Formula sheet: Structure of Atom

Structure of Atom12 min readLesson 1 of 8

1.1 What changes from one model to the next Each model in this chapter answers a question the previous one could not. Thomson explained neutrality but not the nucleus. Rutherford found the nucleus but not why the atom survives. Bohr saved the hydrogen spectrum with fixed orbits. Quantum mechanics replaced the orbit with an orbital, a probability picture. Every formula below belongs to one of those steps, so learn the question with the equation. 1.2 Light as a wave and as a particle The wave relation links the three properties of any electromagnetic radiation. The particle relation gives the energy carried by one quantum of that radiation.

c = ν λ ν ~ = \frac{1}{λ}

Wave relation, with frequency in Hz and wavelength in m

E = h ν = \frac{h c}{λ}

Energy of one photon, h = 6.626 \times 10^{-34} J s

h ν = h ν_{0} + K_{max} K_{max} = e V_{0}

Photoelectric equation, with work function W_0 = h\nu_0 and stopping potential V_0

E (eV) = \frac{1240}{λ ( nm )}

Useful light constants: hc = 1.986 \times 10^{-25} J m = 1240 eV nm

1.3 The hydrogen spectrum The Rydberg relation gives the wavelength in any hydrogen-like spectrum, with Z the nuclear charge. The Bohr energy formula is the same statement written in terms of the levels.

\frac{1}{λ} = R Z^{2} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}}), R = 1.097 \times 1 0^{7} m^{- 1}

Rydberg equation for a hydrogen-like atom, n_2 the upper level

r_{n} = 52.9 \frac{n ^{2}}{Z} pm E_{n} = - 13.6 \frac{Z ^{2}}{n ^{2}} eV

Bohr orbit radius and energy for hydrogen, with Z = 1

Δ E = 2.18 \times 1 0^{- 18} Z^{2} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}}) J = h ν

Energy change between two Bohr levels, in joules

m v r = \frac{nh}{2 π} v_{n} = \frac{2.18 \times 1 0 ^{6}}{n} m s^{- 1}

Bohr quantisation condition and orbital velocity

1.4 Matter waves and uncertainty

λ = \frac{h}{m v} = \frac{h}{2 m K}

de Broglie wavelength of a particle of mass m and speed v

Δ x \cdot Δ p \geq \frac{h}{4 π} Δ x \cdot Δ v \geq \frac{h}{4 π m}

Heisenberg uncertainty principle, position and momentum

1.5 Quantum numbers and nodes

n = 1, 2, 3, \dots l = 0, 1, \dots, (n - 1) m_{l} = - l, \dots, 0, \dots, + l m_{s} = \pm \frac{1}{2}

Possible values of the four quantum numbers for a given n

radial nodes = n - l - 1 angular nodes = l total nodes = n - 1 N_{orbitals} = n^{2}

Node counts for an orbital, and the number of orbitals in a shell

lower (n + l) fills first; for equal (n + l) the lower n fills first

Orbital energy order in multi-electron atoms, by the n + l rule

1.6 Constants and units Constant or quantity Value Speed of light in vacuum 3.0 x 10^8 m s^-1 Planck constant 6.626 x 10^-34 J s Electron charge 1.602 x 10^-19 C Electron mass 9.109 x 10^-31 kg Proton mass 1.672 x 10^-27 kg Neutron mass 1.675 x 10^-27 kg Avogadro constant 6.022 x 10^23 mol^-1 1 eV in joules 1.602 x 10^-19 J 1 unified mass unit 1.66 x 10^-27 kg Bohr radius (hydrogen, n = 1) 52.9 pm Ground state energy of hydrogen -13.6 eV

Constant or quantity	Value
Speed of light in vacuum	3.0 x 10^8 m s^-1
Planck constant	6.626 x 10^-34 J s
Electron charge	1.602 x 10^-19 C
Electron mass	9.109 x 10^-31 kg
Proton mass	1.672 x 10^-27 kg
Neutron mass	1.675 x 10^-27 kg
Avogadro constant	6.022 x 10^23 mol^-1
1 eV in joules	1.602 x 10^-19 J
1 unified mass unit	1.66 x 10^-27 kg
Bohr radius (hydrogen, n = 1)	52.9 pm
Ground state energy of hydrogen	-13.6 eV

Exam

Convert the wavelength to metres and the energy to joules before substitution, and check the order of magnitude. A photon of visible light carries about 2 to 3 eV, which is why answers near 10^-19 J are plausible for single photons and answers in kJ mol^-1 belong to bulk samples.

Write c = nu times lambda before using either c or lambda.
Use E = hc/lambda for a photon and E = h nu only when frequency is given.
Keep n2 greater than n1 in the Rydberg formula and subtract the smaller term from the larger.
Use the n + l rule to order orbitals in multi-electron atoms.
Count radial nodes as n minus l minus 1 and angular nodes as l.
Remember that m_s can take only two values for an electron.

in hinglish

Structure of Atom mein pehla equation c = νλ hai, jo light ki wave nature deta hai; photon ki energy ke liye E = hν = hc/λ use karo. Photoelectric effect mein hν = hν0 + K_max, aur K_max = eV0. Hydrogen spectrum ke liye Rydberg equation 1/λ = RZ²(1/n1² - 1/n2²), jahan R = 1.097 \times 10^7 m^-1. Bohr model ki do formulas yaad rakho: r_n = 52.9 n²/Z pm aur E_n = -13.6 Z²/n² eV; energy change ΔE = 2.18 \times 10^-18 Z²(1/n1² - 1/n2²) J. Matter waves ke liye de Broglie λ = h/mv, aur Heisenberg ka Δx\cdotΔp \geq h/4π. Quantum numbers n, l, m_l, m_s likhna aata ho; radial nodes n-l-1, angular nodes l, total nodes n-1. Multi-electron atoms mein n+l rule se orbitals ka order banao. Numerical mein unit convert karo, significant figures rakho, aur answer ka order of magnitude check karo. Wavelength ke liye metre, energy ke liye joule, aur photon energy eV mein ho to 1 eV = 1.602 \times 10^-19 J ka conversion mat bhoolo.