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23 Nov 2019
The emergence of line spectra from a gas when a current passesthrough it was an observed phenomenon waiting for an explanation inthe early 20th century. The atomic line spectra coming fromelements such as hydrogen had been analyzed since the late 19thcentury. By studying the wavelength of the emerging radiation ofhydrogen, experimenters found (often by trial and error) that thewavelengths in those spectra were described by the formula
\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2}\right),
where R is known as the Rydberg constant. It has a value of R =1.097 \times 10^{7}\; \rm m^{-1}. The variables n_1 and n_2 areinteger numbers (n_1 = 1,2,3,4,\dots). As experiments continued,scientists began to see more and more characteristic lines emergingfrom the hydrogen spectrum. Each of them corresponded to awavelength predicted by this formula with some integer values forn_1 and n_2.
In 1913, Niels Bohr provided an explanation for the observationsmade in experiments by proposing that each electron in an atom hadonly certain allowable energy levels. Furthermore, he postulatedthat when an electron changed its energy level, it must emit orabsorb a single photon with the energy equal to the energydifference between the two levels.
With this interpretation, the variables n_1 and n_2 acquired newsignificance. Bohr's model predicted that n_2 was the initialenergy state of the atom (n_initial) and that n_1 was the finalstate of the atom (n_final). A photon with the predicted wavelengthwas created when an atom made a transition from its initial energystate to a lower final energy state. The equation now read
\frac{1}{\lambda} = R \left( \frac{1}{n_{\rm final}^2} -\frac{1}{n_{\rm initial}^2} \right).
Bohr's model also predicted that a photon of the appropriatewavelength (given by the same formula) could promote an atom from alower energy level to a higher one.
In this problem, we explore the photon characteristics necessaryfor transitions between energy levels in hydrogen and singlyionized {\rm He}^+, whose energy levels are accurately predicted bythe Bohr model.
What is the change in energy DeltaE of the hydrogen atom as theelectron makes the transition from the n=3 energy level to the n=1energy level?
Express your answer numerically in electron volts.
b. When an atom makes a transition from a higher energy level to alower one, a photon is released. What is the wavelength of thephoton that is emitted from the atom during the transition from n=3to n=1?
Express your answer numerically in angstroms.
c.
When the hydrogen atom makes the transition from the n=2 to the n=1energy level, it emits a photon. This photon can be absorbed by toa singly ionized helium atom ({\rm He}^+) in its second energylevel (n_{\rm initial} = 2), causing it to move to a higher energylevel. To what final energy level number n_final can the photonpromote the helium ion?
Note that {\rm He}^+ has two protons in its nucleus and a singleelectron.
Express your answer as an integer.
Please answer fully, Thank you
a.
The emergence of line spectra from a gas when a current passesthrough it was an observed phenomenon waiting for an explanation inthe early 20th century. The atomic line spectra coming fromelements such as hydrogen had been analyzed since the late 19thcentury. By studying the wavelength of the emerging radiation ofhydrogen, experimenters found (often by trial and error) that thewavelengths in those spectra were described by the formula
\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2}\right),
where R is known as the Rydberg constant. It has a value of R =1.097 \times 10^{7}\; \rm m^{-1}. The variables n_1 and n_2 areinteger numbers (n_1 = 1,2,3,4,\dots). As experiments continued,scientists began to see more and more characteristic lines emergingfrom the hydrogen spectrum. Each of them corresponded to awavelength predicted by this formula with some integer values forn_1 and n_2.
In 1913, Niels Bohr provided an explanation for the observationsmade in experiments by proposing that each electron in an atom hadonly certain allowable energy levels. Furthermore, he postulatedthat when an electron changed its energy level, it must emit orabsorb a single photon with the energy equal to the energydifference between the two levels.
With this interpretation, the variables n_1 and n_2 acquired newsignificance. Bohr's model predicted that n_2 was the initialenergy state of the atom (n_initial) and that n_1 was the finalstate of the atom (n_final). A photon with the predicted wavelengthwas created when an atom made a transition from its initial energystate to a lower final energy state. The equation now read
\frac{1}{\lambda} = R \left( \frac{1}{n_{\rm final}^2} -\frac{1}{n_{\rm initial}^2} \right).
Bohr's model also predicted that a photon of the appropriatewavelength (given by the same formula) could promote an atom from alower energy level to a higher one.
In this problem, we explore the photon characteristics necessaryfor transitions between energy levels in hydrogen and singlyionized {\rm He}^+, whose energy levels are accurately predicted bythe Bohr model.
What is the change in energy DeltaE of the hydrogen atom as theelectron makes the transition from the n=3 energy level to the n=1energy level?
Express your answer numerically in electron volts.
b. When an atom makes a transition from a higher energy level to alower one, a photon is released. What is the wavelength of thephoton that is emitted from the atom during the transition from n=3to n=1?
Express your answer numerically in angstroms.
c.
When the hydrogen atom makes the transition from the n=2 to the n=1energy level, it emits a photon. This photon can be absorbed by toa singly ionized helium atom ({\rm He}^+) in its second energylevel (n_{\rm initial} = 2), causing it to move to a higher energylevel. To what final energy level number n_final can the photonpromote the helium ion?
Note that {\rm He}^+ has two protons in its nucleus and a singleelectron.
Express your answer as an integer.
Please answer fully, Thank you
a.
30 Jun 2023
