Question

Which of the following are in the ascending order of the wavelength?
This question has multiple correct options.
A. ${H_\alpha }$ , ${H_\beta }$ , ${H_\gamma }$ , … lines in the Balmer series of hydrogen atoms.
B. Lyman limit, Balmer limit, and Paschen limit in the hydrogen spectrum.
C. Violet, blue, yellow and red colours in the solar spectrum.
D. None of the above

Answer 1

First of all, we will use the formula, which says energy is inversely proportional to the wavelength. From this equation, we can directly compare the energy levels and their wavelength.
After that we will again use the equation which relates wavelength, energy level and Rydberg’s constant. From this equation too we will directly compare the wavelengths of the corresponding energy level.

Formula used:
The formula which gives the energy which relates wavelength and speed of light is as follows:
$E = \dfrac{{hc}}{\lambda }$ …… (1)
Where,
$E$ indicates the energy associated with the wavelength.
$h$ indicates the Planck’s constant.
$c$ indicates the speed of light.
$\lambda$ indicates the wavelength of the light.
The formula which gives relation between the wavelength and the energy level is given below:
$\dfrac{1}{\lambda } = R \times \left( {\dfrac{1}{{{n^2}}}} \right)$ …… (2)
Where,
$\lambda$ indicates the wavelength of the light.
$R$ indicates the Rydberg’s constant.
$n$ indicates the energy level.

Complete answer:
In the given question, we are supplied with the following data:
We are required to find in which of the following cases, there is the ascending order of the wavelength.First, we are given the spectral lines of the Balmer series of hydrogen atoms.Then, we are given the Lyman limit, Balmer limit, and Paschen limit in the hydrogen spectrum.After that we are given the violet, blue, yellow and red colours in the solar spectrum.We are required to select which of the following options contains the increasing order of the wavelength.

To begin with, we will use the equation (1) and try to find the order of the wavelength.From equation (1), we can see that the energy is inversely proportional to the wavelength.We can also say that energy associated with the first level has low energy and hence it will be associated with higher wavelength. The energy associated with the second energy level has more energy than the first level and hence it will be associated with lower wavelength than compared to the first level. So, the decreasing order of the wavelength is:
${H_\alpha } > {H_\beta } > {H_\gamma }$

Again, we have equation (2), which says that the wavelength is directly proportional to the energy level. Since, the increasing order of the energy levels of the series are Lyman, Balmer, and Paschen. So, the wavelength of the following series also increases as the value of $n$ increases. Again, we know that the VIBGYOR is the spectrum of light, which corresponds to the decreasing energy. We know that lower energy is associated with higher wavelength and vice-versa. Hence, the increasing order of the wavelength is violet, blue, yellow and red.

The correct options are B and C.

Note: While solving this problem, most of the students seem to have confusion regarding the energy and wavelength. It is important to remember that higher the energy is, lower is the wavelength and vice-versa. The blue flame of a burner has more energy than the red flame. Again, red light is used in vehicles, because of larger wavelengths, so that they are visible from a large distance.