Atoms — JEE Mains Physics

Formulas (2)

• Rutherford scattering and the distance of closest approach · Closest approach and impact parameter
  $$r_0 = \frac{1}{4\pi\epsilon_0}\,\frac{2Ze^2}{K}, \qquad b = \frac{r_0}{2}\cot\frac{\theta}{2}$$
• Bohr's postulates and quantised angular momentum · Bohr's quantisation and frequency condition
  $$L = mvr = \frac{nh}{2\pi}, \qquad h\nu = E_{\text{upper}} - E_{\text{lower}}$$

Watch out for (8)

• The alpha carries charge 2e
  → Rutherford scattering and the distance of closest approach
• Convert MeV to joules before using k = 9 × 10⁹
  → Rutherford scattering and the distance of closest approach
• Closest approach is an upper limit on the radius
  → Rutherford scattering and the distance of closest approach
• Rare rebounds are not caused by faster alphas
  → Rutherford scattering and the distance of closest approach
• A multiple of h/2π, not of h
  → Bohr's postulates and quantised angular momentum
• Read n off L before anything else
  → Bohr's postulates and quantised angular momentum
• Higher orbit minus lower orbit
  → Bohr's postulates and quantised angular momentum
• Bohr's model is for one electron only
  → Bohr's postulates and quantised angular momentum

Formulas (2)

• Bohr radius and orbital speed in hydrogen-like ions · Bohr radius and speed
  $$r_n = 0.529\,\frac{n^2}{Z}\ \text{Å}, \qquad v_n = 2.19 \times 10^{6}\,\frac{Z}{n}\ \text{m/s}$$
• Energy levels and the kinetic–potential split · Energy of level n
  $$E_n = -13.6\,\frac{Z^2}{n^2}\ \text{eV}, \qquad K = -E, \quad U = 2E$$

Watch out for (6)

• Radius goes as n², speed as 1/n
  → Bohr radius and orbital speed in hydrogen-like ions
• A larger Z makes the orbit smaller
  → Bohr radius and orbital speed in hydrogen-like ions
• Build a derived quantity from r and v
  → Bohr radius and orbital speed in hydrogen-like ions
• Z is squared in the energy
  → Energy levels and the kinetic–potential split
• Excited-state numbers are one behind n
  → Energy levels and the kinetic–potential split
• Kinetic energy falls as the electron moves out
  → Energy levels and the kinetic–potential split

Formulas (1)

• Ratios of two spectral lines · Rydberg formula and combining lines
  $$\frac{1}{\lambda} = RZ^2\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right), \qquad \frac{1}{\lambda_{AC}} = \frac{1}{\lambda_{AB}} + \frac{1}{\lambda_{BC}}$$

Reference tables (1)

Watch out for (6)

• The series limit is the shortest wavelength
  → Hydrogen spectral series, first lines and series limits
• Each series has its own lower level
  → Hydrogen spectral series, first lines and series limits
• Count members up from the longest line
  → Hydrogen spectral series, first lines and series limits
• Turn the bracket ratio upside down for wavelength
  → Ratios of two spectral lines
• Energies add; wavelengths do not
  → Ratios of two spectral lines
• Z cancels only within one ion
  → Ratios of two spectral lines

Formulas (3)

• Photon energy and wavelength of a transition · Energy and wavelength of a transition
  $$\Delta E = 13.6\,Z^2\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)\ \text{eV}, \qquad \lambda\,(\text{nm}) = \frac{1240}{\Delta E\,(\text{eV})}$$
• Excitation, absorption and the number of spectral lines · Lines from level n and the gap from the ground state
  $$N = \frac{n(n-1)}{2}, \qquad E_n - E_1 = 13.6\,Z^2\left(1 - \frac{1}{n^2}\right)\ \text{eV}$$
• Recoil of the emitting atom and X-ray photons · Photon momentum, recoil and the X-ray cut-off
  $$p = \frac{E}{c} = \frac{h}{\lambda}, \qquad v_{\text{recoil}} = \frac{E}{Mc}, \qquad \lambda_{\min} = \frac{hc}{eV}$$

Watch out for (9)

• Use the constants the question gives
  → Photon energy and wavelength of a transition
• The gap scales as Z²
  → Photon energy and wavelength of a transition
• Big frequency means a jump near the ground state
  → Photon energy and wavelength of a transition
• A photon must match; an electron need not
  → Excitation, absorption and the number of spectral lines
• A sample and a single atom count differently
  → Excitation, absorption and the number of spectral lines
• Balmer lines need n = 3 first
  → Excitation, absorption and the number of spectral lines
• Recoil uses the atom's mass
  → Recoil of the emitting atom and X-ray photons
• The cut-off wavelength ignores the target
  → Recoil of the emitting atom and X-ray photons
• Kα energy is a difference of two hole energies
  → Recoil of the emitting atom and X-ray photons