Question

a) Calculate the energy associated with the first orbit of $H{{e}^{+}}$. What is the radius of this orbit?
(b) What is the lowest value of n that allows ‘g’ orbitals to exist?

Answer 1

Hint: The $H{{e}^{+}}$ is hydrogen like atom. Energy of the hydrogen like atom is dependent on the atomic number and number of electrons in the ion present. Aufbau Principle originates from Pauli's exclusion principle.

Complete step by step answer: (a) For hydrogen like atom, the energy level for given quantum number n is :
\[{{E}_{n}}=\dfrac{{{Z}^{2}}}{{{n}^{2}}}(-13.6eV)\]
A hydrogen like atom/ion is any atomic nucleus bound to one electron and thus is isoelectronic with hydrogen.
For $H{{e}^{+}}$ atom, Z=2 and and since first orbit is asked, therefore, n=1,
\[\begin{align}
  & E=-\dfrac{{{(2)}^{2}}}{{{1}^{2}}}13.6 \\
 & E=-13.6\times 4 \\
 & E=54.4eV \\
\end{align}\]
Therefore, energy for $H{{e}^{+}}$ atoms is 54.4eV.
(b) This question can be explained through Aufbau Principle. The Aufbau principle dictates the manner in which electrons are filled in the atomic orbitals of an atom in its ground state. It states that electrons are filled into atomic orbitals in the increasing order of orbital energy level.According to the Aufbau principle, the available atomic orbitals with the lowest energy levels are occupied before those with higher energy levels. Azimuthal numbers are assigned to orbitals, such as, for s, l=0, similarly for g, l=4, then, the relation of principal quantum number (n) and azimuthal number (l) is l=n-1 , as value of l is from, 0 to (n-1), therefore, l for g is 4, then value of n will be,
\[\begin{align}
  & l=n-1 \\
 & 4=n-1 \\
 & n=5 \\
\end{align}\]

Therefore, the minimum value of n for g to exist is 5.

Note: Aufbau comes from the german word, “Aufbauen” which means to build.Pauli exclusion principle states that no two fermions in an atom can have the same set of quantum numbers, hence they have to pile up or build up into higher energy levels.