Respuesta :
Answer
the least energetic spectral line amongst the infrared series of the H atom has a wavelength of 656.2nm, the red spectral line.
Explanation
Spectral lines are bright or dark lines appearing in an otherwise continuous spectrum. These can be bright lines in an emmision spectrum (continous black), or dark lines for absorption spectrum (continuous 'color' spectrum). The spectral lines appear at particular wavelengths distinct to every atom. These wavelengths are related to the energy associated with it. Spectral patterns are sometimes referred to as the atomic fingerprint as they can be used to determine the identity of atoms. In fact, these have been used during the discovery of helium (in space!) as well as other elements.
The spectral lines observed for hydrogen are:
410.0 nm (Violet)
434.0 nm (Blue-Violet)
486.1 nm (Blue-Green)
656.2 nm (Red)
Since wavelength is inversely proportional to energy, the least energetic among these is the 656.2nm, red spectral line.
It can be shown mathematically thus,
Photon wavelength is inversely proportional to energy. So if we want the least energetic spectral line of the hydrogen atom, we should determine the longest wavelength possible. We can do this by minimizing the inverse wavelength given by the Rydberg formula:
1/λ = RH(1/nf^2−1/n^2)
Now we can minimize this expression when we set:
n=3
nf=2
So now we substitute:
1/λ=RH(1/(2^2)−1/(3^2))
1/λ=RH(1/4−1/9)
1/λ=RH(0.25−0.1111)
We substitute our Rydberg constant:
1/λ=(1.09737×10^7m−1)(0.1389)
We get:
1/λ=(1.5242×10^6)m−1
Now we take the reciprocal to get:
λ=(6.5607×10^−7) m
We thus get:
λ=656.07 nm
Answer:
The least energetic spectral line in the infrared series of the H atom is 656.1 nm
Explanation:
Photon wavelength is inversely proportional to energy. To obtain the least energetic spectral line of the hydrogen atom (H), we determine the longest wavelength possible.
[tex]\frac{1}{\lambda} = R_H[\frac{1}{n_f^2} -\frac{1}{n^2}][/tex]
Where;
nf = 2
n = 3
RH is Rydberg constant = 1.09737 × 10⁷m⁻¹
λ is the wavelength of the least energetic spectral line
Substituting the above values into the equation, we will have
[tex]\frac{1}{\lambda} = 1.09737 X 10^7[\frac{1}{2^2} -\frac{1}{3^2}][/tex]
[tex]\frac{1}{\lambda} = 1.09737 X 10^7[\frac{1}{4} -\frac{1}{9}][/tex]
[tex]\frac{1}{\lambda} = 1.09737 X 10^7[0.25 -0.1111][/tex]
[tex]\frac{1}{\lambda} = 1.09737 X 10^7[0.1389][/tex]
[tex]\frac{1}{\lambda} = 1524246.93[/tex]
[tex]\lambda} = \frac{1}{1524246.93}[/tex]
[tex]\lambda} = 6.561 X10^{-7} m[/tex]
λ = 656.1 X10⁻⁹ m
In (nm): λ = 656.1 nm
Therefore, the least energetic spectral line in the infrared series of the H atom is 656.1 nm
