Question

Wavelengths \[{\lambda _1}\] and \[{\lambda _2}\] are of the first line of Balmer series of deuterium and hydrogen respectively, then:
A.\[{\lambda _1} > {\lambda _2}\]
B.\[{\lambda _1} < {\lambda _2}\]
C.\[{\lambda _1} = {\lambda _2}\]
D.\[{\lambda _1} \geqslant {\lambda _2}\]

Answer 1

Hint: We will use the Rydberg’s formula for the comparison of the two wavelengths. The value of Rydberg constant for deuterium is greater than that of the value of Rydberg constant for hydrogen.

Formula used: \[\dfrac{1}{\lambda } = {{\text{R}}_{\text{H}}}.{{\text{Z}}^2}\left[ {\dfrac{1}{{{\text{n}}_1^2}} - \dfrac{1}{{{\text{n}}_2^2}}} \right]\]
Here \[\lambda \] is wavelength, \[{{\text{R}}_{\text{H}}}\] is Rydberg constant, \[{\text{Z}}\] is nuclear charge, \[{{\text{n}}_1}{\text{ and }}{{\text{n}}_2}\] are initial and final energy states respectively.

Complete step by step solution:
Hydrogen and deuterium are both isotopes that mean they have the same atomic number but have different mass numbers. The atomic number of hydrogen and deuterium is 1 but the mass of hydrogen is 1 and the mass of deuterium is 2. The relation given by Rydberg is as follow:
\[\dfrac{1}{\lambda } = {{\text{R}}_{\text{H}}}.{{\text{Z}}^2}\left[ {\dfrac{1}{{{\text{n}}_1^2}} - \dfrac{1}{{{\text{n}}_2^2}}} \right]\]
The value of Z is the same for both deuterium and hydrogen. This is because Z represents the nuclear charge and the charge on the nucleus is due to the presence of protons. The number of protons in both hydrogen and deuterium is 1 whereas deuterium has one neutron and hydrogen has zero neutrons. The value of initial and final energy state will remain the same for the Balmer series for both deuterium and hydrogen will be the same. The wavelength and Rydberg constant are inversely proportional to each other.
The value of Rydberg constant for deuterium is \[109708{\text{ c}}{{\text{m}}^ -1 }\] and for hydrogen is \[109678{\text{ c}}{{\text{m}}^ -1 }\].
Hence, the wavelength of deuterium will be less than the wavelength of hydrogen.

Thus, the correct option is B.

Note: There is one more isotope of hydrogen that is Tritium which has the same atomic number but has mass number 3. It has 2 neutrons and 1 proton. The ground state starts from the n value 2 for Balmer series.