• Z $-$ number of protons
• $Z_e$ $-$ charge of nucleus

• Electron 1 → Z
• Electron 2 → Z-a
• Electron 3 → Z-b

$Z_{eff} = z -\sigma \rightarrow$ screening constant

As n (principal quantum number) increases $\sigma$ increases - $Z_{eff}$ < Z

• $\sigma \quad 1S < 2s < 3s$
• $Z_{eff} \quad 1s > 2s >3s$

• No two electron in an atom can have the same set of four quantum numbers
• Quantum numbers $=$ $n\hspace{3mm}l\hspace{3mm}m\hspace{3mm}m_s$

For 1s

• n =1

• l =0

• m =0

• $m_s=+\frac {1}{2} , -\frac {1}{2}$

An orbital can have up to 2 electron and of opposite spin

Building up principle (aufbau principle)

• The orbitals are filled in the order of their increasing energy

• $1s\hspace{1mm}2s \hspace{1mm} 2p \hspace{1mm} 3s \hspace{1mm} 3p \hspace{1mm} 4s \hspace{1mm} 3d\hspace{1mm} 4p \hspace{1mm} 5s \hspace{1mm} 4d \hspace{1mm} 5p \hspace{1mm}6s \hspace{1mm} 4f \hspace{1mm} 5d \hspace{1mm} 6p$

  • H -$1s^1$
  • He -$1s^2$
  • Li -$1s^2 2s^1$
  • Na -$1s^2 2s^2 2p^6 3s^1$

$7^N - 1S^2 2s^2 2p^3$

$p^3 , d^5$&$f^7$ shells - half-filled shells

half-filled shells & full-filled shells are most stable

• $24^{Cr} - 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^4$

• $\underbrace{[Ar]}_{core}$ $\underbrace {4s^2 3d^4} _{valence}$ ( less stable )

• $[Ar] 4s^1 3d^5$ ( more stable )

$29^{Cu} - [Ar] 4s^2 3d^9 \rightarrow [Ar]4s^1 3d^{10}$

$64^{Gd} - [54^{Xe}] 10 e^-$

Problem 1 - The hydrogen-like species $Li^{2+}$ is in a spherically symmetric state $S_1$ with one radial node. Upon abosrbing light the ion undergoes transition to a state $S_2$. The state $S_2$ has one radial node and its energy is equal to the ground state energy of the hydrogen atom

i) The state $S_1$ is

A) 1s

B) 2s

C) 2p

D) 3s

Ans B) 2s

ii) Energy of the state $S_1$ in units of the hydrogen atom ground state energies is

A) 0.75 $\quad$

B) 1.50 $\quad$

C) 2.25 $\quad$

D) 4.50 $\quad$

Ans C) 2.25

iii) The orbital angular momentum quantum number of the state $S_2$ is

A) 0 $\quad$

B) 1 $\quad$

c) 2 $\quad$

D) 3 $\quad$

Ans B) 1

• l = 0
• l = 1 (-1) , 0 , (1)
• l = 2 -2 , -1 , 0 , (1) , 2
• l = 3 -3 , -2 , -1 , 0 , (1) , 2 , 3
• The total number of electrons is 6

Problem 6 - The atomic masses of He and Ne are 4 and 20 amu respectively. The value of the de broglie wavelength of He gas at $-73^\circ C$ is "M" times that of the de broglie wavelength of Ne at $727^\circ C$. M is

Ans -

• $T_{Ne}= 727^\circ C$= 1000Kv
• $T_{He} = -73^\circ C = 200Kv$
• $\lambda = \frac {h}{mv}=\frac {h}{p}$
• $\frac {\lambda_{He}}{\lambda_{Ne}} =?=\frac {P_{Ne}}{P_{He}}$
• $E_k = \frac {3}{2}KT$
• $E_k = \frac {p^2}{2m}\rightarrow p=\sqrt {2mE_k}$
• $\frac {P_Ne}{P_He}=\sqrt{\frac{2.m_{Ne}.\frac{3}{2}k.1000}{2.m_{He}.\frac{3}{2}k.200}}$
• =$\sqrt{\frac {20}{4}.5}$ $\quad$ $=5$