# How come the number of wandering electrons is same as the number of the positive ions?

My book mentions the following:

Cause of resistance : When an ion of a metal is formed , its atoms lose electrons from its outer orbit . A metal ( or conductor ) has a large number of wandering electrons and an equal number of fixed positive ions . The positive ions do not move , while the electrons move almost freely inside the metal These electrons are called free electrons . They move at random , colliding amongst themselves and with the positive ions in any direction as shown

The book mentions that :A metal has a large number of wandering electrons and an equal number of fixed positive ions. My doubt arises that lets says the metal is aluminium since aluminium has 3 valence electrons a single atom will loose 3 electrons which becomes the free electrons in the metal, so since a atom looses 3 electrons to form a cation so in this case should not the number of wandering electrons be three times the number of positive ions. So how come the number of wandering electrons is same as the number of the positive ions

• I guess on an average per unit volume number of electrons = number of protons. Dec 26, 2021 at 3:24
• @Ha'Penny Sounds like an answer. A block of material is neutrally charged. Dec 26, 2021 at 4:03
• Also note that the "fee electrons" in the paragraph are simplifying a quantum mechanical model. In the band theory of solids for metals the moving electrons are bound to the lattice, and belong to the conduction band hyperphysics.phy-astr.gsu.edu/hbase/Solids/band.html Dec 26, 2021 at 5:39
• I am not sure Al loses all its 3 valence electrons into the electron gas. A single electron per atom is imho a better estimate. Dec 26, 2021 at 16:23

The paragraph should be read as "The total charge of the wandering electrons equals the total charge of positive ions". The actual number of positive atoms may be less, but that is offset by the any that are doubly or triply ionized.

The important idea is that the conductor is usually electrically neutral, with charge separation possible, but not a bulk excess or deficit of charge.

I think that paragraph is badly worded and, at face value, wrong.

Most probably the author meant something like:

A metal ( or conductor ) has a large number of wandering electrons and a number of fixed positive ions that amount to the same charge, but of opposite sign.

I guess in the process of making the sentence more compact and fluid the error crept-in.

• To the downvoter: care to explain why? Dec 27, 2021 at 11:18

Strictly speaking, it should be charge of ions, rather than number of ions. However, the ionization energy of an atom increases the greater the charge of the atom. For instance, the first ionization energy of Al is $$577.5 \text{kJ mol}^{-1}$$, while the second ionization energy is $$1816.7$$, more than three times as much. So once one atom loses (note spelling) an electron, it's easier to take the next electron from an atom that hasn't lost any electrons. In the case of, say, a typical capacitor, the number of net electrons missing from one plate will be large in an absolute sense, but as a proportion of total atoms, it will be miniscule, so we can model the system in terms of ions with single charge each.