Atomic Spectrum

Planck proposed the following three main points in his quantum theory:

1. Any radiation should be associated with energy.
2. The energy is released in the form of radiation, occurring in small packets called quanta.
3. The energy is directly proportional to the frequency.

The Rydberg equation helps determine the wavelength of the spectrum. The wavelength of the H-spectrum can be used to calculate the frequency and energy of the H-spectrum. The wave number is inversely proportional to the square of energy level differences ($n_2$).

Since $V = 1/\lambda$

The values of $n_1$ and $n_2$ for the H-spectrum can be obtained if the number of lines and series are given. The value of $n_1$ is equal to the number of series. The value of $n_2$ is equal to the number of lines plus the number of series. For example, for the third line of the Balmer series:

1. Number of line = Third line
2. Number of series = Balmer's series
3. $n_1 = 2$ (Number of series)
4. $n_2 = 3 + 2 = 5$

Transition energy is the energy required to shift an electron from one shell to another. The energy required to shift electrons from one shell to another equals the energy difference between the two shells. The energy difference between the lowest energy level ($E_1$) and the highest energy level ($E_2$) is given by the following equation:

$E = E_2 - E_1$

Transition trends:

1. $E = E_2 - E_1$: Transition takes place from $E_1$ to $E_2$ exactly.
2. $E > E_2 - E_1$: Transition takes place from $E_1$ towards above $E_2$.
3. $E < E_2 - E_1$: Transition takes place from $E_1$ but hangs between $E_1$ and $E_2$.

If electrons gain enough energy equal to the ionization energy, the electron will jump completely from the ground state to infinity, ionizing the atom positively. This energy is used as ionization and kinetic energy. The ionization energy of electrons is equal to the energy associated with the electron in the shell or quantum number it belongs to.

Energy levels or shells in an atom are regions around the nucleus where electrons are likely to be found. The energy values of these shells are negative, indicating that energy must be absorbed for an electron to move to a higher shell. The energy at an infinite distance from the nucleus is considered zero.

Energy values of shells

• $n_\infty = 0.00 \ eV$
• $n_7 = -0.15 \ eV$
• $n_6 = -0.21 \ eV$
• $n_5 = -0.54 \ eV$
• $n_4 = -0.84 \ eV$
• $n_3 = -1.50 \ eV$
• $n_2 = -3.40 \ eV$
• $n_1 = -13.60 \ eV$

Calculation of energy changes

The energy associated with an electron transition between two shells can be determined using the formula:

$E = E_2 - E_1$

Example 1

Find the first ionization energy of potassium:

$E = E_{\infty} - E_1 = 0.00 \ eV - (-13.60 \ eV) = 13.60 \ eV$

Example 2

Calculate the wavelength of the third line of the Balmer series:

$\frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{n^2} \right)$

For the third line ($n = 5$):

$\frac{1}{\lambda} = 1.09 \times 10^7 \left( 0.25 - 0.04 \right)$

$\lambda \approx 4.37 \times 10^{-7} \ m$

Transition rules

1. If energy is greater than $E_4 - E_2$, the electron can move to a higher energy level.
2. If energy is equal to $E_4 - E_2$, the electron transitions from $E_2$ to $E_4$.