Chapter 3.8, Problem 3.8.2SR

Interpretation Introduction

Interpretation:

The quantum number which desires the energy of an orbital in the hydrogen atom should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

The distribution of electron density in an atom is defined by Quantum numbers.  They are derived from the mathematical solution of Schrodinger’s equation in the hydrogen atom.  The four 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 characterized by a unique set of the quantum numbers.

Principal Quantum Number ($n$)

The size of an orbital and the energy of an electron are specified by the principal quantum number ( $n$).  If the value of $n$ is larger, then the average distance from the nucleus to the specified orbital of an electron will be greater.  Hence, the orbital’s size is large with the increasing energy. The principal quantum numbers get the integral values of $1,2,3$ and so on.  If the same value of ‘ $n$’ is present in the orbitals, then, all the electrons are occupied in the same shell (level).  The total number of the orbitals corresponding to a given $n$ value is found by $n^2$.

Angular Momentum Quantum Number ($l$)

The shape of the atomic orbital is given by the angular momentum quantum number ( $l$) which are integers and its values depend on the integral value of the principal quantum number, $n$.  The probable values of $l$ range are given from $0\text{ to }n-1$ for $n$ value.  There is one possible value of $l$ ($l\text{ = 0}$) for $n=1$.  There are two values of $l$ which are $0\text{ and }1$ for $n\text{ = 2}$.  There are three values of $l$ which are $\text{0, 1 and 2}$ for $n\text{ = 3}$.   The $l$ value gives the type of orbitals namely, $s,p,dandf$.  s orbital comes for $l\text{ = 0}$; p orbital for $l\text{ = 1}$; d orbital for $l\text{ = 2}$; f orbital  for $l\text{ = 3}$.  If the orbitals have the same $n$ and $l$ values, they are present in the same subshell (sublevel).  A smaller amount of energy is contributed by the $l$ values which increase with the subshell levels $\left(s<p<d<f\right)$.

Magnetic Quantum Number ($m_l$)

The orientation of the orbital in space is given the magnetic quantum number ( $m_l$).  The value of $m_l$ depends on the $l$ value in a subshell.  It divides the subshell into the individual orbitals which have the electrons.  There are $\left(2l+1\right)$ integral $m_l$ values for a $l$ value which is explained as follows:

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

There is one possible $m_l$ value which is $0$ for $l\text{ = 0}$.

There are three $m_l$ values which are $-1,0\text{ and }+1$ for $l\text{ = 1}$.

There are five $m_l$ values which are $-2,-1,0,+1\text{ and }+2$ for $l\text{ = 2}$.

There are seven $m_l$ values which are $-3,-2,-1,0,+1,+2\text{ and }+3$ for $l\text{ = 3}$ and so on.

For a particular $l$ value,  the number of $m_l$ values specifies the number of orbitals in a subshell.  Therefore, each $m_l$ value gets a different orbital.

Electron Spin Quantum Number ($m_s$)

For an electron, the orientation of the spin axis is given by it.  An electron can spin in 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, two electrons should not have the same spin quantum number.

To find: The quantum number which desires the energy of an orbital in the hydrogen atom