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Structure of Atoms and Nuclei formulas

8 formulas and 15 common traps for MHT-CET Physics Structure of Atoms and Nuclei, grouped by subtopic.

Full notes with worked examples

Bohr's Model: How Each Orbit Scales With n

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Radius, Speed and Everything Built From Them

Bohr orbit

rn∝n2Z,vn∝Zn,mvr=nh2πr_n \propto \frac{n^2}{Z}, \qquad v_n \propto \frac{Z}{n}, \qquad mvr = \frac{nh}{2\pi}

Energy Levels and the Photon Between Them

Energy of the nth level

En=−13.6 Z2n2 eV,hν=Ei−EfE_n = -\frac{13.6\,Z^2}{n^2}\ \text{eV}, \qquad h\nu = E_i - E_f

The Quantum Condition Outside the Atom

Rotor and charge in a field

Erot=n2h28π2I,EB=n qBh4πmE_{\text{rot}} = \frac{n^2 h^2}{8\pi^2 I}, \qquad E_{B} = \frac{n\,qBh}{4\pi m}

Common traps

Numbering the excited states from 1

The ground state is n = 1, so the FIRST excited state is n = 2 and the third excited state is n = 4. Counting from 1 puts every ratio one orbit out.

Stopping at the current for the field at the nucleus

The current goes as n⁻³, but the field is μ₀I/2r, and r grows as n². The field therefore goes as n⁻⁵, not n⁻³.

Saying the potential energy falls as the electron moves out

The potential energy is negative and becomes LESS negative outward: −6.8 eV at n = 2 is greater than −27.2 eV at n = 1. It increases, while the kinetic energy decreases.

Forgetting Z² when changing the atom

He⁺ and Li²⁺ are hydrogen-like but their levels are Z² deeper. Compare two atoms with E ∝ Z²/n², both factors at once.

Writing L = nh

Bohr's condition is L = nh/2π. Dropping the 2π puts a factor 4π² into every energy, and the options include that wrong answer.

The Hydrogen Spectrum and its Series

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The Series, Their First Lines and Their Limits

Series limit

1λ∞=Rnf2,ν∞=Rcnf2\frac{1}{\lambda_\infty} = \frac{R}{n_f^2}, \qquad \nu_\infty = \frac{Rc}{n_f^2}

Ratios of Two Lines

Rydberg formula

1λ=RZ2(1nf2−1ni2)\frac{1}{\lambda} = RZ^2\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)

Common traps

Calling the series limit the longest wavelength

The limit comes from n = ∞, the biggest jump into that level, so it has the most energy and the SHORTEST wavelength. The first line is the longest.

Using n_f = 1 for every series

Only Lyman ends on n = 1. The Balmer limit is R/4, not R; Paschen's is R/9. Set the lower level from the series name before substituting.

Dividing wavelengths the way the brackets divide

The bracket is 1/λ. A larger bracket means a SHORTER wavelength, so the wavelength ratio is the brackets' ratio turned upside down.

Counting an upward jump as emission

n = 1 to n = 2 absorbs a photon. 'Which transition emits the highest frequency' excludes every jump upward, however large.

Nuclei: Radioactive Decay and Binding Energy

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The Decay Law and Half-Life

Decay law

N=N0e−λt,T1/2=ln⁡2λ,A=λNN = N_0 e^{-\lambda t}, \qquad T_{1/2} = \frac{\ln 2}{\lambda}, \qquad A = \lambda N

Counting Alpha and Beta Decays

Counting decays

nα=Ai−Af4,nβ=2nα−(Zi−Zf)n_\alpha = \frac{A_i - A_f}{4}, \qquad n_\beta = 2n_\alpha - (Z_i - Z_f)

Binding Energy and a Nucleus That Splits

Binding energy

BE=[Zmp+(A−Z)mn−M]c2BE = \left[Zm_p + (A - Z)m_n - M\right]c^2

Common traps

Answering 'left' when the question asks 'decayed'

(1/2)ⁿ is what REMAINS. The decayed fraction is 1 − (1/2)ⁿ, and the options carry both.

Multiplying N by T for activity

A = λN = N ln 2/T: a LONGER half-life means a LOWER activity. The ratio of activities is N₁T₂ : N₂T₁, not N₁T₁ : N₂T₂.

Pairing beta-plus with an antineutrino

Beta-minus (Z rises) comes with an ANTIneutrino; beta-plus (Z falls, a positron) comes with a neutrino.

Counting betas before alphas

Fix the alphas from the mass number first; only then does the change in Z tell you the betas. Starting from Z gives a wrong count.

Taking the radius ratio as the mass ratio

Mass goes as A and radius as A^(1/3), so the mass ratio is the radius ratio CUBED. Radii 1 : 2 give masses 1 : 8.

Writing the binding energy with the nucleus first

Binding energy is positive: nucleons minus nucleus. One paper prints (M_O − 8M_P − 9M_N)c², which is negative; the magnitude is what is meant.

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