carbon form vanadium carbide, an interstitial alloy. Given the atomic radii of \(\mathrm{V}(135 \mathrm{pm})\) and \(\mathrm{C}(77 \mathrm{pm}),\) which holes in a cubic closest-packed array of vanadium atoms do you think the carbon atoms are more likely to occupy-octahedral or tetrahedral?

In the fullerene known as buckminsterfullerene, molecules of \(\mathrm{C}_{60}\) form a cubic closest-packed array of spheres with a unit cell edge length of \(1410 \mathrm{pm}\). a. What is the density of crystalline \(C_{60} ?\) b. If we treat each \(\mathrm{C}_{60}\) molecule as a sphere of 60 carbon atoms, what is the radius of the \(C_{60}\) molecule? c. \(C_{60}\) reacts with alkali metals to form \(\mathrm{M}_{3} \mathrm{C}_{60}\) (where \(\mathrm{M}=\mathrm{Na} \text { or } \mathrm{K}) .\) The crystal structure of \(\mathrm{M}_{3} \mathrm{C}_{60}\) contains cubic closest-packed spheres of \(C_{60}\) with metal ions in holes. If the radius of a \(\mathrm{K}^{+}\) ion is \(138 \mathrm{pm},\) which type of hole is a \(\mathrm{K}^{+}\) ion likely to occupy? What fraction of the holes will be occupied? d. Under certain conditions, a different substance, \(\mathrm{K}_{6} \mathrm{C}_{60}\) can be formed in which the \(\mathrm{C}_{60}\) molecules have a bcc unit cell. Calculate the density of a crystal of \(\mathrm{K}_{6} \mathrm{C}_{60}\)
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