# Atomic Structure Formulas

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## Important Formulas of Atomic Structure

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1. Works Related with Scientists: 2. $$\frac{\text { massof } \mathrm{e}^{-}}{\text {mass of } \mathrm{H} \text { atom }}=\frac{1}{1837}$$
(Speed) Anode rays < < cathode rays
& (deflection) Anode rays < cathode rays
(e/m)anode rays < < (e/m)cathode rays ( due to mp > me)
For neutral atoms
Z = P = e & A = P + n = Z + n = e + n 3. Some formulas Related with Bohr Model:
$$\frac{m v^{2}}{r}=\frac{Z e^{2}}{r^{2}}$$
& mvr = $$\frac{\mathrm{nh}}{2 \pi}$$ & c = υλ
& $$\bar{υ}$$ = $$\frac{1}{\lambda}$$ & E = hυ & E = mc2
& υ = c$$\bar{υ}$$
& v = $$\frac{2 \pi \mathrm{Ze}^{2}}{\mathrm{nh}}$$ = 2.188 × 108Z/n cm/sec
& r = $$\frac{n^{2} h^{2}}{4 \pi^{2} M Z e^{2}}$$ = 0.529$$\left(\frac{n^{2}}{Z}\right)$$Å
Er = $$\frac{-2 \pi^{2} m Z^{2} e^{4}}{n^{2} h^{2}}$$ = -KE = $$\frac{\mathrm{PE}}{2}$$
= $$\frac{-Z e^{2}}{2 r}$$ = K.E. + P.E. In SI unit
Er = $$\frac{-2 \pi^{2} m K^{2} Z^{2} e^{4}}{n^{2} h^{2}}$$

 I from E – 21.8 × 10-12 Z2/n2 erg II from E – 13.6 Z2/n2 erg III from E – 313.6 Z2/n2 Kcal Iv from E – 109700 Z2/n2 cm-1

$$\frac{E_{A}}{E_{B}}=\frac{Z_{A}^{2}}{Z_{B}^{2}} \frac{n_{B}^{2}}{n_{A}^{2}}$$
E1 < E2 < E3 < ……. < E (0), (EP)1 < (EP)2 < (EP)3 < …….
(E2 – E1) > (E3 – E2) > (E4 – E3) > ………, (SE)1 > (SE)2 > (SE)3 ……..
E1 = -13.6 eV
& E2 = – 3.4 eV
& E3 = -1.5 eV
& E4 = -0.85 eV
& E5 = -0.54 eV RH = $$\frac{2 \pi^{2} m e^{4}}{c h^{3}}$$
Total no. of spectrum line = $$\frac{n(n-1)}{2}$$

4. Various Series of spectrum lines of H-Spectrum: 5. Order of wave length & Frequency of electromagnetic radiation –

 Visible zone Cosmic Rays γ Rays x Rays UV Zone VR IR Zone Micro Wave Radio Wave λ↑ υ↓ E↓

6. De Broglie Wave Equation:
λ = h/mv = h/p
2πr = nλ & $$\frac{1}{2}$$ mv2 = ev
& λ = $$\frac{h}{\sqrt{2 e m V}}$$

7. Heisenberg Uncertainty Principle:
(Δx) (Δp) ≥ h/4π
& (Δx) (Δv) ≥ h/4πm
ΔE. Δt > ≥ $$\frac{h}{4 \pi}$$ (For energy & time)
& ΔΦ. Δθ ≥ $$\frac{h}{4 \pi}$$ (For angular motions)
Photo electric effect:
hv = w + $$\frac{1}{2}$$ mv2
w = hv0 work function

8. In sommer field concept:
PΦ = nΦ$$\frac{h}{2 \pi}$$
& Pr = nr$$\frac{h}{2 \pi}$$
& P = Pr + PΦ
or n = nΦ + nr
$$\frac{\text { Semi major axis }}{\text { Semi minor axis }}=\frac{a}{b}=\frac{n}{\ell}$$

9. Shrodinger wave Equation:
Δ2ψ + $$\frac{8 \pi^{2} m}{h^{2}}$$(E – V)ψ = 0

10. In Moseleys’ experiment
$$\sqrt{v}$$ = a (Z – b) & a = $$\sqrt{\frac{3 \mathrm{R}_{\mathrm{C}}}{4}}$$

11. (a) No. of electrons in any subshell = 2(2l + 1)
(b) No. of orbitals in any subshell = (2l + 1)
(c) Orbital angular momentum of e = $$\sqrt{[}$$l(l + 1)]h/2π
(d) Spin angular momentum of e = $$\sqrt{[}$$s(s + 1)]h/2π
(e) No. of Max. electrons in any shell = 2n2
(f) Max. no. of orbitals in any shell = n2 (valid upto n = 4)
(g) Max. no. of subshell in any shell = n

12. Nodes [(n – 1) = total nodes, t = angular nodes,
(n – L – 1) = Radial nodes, nodel plane = l]
Nodals surfaces [for s-orbitals = n – 1]

13. Unstable Particle of atom: 14. 15. M = $$\frac{M_{1} x+M_{2} y+M_{3} z}{x+y+z}$$ & (energy) s < p < d < f & Mμ = 200 Me order of filling e in various orbitals
1s. 2s, 2p, 3s, 3p. 4s, 3d, 4p, 5s ……

16. 3rd excited slate = E4, 4th excited state = E5,
3rd excitation potential = E4 – E1, 4th excitation potential = E5 – E1,
3rd ionisation potential = E – E3, 4th ionisation potential = E – E4,
3rd separation energy = E – E4, 4th separation energy = E – E5,

17. (i) Kernel of an atom:
Part remained after removing the outer most orbit is known as kernel,
eg. electrons in Kernel of O → 2e, P → 10e, Li → 2e

(ii) Core charge:
No. of electrons in outermost orbit
eg. core charge of O → 6, Na 1, N → 5

18. Paramagnetic Moment = $$\sqrt{n(n+2)}$$ B.M,
1 B.M. = $$\frac{\mathrm{eh}}{4 \pi \mathrm{mc}}$$, n = total no. of unpaired electrons.

19. Importance of (n – 1):

• Excited state of electrons is always equals to (n – 1)
• According to sommerfield no. of elliptical orbit is equal to (n – 1)
• Nodal Surface = (n – 1)
• Value of l = 0 to (n – 1)
• Penultimate shell = (n – 1)

20. Exceptions of Aufbau Principle: 21. (a) Atomic weight, x specific heat = 6.4\

(b) Radius of nucleus 10-13 cm = 10-15 m
or rn = 1.33 × 10-13 (A)1/3 A → Mass No.

(c) Radius of atom = 10-8 cm = 10-10 m

22. (i) Aufbau Principle:
According to this principle, “In the ground state, the atomic orbitals are filled in order of increasing energies”, i.e. in the ground state the electrons occupy the lowest orbitals available to them.

(ii) Pauli’s Exclusion Principle:
According to this principle, “No two electrons in an atom can have all the four quantum numbers n, l, m and s identical”.

(iii) Hund’s Rule of Maximum Multiplicity:
According to this rule “Electron filling will not take place in orbitals of same energy untill all the available orbital of a given sub shell contain one contain one electron each with parallel spin”.

(iv) Isosters:
Substance which have same number of electron and atoms called Isosters.
eg.
CO2 N2O
22 22

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