Chapter 3.9, Problem 10PPC

Interpretation Introduction

Interpretation:

The maximum numbers of electrons in the alternate universe with the principal quantum number 3 in a given atom should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom.  The types of quantum numbers are the principal quantum number ($n$), the angular momentum quantum number ($l$), the magnetic quantum number ($m_l$) and the electron spin quantum number ($m_s$). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number ($n$)

The principal quantum number ( $n$) assigns the size of the orbital and specifies the energy of an electron.  If the value of $n$ is larger, then the average distance of an electron in the orbital from the nucleus will be greater.  Therefore the size of the orbital is large. The principal quantum numbers have the integral values of $1,2,3$ and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom.  If all orbitals have the same value of ‘ $n$’, they are said to be in the same shell (level).  The total number of orbitals for a given $n$ value is $n^2$.  As the value of ‘ $n$’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number ($l$)

The angular momentum quantum number ( $l$) explains the shape of the atomic orbital.  The values of $l$ are integers which depend on the value of the principal quantum number, $n$.  For a given value of $n$, the possible values of $l$ range are from $0\text{ to }n-1$.  If $n=1$, there is only one possible value of $l$ ($l\text{ = 0}$).  If $n\text{ = 2}$, there are two values of $l:0\text{ and }1$.  If $n\text{ = 3}$, there are three values of $l:\text{0, 1 and 2}$.  The value of $l$ is selected by the letters $s,p,dandf$.  If $l\text{ = 0}$, we have an s orbital; if $l\text{ = 1}$, we have a p orbital; if $l\text{ = 2}$, we have a d orbital and finally if $l\text{ = 3}$, we have a f orbital.  A collection of orbitals with the same value of $n$ is called a shell.  One or more orbitals with the same $n$ and $l$ values are referred to a subshell (sublevel).  The value of $l$ also has a slight effect on the energy of the subshell; the energy of the subshell increases with $l$ $\left(s<p<d<f\right)$.

Magnetic Quantum Number ($m_l$)

The magnetic quantum number ( $m_l$) explains the orientation of the orbital in space.  The value of $m_l$ depends on the value of $l$ in a subshell.  This number divides the subshell into individual orbitals which hold the electrons.  For a certain value of $l$, there are $\left(2l+1\right)$ integral values of $m_l$ which is explained as follows:

${}_{ }{m}_l =-l\mathrm{...0...}+l$

If $l\text{ = 0}$, there is only one possible value of $m_l:0$.

If $l\text{ = 1}$, then there are three values of $m_l:-1,0\text{ and }+1$.

If $l\text{ = 2}$, there are five values of $m_l$, namely, $-2,-1,0,+1\text{ and }+2$.

If $l\text{ = 3}$, there are seven values of $m_l$, namely, $-3,-2,-1,0,+1,+2\text{ and }+3$ and so on.

The number of $m_l$ values indicates the number of orbitals in a subshell with a particular $l$ value.  Therefore, each $m_l$ value refers to a different orbital.

Electron Spin Quantum Number ($m_s$)

It specifies the orientation of the spin axis of an electron.  An electron can spin in only one of two directions.  There are two possible ways to represent $m_s$ values.  They are +½ and ‒½.  One electron spins in the clockwise direction.  Another electron spins in the anticlockwise direction.  But, no two electrons should have the same spin quantum number.

Pauli exclusion principle

No two electrons in an atom should have the four same quantum numbers.  Two electrons are occupied in an atomic orbital because there are two possible values of $m_s$.  As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins.  If two electrons have the same values of $n,l{\text{ and }}_{ }{m}_l$ values, they should have different values of $m_s$.

To find: Count the maximum number of electrons in the alternate universe with the principal quantum number 3 in a given atom