Silver Number Of Electrons

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'Fermi level' is the term used to describe the top of the collection of electron energy levels at absolute zero temperature. This concept comes from Fermi-Dirac statistics. Electrons are fermions and by the Pauli exclusion principle cannot exist in identical energy states. So at absolute zero they pack into the lowest available energy states and build up a 'Fermi sea' of electron energy states. The Fermi level is the surface of that sea at absolute zero where no electrons will have enough energy to rise above the surface. The concept of the Fermi energy is a crucially important concept for the understanding of the electrical and thermal properties of solids. Both ordinary electrical and thermal processes involve energies of a small fraction of an electron volt. But the Fermi energies of metals are on the order of electron volts. This implies that the vast majority of the electrons cannot receive energy from those processes because there are no available energy states for them to go to within a fraction of an electron volt of their present energy. Limited to a tiny depth of energy, these interactions are limited to 'ripples on the Fermi sea'.
  • (a) Calculate the number of electrons in a small, electrically neutral silver pin that has a mass of 8.0 g. Silver has 47 electrons per atom, and its molar mass is 107.87 g/mol. (b) Imagine adding electrons to the pin until the negative charge has the very large value 3.00 mc.
  • The allowed energies of free electrons in a metal depend on electron mass and on the electron number density of the metal. The density of states of an electron in a metal increases with energy, because there are more ways for an electron to fill a high-energy state than a low-energy state.
  • Silver, #'Ag'#, is located in period 5, group 11 of the periodic table, and has an atomic number equal to #47#. This tells you that a neutral silver atom will have a total of #47# electrons surrounding its nucleus.

Number of electrons in each shell Each subshell is constrained to hold 4ℓ + 2 electrons at most, namely: Each s subshell holds at most 2 electrons Each p subshell holds at most 6 electrons.

At higher temperatures a certain fraction, characterized by the Fermi function, will exist above the Fermi level. The Fermi level plays an important role in the band theory of solids. In doped semiconductors, p-type and n-type, the Fermi level is shifted by the impurities, illustrated by their band gaps. The Fermi level is referred to as the electron chemical potential in other contexts.

In metals, the Fermi energy gives us information about the velocities of the electrons which participate in ordinary electrical conduction. The amount of energy which can be given to an electron in such conduction processes is on the order of micro-electron volts (see copper wire example), so only those electrons very close to the Fermi energy can participate. The Fermi velocity of these conduction electrons can be calculated from the Fermi energy.

Table

This speed is a part of the microscopic Ohm's Law for electrical conduction. For a metal, the density of conduction electrons can be implied from the Fermi energy.

The Fermi energy also plays an important role in understanding the mystery of why electrons do not contribute significantly to the specific heat of solids at ordinary temperatures, while they are dominant contributors to thermal conductivity and electrical conductivity. Since only a tiny fraction of the electrons in a metal are within the thermal energy kT of the Fermi energy, they are 'frozen out' of the heat capacity by the Pauli principle. At very low temperatures, the electron specific heat becomes significant.

Fermi energies for metals
Table of Fermi energies

Electrolysis

Electrolysis involves passing an electric current through either a molten salt or an ionic solution. The ions are 'forced' to undergo either oxidation (at the anode) or reduction (at the cathode). Most electrolysis problems are really stoichiometry problems with the addition of an amount of electric current. The quantities of substances produced or consumed by the electrolysis process is dependent upon the following:

  • electric current measured in amperes or amps
  • time measured in seconds
  • the number of electrons required to produce or consume 1 mole of the substance
Amps, Time, Coulombs, Faradays, and Moles of Electrons

Three equations relate these quantities:

  • amperes x time = Coulombs
  • 96,485 coulombs = 1 Faraday
  • 1 Faraday = 1 mole of electrons
Valence electrons in ag The thought process for interconverting between amperes and moles of electrons is:

amps & time ' no save> Coulombs ' no save> Faradays ' no save> moles of electrons

Use of these equations are illustrated in the following sections.

Calculating the Quantity of Substance Produced or Consumed

Gold Number Of Electrons

To determine the quantity of substance either produced or consumed during electrolysis given the time a known current flowed::

Number Of Electrons In Silver Ion

  • Write the balanced half-reactions involved.
  • Calculate the number of moles of electrons that were transferred.
  • Calculate the number of moles of substance that was produced/consumed at the electrode.
  • Convert the moles of substance to desired units of measure.
Example:

Silver Element Number Of Electrons

A 40.0 amp current flowed through molten iron(III) chloride for 10.0 hours (36,000 s). Determine the mass of iron and the volume of chlorine gas (measured at 25oC and 1 atm) that is produced during this time.
  • Write the half-reactions that take place at the anode and at the cathode.
anode (oxidation): 2 Cl- ' nosave> Cl2(g) + 2 e-

cathode (reduction) Fe3+ + 3 e- ' nosave> Fe(s)

  • Calculate the number of moles of electrons.
  • Calculate the moles of iron and of chlorine produced using the number of moles of electrons calculated and the stoichiometries from the balanced half-reactions. According to the equations, three moles of electrons produce one mole of iron and 2 moles of electrons produce 1 mole of chlorine gas.

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  • Calculate the mass of iron using the molar mass and calculate the volume of chlorine gas using the ideal gas law (PV = nRT).
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Calculating the Time Required

To determine the quantity of time required to produce a known quantity of a substance given the amount of current that flowed:

  • Find the quantity of substance produced/consumed in moles.
  • Write the balanced half-reaction involved.
  • Calculate the number of moles of electrons required.
  • Convert the moles of electrons into coulombs.
  • Calculate the time required.
Example: How long must a 20.0 amp current flow through a solution of ZnSO4 in order to produce 25.00 g of Zn metal.
  • Convert the mass of Zn produced into moles using the molar mass of Zn.
  • Write the half-reaction for the production of Zn at the cathode.

    • Zn2+(aq) + 2 e- ' nosave> Zn(s)

  • Calculate the moles of e- required to produce the moles of Zn using the stoichiometry of the balanced half-reaction. According to the equation 2 moles of electrons will produce one mole of zinc.
  • Convert the moles of electrons into coulombs of charge using Faraday's constant.
  • Calculate the time using the current and the coulombs of charge.
Silver Number Of Electrons

Calculating the Current Required

To determine the amount of current necessary to produce a known quantity of substance in a given amount of time:

  • Find the quantity of substance produced/or consumed in moles.
  • Write the equation for the half-reaction taking place.
  • Calculate the number of moles of electrons required.
  • Convert the moles of electrons into coulombs of charge.
  • Calculate the current required.
Example: What current is required to produce 400.0 L of hydrogen gas, measured at STP, from the electrolysis of water in 1 hour (3600 s)?
  • Calculate the number of moles of H2. (Remember, at STP, 1 mole of any gas occupies 22.4 L.)
  • Write the equation for the half-reaction that takes place.

    • Hydrogen is produced during the reduction of water at the cathode. The equation for this half-reaction is:

    4 e- + 4 H2O(l) ' nosave> 2 H2(g) + 4 OH-(aq)

  • Calculate the number of moles of electrons. According to the stoichiometry of the equation, 4 mole of e- are required to produce 2 moles of hydrogen gas, or 2 moles of e-'s for every one mole of hydrogen gas.
  • Convert the moles of electrons into coulombs of charge.
  • Calculate the current required.
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