Question

Show that the Balmer series occurs between $3647{A^\circ }$ and $6562A^\circ$. ($R = 1.0968 \times {10^7}{m^{ - 1}}$).

Answer 1

The Balmer series describes the spectral line of emissions of hydrogen atom and it occurs when the electron does transition from a higher energy level (energy level above $n = 2$) to the lower energy level of $n = 2$.

Complete step by step answer:
As we know that when an electron is travels from low energy level to higher energy level then it absorbs energy and when the electron comes back from higher energy level to lower energy level then this process in case of hydrogen is known as emission and the spectral lines produced through emission is known as spectral lines of hydrogen. There are mainly five series of emissions of hydrogen. These are Lyman series, Balmer series, Paschen series, Bracket series and P fund series.
In the Balmer series the transition of the electron occurs from any higher energy state to the lower energy state. Here, the higher energy state may be $n = 3,4,5,6.......\infty$ to the lower state of $n = 2$.
Also, we know that the general formula for wavelength of hydrogen spectrum emission can be given as:
$\dfrac{1}{\lambda } = R\left[ {\dfrac{1}{{{2^2}}} - \dfrac{1}{{{n^2}}}} \right]$
Where, $\lambda =$ wavelength
$R =$ Rydberg constant
$n =$ higher energy state
As we know that Balmer series starts when we will have minimum value of higher state that is $n = 3$,
Therefore, $\dfrac{1}{\lambda } = R\left[ {\dfrac{1}{{{2^2}}} - \dfrac{1}{{{3^2}}}} \right]$
$\dfrac{1}{\lambda } = R\left[ {\dfrac{5}{{36}}} \right] \\\ \lambda = \dfrac{{36}}{{R \times 5}} \\\ \lambda = \dfrac{{36}}{{1.0968 \times {{10}^7}}} = 6563A^\circ \\\$
Also, we know that Balmer series starts when we will have maximum value of higher state that is $n = \infty$,
Therefore, $\dfrac{1}{\lambda } = R\left[ {\dfrac{1}{{{2^2}}} - \dfrac{1}{{{\infty ^2}}}} \right]$
$\dfrac{1}{\lambda } = R\left[ {\dfrac{1}{4}} \right] \\\ \lambda = \dfrac{4}{R} \\\ \lambda = \dfrac{4}{{1.0968 \times {{10}^7}}}m = 3647A^\circ \\\$

Hence, the Balmer series occurs between $3647{A^\circ }$ and $6562A^\circ$.

Note:

Always remember that Balmer series lies in visible region therefore they are visible, Lyman series (in which electron transition occurs form higher energy level to $n = 2$) occurs in ultraviolet region and the Paschen series (in which electron transition occurs form higher energy level to $n = 3$) occurs in the infrared region.