Question

Which one of the following statements is not correct?
A) Rydberg’s constant and wavenumber have the same units.
B) Lyman series of hydrogen spectrum occur in the ultraviolet region.
C) The angular momentum of the electron in the ground state of a hydrogen atom is equal to \[\dfrac{h}{{2\pi }}\].
D) The radius of the first Bohr orbit of hydrogen atom is \[2.116{\text{ }} \times {10^{ - 8}}{\text{cm}}\].

Answer 1

Hint: In the hydrogen spectrum, Rydberg formula relates the wavenumber of wavelength with the principal quantum numbers of two energy levels involved in the electron transition.
The wavenumber of a particular radiation is inversely proportional to the reciprocal of the wavelength.

Complete step by step answer:
Write the Rydberg formula as shown below:
\[\dfrac{1}{\lambda } = R \times \left( {\dfrac{1}{{n_1^2}} - \dfrac{1}{{n_2^2}}} \right)\]
In the Rydberg formula, \[\lambda \] is the wavelength, R is the Rydberg constant, \[{n_1},{n_2}\] are the principal quantum numbers of two energy levels involved in the electron transition.
According to the Rydberg formula, \[\dfrac{1}{\lambda } = R \times \left( {\dfrac{1}{{n_1^2}} - \dfrac{1}{{n_2^2}}} \right)\] the units of the Rydberg constant and \[\dfrac{1}{\lambda }\] will be same. Here \[\dfrac{1}{\lambda }\] represents the wavenumber. The wavenumber is the reciprocal of the wavelength.
Hence, Rydberg’s constant and wavenumber have the same units.
Hence, the option (A) represents the correct statement.
In the Lyman series the lowest energy is \[10.4{\text{ev}}\]. This energy belongs to the ultraviolet region.
Lyman series of hydrogen spectrum occur in the ultraviolet region.
Hence, the option (B) represents the correct statement.
The angular momentum of the electron in the ground state of a hydrogen atom is equal to \[\dfrac{h}{{2\pi }}\]
Hence, the option (C) represents the correct statement.
The radius of first Bohr orbit of hydrogen atom is \[{\text{0}}{\text{.529 }} \times {10^{ - 8}}{\text{cm}}\]
\[{\text{r = 0}}{\text{.529 }} \times {10^{ - 8}}{\text{cm}} \times \dfrac{{{n^2}}}{Z} \\
{\text{r = 0}}{\text{.529 }} \times {10^{ - 8}}{\text{cm}} \times \dfrac{{{1^2}}}{1} \\
{\text{r = 0}}{\text{.529 }} \times {10^{ - 8}}{\text{cm}}\]
For hydrogen atoms, the atomic number Z is 1. For hydrogen atoms, the principal quantum number n is also 1.
Hence, the option (D) represents incorrect statements.

Note: Different series in the hydrogen spectrum are named as lyman, balmer, paschen, brackett and pfund. For these series the value of \[{n_1}\] is respectively 1,2,3,4 and 5. They belong to the UV, visible, IR, IR and IR region respectively.