Study Notes on Energy Levels in Atoms and Photon Transitions

Expression and Formulas Related to Energy Levels in Atoms

Expression of Energy

• The energy expression for an electron in an atom is given as: $E_n = -k rac{z^2}{n^2}$ where:

  • $n$ = principal quantum number,

  • $z$ = atomic number,

  • $k$ = a constant related to the electron's charge.

Allowed Energy Values

• For hydrogen (where $z = 1$), the energy at different quantum levels can be calculated using:

  • For the ground state (n=1):
    $E_1 = -13.6 ext{ eV}$

  • For the second energy level (n=2):
    $E_2 = - rac{13.6}{2^2} = -3.4 ext{ eV}$

  • For the third energy level (n=3):
    $E_3 = - rac{13.6}{3^2} = -1.51 ext{ eV}$

Calculation of Velocity of Electron

• The velocity of the electron in the nth orbit can also be expressed as:
  $v_n = rac{z e^2}{h} rac{1}{n}$

• For hydrogen: $v_n = rac{e^2}{h}$ is approximately equal to:

  • For 1st orbit (n=1):
    $v_1 ext{ in 1st orbit} <br>ightarrow 2.18 imes 10^6 ext{ m/s}$

Time Period and Frequency of Revolution

• The time period for one complete revolution (Tn) of the electron can be calculated as:
  $T_n = rac{2 imes ext{π}}{v_n}$

• From the expression of energy:
  $T_n = rac{2 imes ext{π} imes 0.529}{v_n}$

• The frequency of revolution (fr) can be expressed as:
  $f_n = rac{1}{T_n}$

Kinetic Energy, Potential Energy, and Total Energy

• The total mechanical energy (T.E) is given by the sum of kinetic energy (K.E) and potential energy (P.E) of the electron:
  $T.E = K.E + P.E$

• The kinetic energy can be calculated using:
  $K.E = rac{1}{2} mv^2$
  where $m$ is the mass of the electron and $v$ is its velocity.

• The potential energy expression for an electron in an atom is:
  $P.E = - rac{k e^2}{r}$

Key Energy Calculations for Hydrogen Atom

• For various orbits:

  • For the 1st orbit:
    $K.E = +13.6 ext{ eV}, \ P.E = -27.2 ext{ eV}, \ T.E = -13.6 ext{ eV}$

Energy Level Diagram and Quantization

• The energy levels are quantized, meaning electrons can only occupy certain finite energy levels. Each transition corresponds to the absorption or emission of a photon:
  $E_{ ext{photon}} = E_n - E_m$
  where n > m.

Lyman Series and Line Spectrum

• The Lyman series corresponds to transitions that occur when an electron falls to the n=1 level from higher levels (n=2,3,4,…). The wavelengths for transitions can be calculated:

  • The first line of the Lyman series corresponds to:
    $rac{1}{ ext{λ}} = R imes (1^2 - rac{1}{n^2})$
    The maximum wavelength of line transitions for the hydrogen atom can be calculated as:
    $1.216 ext{ Å}$ (for n=1 to n=2)