Atoms, Molecules and Nuclei

(1) Bohr’s Postulate:
(i) ${mv^2}/{r} = 1/{4πε_o} {e^2}/{r^2}$
(ii) $mvr = {nh}/{2π}$
(iii) $E = E_h – E_p$

(2) $r = ({h^2 ε_o}/{πm e^2}) n^2$         i.e. $r α n^2$

(3) $v = {e^2}/{2ε_o h n}$       i.e. $v α 1/n$ … speed of electron

(4) $E_n = ({m e^4}/{8 ε_o h^2}) 1/n^2$       i.e. $E α 1/n^2$

here, $PE = – {e^2}/{4πε_o h^2}$ = $- {m e^4}/{4πε_o_^2 h^2 n^2}$
$KE = – {e^2}/{8πε_o h^2}$ = $- {m e^4}/{8πε_o_^2 h^2 n^2}$

(5) $1/λ = R [1/p^2 – 1/n^2]$ R = 1.097 x $10^7$ $m^{-1}$

(6) Decay Laws:
(i) ${dN}/{dt} = – λN$
(ii) $N = N_o e^{-λt}$

(7) Half Life Period: $T = {0.693}/{λ}$

(8) De Broglie Hypothesis: (i) $E = mc^2$ & $E = {hc}/{λ}$
p = mv
(ii) $p = √{2 m e v}$
(iii) $λ = h/p$ = $h/{√{2 m e v}}$
(iv) $λ = {12.27}/{√v}$ A° v -> acc. voltage

(9) Ionisation Energy = $E_∞ – E_n$

(10) Series Limit: $1/λ = R (1/n^2 – 1/∞^2)$