Hydrogen atoms are excited to the stationary state designated by the principal quantum number, n= 4. What will be the number of lines(frequencies) emitted by the hydrogen atoms?
4 because number of spectral lines are directly proportional to principal quantum number
To determine the number of lines (frequencies) emitted by hydrogen atoms in the excited state with the principal quantum number, n = 4, we need to understand the concept of electronic transitions in hydrogen atoms.
In hydrogen atoms, when an electron transitions from a higher energy level to a lower energy level, it emits electromagnetic radiation in the form of light. Each transition corresponds to a specific frequency or wavelength of light.
The energy of an electron in a hydrogen atom is given by the equation:
E = -13.6 eV / n^2
Where:
E = Energy of the electron
-13.6 eV = Ionization energy of hydrogen
n = Principal quantum number
When an electron transitions from a higher energy level to a lower energy level, the energy difference between the two levels is equal to the energy of the emitted photon of light.
For a hydrogen atom in the excited state with n = 4, the possible electronic transitions can occur from the levels:
n = 4 to n = 3
n = 4 to n = 2
n = 4 to n = 1
Using the equation mentioned above, we can calculate the energy differences between these levels.
E(4→3) = -13.6 eV / (4^2) - (-13.6 eV / (3^2))
E(4→2) = -13.6 eV / (4^2) - (-13.6 eV / (2^2))
E(4→1) = -13.6 eV / (4^2) - (-13.6 eV / (1^2))
Next, we can convert the energy differences into frequencies (or wavelengths) using the equation:
E = h * f
Where:
E = Energy difference between levels
h = Planck's constant (6.626 x 10^-34 J.s)
f = Frequency of light emitted
Using this equation, we can calculate the frequencies for each transition.
f(4→3) = E(4→3) / h
f(4→2) = E(4→2) / h
f(4→1) = E(4→1) / h
Finally, by counting the number of distinct frequencies obtained, we can determine the number of lines emitted by hydrogen atoms in the excited state with the principal quantum number, n = 4.
n =4 to n = 3
n = 4 to n = 2
n = 4 to n = 1
n = 3 to n = 2
n = 3 to n = 1
m = 2 to n = 1