Question

The electrons, identified by quantum numbers $n$ and $l$ (i) $n = 4,l = 1$ (ii) $n = 4,l = 0$ (iii) $n = 3,l = 2$ (iv) $n = 3,l = 1$ can be placed in order of increasing order of increasing energy, from the lowest to highest, as.

Answer 1

We are given the Principal quantum numbers and Azimuthal quantum numbers of the electron respectively. To calculate the energy of the electrons placed in these orbitals we need to apply the $(n + l)$ rule and then arrange them in the increasing order of their energy.

Complete step by step solution:
The $(n + l)$ rule governs the order in which electrons are filled in orbitals in the successive atoms of the periodic table. Electrons are filled in increasing order of the energy of orbitals, from lowest energy orbital to highest energy orbital. The energy of electrons increases as their $(n + l)$ values increase.
Now, (i) Given, Principal quantum number, $n = 4$ and Azimuthal quantum number, $l = 1 = 4p$, so the value will be $(n + l) = 4 + 1 = 5$
(ii) Principal quantum number, $n = 4$ and Azimuthal quantum number, $l = 0 = 4s$ , so the value will be $(n + l) = 4 + 0 = 4$
(iii) Principal quantum number, $n = 3$ and Azimuthal quantum number, $l = 2 = 3d$, so the value will be $(n + l) = 3 + 2 = 5$
(iv) Principal quantum number, $n = 3$ and Azimuthal quantum number, $l = 1 = 3p$ , so the value will be $(n + l) = 3 + 1 = 4$
Now, a lower value of $(n + l)$ means lesser energy, and for those subshells whose $(n + l)$ value is the same then according to the rule, the orbital which has a lower value of $n$ will have lower energy.
Therefore, the increasing order of energies will be $(iv) > \left( {ii} \right) > (iii) > (i)$.

Note: According to the Aufbau principle, In the ground state of atom electrons are filled in the increasing order of their energies. $1s < 2s < 2p < 3s < 3p$ which is in the order of increasing $(n + l)$. Also, energies of the orbitals in the same subshell tend to decrease with an increase in the atomic number. The value of the Principal quantum number will always be an integer from $n = 1,2,3....$ and for a given value of $n$ the value of Azimuthal quantum number $(l)$ can vary from zero to $n - 1$.