Chapter 36: Problem 58

Show that the maximum number of electrons in an atom's \(n\) th shell is \(2 n^{2}.\)

Short Answer

Expert verified

This shows that the maximum number of electrons in any \(n\)th shell of an atom is \(2n^{2}\), proving the original statement.

Step by step solution

Understand the Quantum Numbers

There are four quantum numbers: the principal quantum number (\(n\)), the azimuthal quantum number (\(l\)), the magnetic quantum number (\(m\)), and the spin quantum number (\(s\)). The principal quantum number (\(n\)) determines the energy level and size of the cell, where \(n\) can take any positive integer value. Each shell is assigned a principal quantum number.

Understand the Azimuthal Quantum Number

The azimuthal quantum number (\(l\)) determines the shape of the electron's orbit. \(l\) can take any value from 0 to \(n-1\). For each shell (\(n\)), there are \(n\) subshells.

Understand the Magnetic Quantum Number

The magnetic quantum number (\(m\)) determines the orientation of the electron's orbit in space. \(m\) can take any value from \(-l\) to \(+l\). For each subshell (\(l\)), there are \(2l + 1\) orbitals.

Understand the Spin Quantum Number

The spin quantum number (\(s\)) describes the spin of the electron which can be either +1/2 or -1/2. For each orbital, there can be 2 electrons with opposite spins.

Proof

A certain shell (\(n\)) has \(n\) subshells, so sum of the number of orbitals for each subshell will give total number of orbitals in the shell which is given by sum of \(2l+1\) for each \(l\) from 0 to \(n-1\). This is equal to \(n^{2}\). Since each orbital holds 2 electrons, the maximum number of electrons in atom's \(n\) th shell would be \(2n^2\).

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Show that the maximum number of electrons in an atom's \(n\) th shell is \(2 n^{2}.\)

Short Answer

Step by step solution

Understand the Quantum Numbers

Understand the Azimuthal Quantum Number

Understand the Magnetic Quantum Number

Understand the Spin Quantum Number

Proof

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