Answer:
The value is [tex]\lambda= 27.3 \ \mu m[/tex]
Explanation:
From the question we are told that
The equilibrium spacing is [tex]r = 0.074 nm= 0.074 *10^{-9} \ m[/tex]
The mass of a hydrogen atom is [tex]m = 1.67 *10^{-27} \ kg[/tex]
The principal quantum number of the second energy level is [tex]l = 3[/tex]
The principal quantum number of the first energy level is [tex]l = 2[/tex]
Given that the hydrogen molecule is a symmetrical diatomic molecule, its moment of inertia is mathematically represented as
[tex]I = \frac{m}{2} r^2[/tex]
Generally for rotational spectrum the energy level is mathematically represented as
[tex]E = l(l + 1) * \frac{h^2}{8 \pi ^2 I}[/tex]
Generally the energy difference between the first energy level and the second energy level is mathematically represented as
[tex]\Delta E = [ l_2(l_2 + I) * \frac{h^2}{8 \pi ^2 I}]- [ l_1(l_1 + I) * \frac{h^2}{8 \pi ^2 I}][/tex]
=> [tex]\Delta E = [ 3(3 + 1) * \frac{h^2}{8 \pi ^2 I}]- [ 2(2 + 1) * \frac{h^2}{8 \pi ^2 I}][/tex]
=> [tex]\Delta E = \frac{6h^2}{8 \pi ^2 I}[/tex]
substituting for I
[tex]\Delta E = \frac{6h^2}{8 \pi ^2 [\frac{m}{2} * r^2]}[/tex]
=> [tex]\Delta E = \frac{3h^2}{2 \pi ^2 m * r^2}[/tex]
Generally this difference in energy level can also be mathematically represented as
[tex]\Delta E = \frac{hc}{\lambda}[/tex]
=> [tex]\frac{3h^2}{2 \pi ^2 m * r^2} = \frac{hc}{\lambda}[/tex]
=> [tex]\lambda= \frac{2 \pi^2 m * r^2 * c}{3h}[/tex]
Here h is Planck's constant with value [tex]h = 6.62607015 * 10^{-34} J \cdot s[/tex]
and c is the speed of light with value [tex]c = 3.0*10^{8} \ m/s[/tex]
So
=> [tex]\lambda= \frac{2* 3.142^2 * 1.67 *10^{-27} * [0.074*10^{-9}]^2 * 3.0*10^{8}}{3*6.62607015 * 10^{-34} }[/tex]
=> [tex]\lambda= 27.3*10^{-6} \ m[/tex]
=> [tex]\lambda= 27.3 \ \mu m[/tex]
In this exercise we have to use the rotational transition knowledge and calculate the wavelength:
The value is [tex]\lambda = 27.3 \mu m[/tex]
From the question we are told that
Given that the hydrogen molecule is a symmetrical diatomic molecule, its moment of inertia is mathematically represented as:
[tex]I=\frac{m}{2}r^2[/tex]
Generally for rotational spectrum the energy level is mathematically represented as:
[tex]E= l(l+1)(\frac{h^2}{8\pi^2 I} )[/tex]
Generally the energy difference between the first energy level and the second energy level is mathematically represented as:
[tex]\Delta E = [l_2 (l_2+I)(\frac{h^2}{8\pi^2 I}) ] - [ l_1(l_1+I)(\frac{h^2}{8\pi^2I} )]\\\Delta E = [3 (3+1)(\frac{h^2}{8\pi^2 I}) ] - [2(2+1)(\frac{h^2}{8\pi^2I} )]\\\\\Delta E = \frac{6h^2}{8\pi^2 I}[/tex]
Substituting for I:
[tex]\Delta E = \frac{6h^2}{8\pi^2 [m/2*r^2]} \\\Delta E = \frac{6h^2}{2\pi^2*m/*r^2]} \\[/tex]
Generally this difference in energy level can also be mathematically represented as:
[tex]\Delta = \frac{hc}{\lambda} \\\frac{3h^2}{2\pi^2 m*r^2}= \frac{hc}{\lambda} \\\lambda= \frac{2\pi^2 m * r^2 * c }{3h}[/tex]
Here h is Planck's constant with value: [tex]h= (6.62607015)*(10)^{-34}[/tex]
and c is the speed of light with value: [tex]c= 3.0 * 10^8[/tex]
[tex]\lambda = 27.3 \mu m[/tex]
See more about rotational transitions at brainly.com/question/17237951
