Respuesta :
Since there is not quite enough energy for transition from n=1 to n=2. So, the photons will pass through the gas chamber without affecting the gas.
Further explanation:
If the energy of photons is enough to excite all the electrons to go in higher state, the electron absorbs the energy of photon and go from a lower energy state to higher energy state. Transition of electron from lower energy state t higher energy state depends on the energy of photons.
Given:
The wavelength of the photons directed through a chamber of hydrogen gas atoms is [tex]656\text{ nm}[/tex].
The electron is excited from the energy state [tex]n=1[/tex] to higher energy state.
Concept:
The energy associated with a photon is given by the following relation.
[tex]E = \dfrac{{hc}}{\lambda}[/tex]
Here, [tex]E[/tex] is the energy associated with photon, [tex]h[/tex] is the plank constant, [tex]c[/tex] is the speed of the light and [tex]\lambda[/tex] is the wavelength associated with photon.
Substitute [tex]6.625\times{10^{-34}}\text{ J}\cdot\text{s}[/tex] for [tex]h[/tex], [tex]3.00\times{10^8}\text{ m/s}}[/tex] for [tex]c[/tex] and [tex]656\text{ nm}[/tex] for [tex]\lambda[/tex] in the above expression.
[tex]\begin{aligned}E&=\frac{{\left( {6.625 \times {{10}^{ - 34}}\,{\text{Js}}} \right)\left( {3.0 \times {{10}^8}\,{\text{m/s}}} \right)}}{{\left( {656 \times {{10}^{ - 9}}\,{\text{m}}} \right)}}\\&=3.0281 \times {10^{-19}}\,{\text{J}}\\\end{aligned}[/tex]
Rydberg equations is used to find out the energy level of an electron while transition from one energy state to another energy state.
[tex]\dfrac{1}{\lambda}=R{z^2}\left({\dfrac{1}{{n_1^2}}-\dfrac{1}{{n_2^2}}}\right)[/tex]
Here, [tex]\lambda[/tex] is the wavelength of the photons directed through a chamber, [tex]R[/tex] is the Rydberg constant, [tex]z[/tex] is the atomic number of hydrogen atom, [tex]{n_1}[/tex] is the lower energy state and [tex]{n_2}[/tex] is the higher energy state.
Substitute [tex]656\times{10^{-9}}\,{\text{m}}[/tex] for [tex]\lambda[/tex], [tex]1.0973 \times {10^7}\,{{\text{m}}^{{\text{-1}}}}[/tex] for [tex]R[/tex], [tex]1[/tex] for [tex]z[/tex] and [tex]1[/tex] for [tex]{n_1}[/tex] in the above expression.
[tex]\dfrac{1}{{\left( {656 \times {{10}^{ - 9}}\,{\text{m}}} \right)}} = \left( {1.0973 \times {{10}^7}{\mkern 1mu} {{\text{m}}^{{\text{-1}}}}} \right){\left( 1 \right)^2}\left( {\dfrac{1}{{{{\left( 1 \right)}^2}}}-\dfrac{1}{{n_2^2}}}\right)[/tex]
Simplify the above expression for [tex]{n_2}[/tex]
[tex]\begin{aligned}{n_2}&=\sqrt{\dfrac{1}{{0.861}}}\\&=1.08\\\end{aligned}[/tex]
From the above result it can be concluded that photons with wavelength [tex]656\text{ nm}[/tex] does not have enough energy to cause an electron to jump even one level higher.
Thus, there is not quite enough energy for transition from n=1 to n=2. So, the photons will pass through the gas chamber without affecting the gas.
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Answer Details:
Grade: College
Subject: Physics
Chapter: Modern physics
Keywords:
Photons, light, energy, 656 nm, 656 times 10^-9 m, 6.56 times 10^-7 m, directed, chamber, hydrogen, gas atom, ground state, ni=1, 1.071, few, absorbed, quite enough, electrons, nf=2.
The electron can not be promoted to a higher energy level due to insufficient energy of the photons hence the photons pass through the gas chamber.
Using the Rydberg formula;
1/λ = R (1/n2^2 - 1/n1^2)
λ = wavelength of photon = 656 × 10^-9 m
R = 1.097 × 10^7 m-1
n2 = ?
n1 = 1
Substituting values;
1/656 × 10^-9 = 1.097 × 10^7(1/1^2 - 1/n2^2)
0.139 = 1 - 1/n2^2
1/n2^2 = 1 - 0.139
1/n2^2 = 0.861
n2 =1
We can see that the electron can not be promoted to a higher energy level due to insufficient energy of the photons hence the photons pass through the gas chamber.
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