“Minus 13.6 over n squared”
Hydrogen energy levels
Ground state −13.6 eV, then −3.4, −1.51, −0.85. The gaps shrink quickly, which is why the spectral lines crowd together at the high end.
Class 12 Physics · Gujarat Board, CBSE, GUJCET, JEE and NEET
Atoms and Nuclei formulas
Atoms and nuclei supplies the Bohr model results and the decay laws, both of which are pure formula work. The energy level diagram for hydrogen is worth sketching until it is automatic, because half the questions here start from it.
22 formulas, 4 memory hooks, and the 5 mistakes that cost the most marks.
The Bohr model
| What | Formula | Symbols and notes |
|---|---|---|
| Quantisation of angular momentum | mvr = nh/2π | n is the principal quantum number |
| Radius of the nth orbit | rn = 0.529n²/Z Å | for hydrogen, Z = 1 |
| Energy of the nth level | En = −13.6Z²/n² eV | negative, because the electron is bound |
| Energy of a transition | ΔE = 13.6(1/n₁² − 1/n₂²) eV | for hydrogen |
| Rydberg formula | 1/λ = R(1/n₁² − 1/n₂²) | R = 1.097 × 10⁷ m−1 |
| Velocity in the nth orbit | vn = 2.18 × 10⁶ Z/n m s−1 | about c/137 for hydrogen's ground state |
Spectral series of hydrogen
| What | Formula | Symbols and notes |
|---|---|---|
| Lyman | n₁ = 1 | ultraviolet |
| Balmer | n₁ = 2 | visible, the series actually seen |
| Paschen | n₁ = 3 | infrared |
| Brackett and Pfund | n₁ = 4 and 5 | far infrared |
The nucleus
| What | Formula | Symbols and notes |
|---|---|---|
| Nuclear radius | R = R₀A1/3 | R₀ = 1.2 × 10−15 m |
| Nuclear density | about 2.3 × 10¹⁷ kg m−3 | the same for every nucleus |
| Mass defect | Δm = [Zmp + (A − Z)mn] − M | the missing mass became binding energy |
| Binding energy | BE = Δm × 931.5 MeV per u | the conversion worth memorising |
| Binding energy per nucleon | highest near A = 56, iron | which is why both fusion and fission release energy |
Radioactivity
| What | Formula | Symbols and notes |
|---|---|---|
| Decay law | N = N₀e−λt | λ is the decay constant |
| Half-life | T1/2 = 0.693/λ | independent of how much you started with |
| Mean life | τ = 1/λ = T1/2/0.693 | always longer than the half-life |
| After n half-lives | N = N₀/2n | the quickest way to answer most questions |
| Activity | A = λN | unit becquerel; 1 curie = 3.7 × 10¹⁰ Bq |
| Alpha decay | A falls by 4, Z falls by 2 | a helium nucleus leaves |
| Beta minus decay | A unchanged, Z rises by 1 | a neutron becomes a proton |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“Balmer is the one you can see”
Which series is visible
Lyman is ultraviolet, Balmer visible, Paschen infrared. Only Balmer ends on n = 2 and only Balmer reaches the eye.
“Halve it n times”
Decay numericals
After n half-lives, N₀ over 2 to the n. For anything that is a whole number of half-lives, you never need the exponential at all.
“Iron is the bottom of the valley”
Why fusion and fission both release energy
Binding energy per nucleon peaks near iron. Light nuclei release energy by joining, heavy ones by splitting, both moving towards iron.
Where marks go
The mistakes that cost the most.
- Forgetting the minus sign on the energy levels, which makes every transition energy come out wrong.
- Mixing up mean life and half-life. Mean life is the longer of the two, by a factor of about 1.44.
- Thinking half-life depends on the amount of the sample or on the temperature. It depends on neither.
- Using A1/3 as A/3 in the nuclear radius formula.
- Saying the Bohr model works for every atom. It works only for hydrogen and hydrogen-like single-electron ions.
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