4-13. The radius of the n = 1 orbit in the hydrogen atom is ao = 0.053 nm. (a) Compute the radius of the n = 6 orbit. (b) Compute the radius of the n = 6 orbit in singly ionized helium (He), which is hydrogenlike, that is, it has only a single electron outside the nucleus. 4-14. Show that Equation 4-19 for the radius of the first Bohr orbit and Equation 4-20 for the magnitude of the lowest energy for the hydrogen atom can be written as ao -a²mc² hc ^c amc² 2πα E₁ where λ Ac = h/mc is the Compton wavelength of the electron and a = ke²/ħc is the fine- structure constant. Use these expressions to check the numerical values of the constants ao and E₁. 4-15. Calculate the three longest wavelengths in the Lyman series (n = 1) in nm, and indicate their position on a horizontal linear scale. Indicate the series limit (shortest wave- length) on this scale. Are any of these lines in the visible spectrum? 4-16. If the angular momentum of Earth in its motion around the Sun were quantized like a hydrogen electron according to Equation 4-17, what would Earth's quantum number be? How much energy would be released in a transition to the next lowest level? Would that energy release (presumably as a gravity wave) be detectable? What would be the radius of that orbit? (The radius of Earth's orbit is 1.50 × 10¹¹ m.) 4-17. On the average, a hydrogen atom will exist in an excited state for about 10-8 s before making a transition to a lower energy state. About how many revolutions does an electron in the n = 2 state make in 10-8 s?
4-13. The radius of the n = 1 orbit in the hydrogen atom is ao = 0.053 nm. (a) Compute the radius of the n = 6 orbit. (b) Compute the radius of the n = 6 orbit in singly ionized helium (He), which is hydrogenlike, that is, it has only a single electron outside the nucleus. 4-14. Show that Equation 4-19 for the radius of the first Bohr orbit and Equation 4-20 for the magnitude of the lowest energy for the hydrogen atom can be written as ao -a²mc² hc ^c amc² 2πα E₁ where λ Ac = h/mc is the Compton wavelength of the electron and a = ke²/ħc is the fine- structure constant. Use these expressions to check the numerical values of the constants ao and E₁. 4-15. Calculate the three longest wavelengths in the Lyman series (n = 1) in nm, and indicate their position on a horizontal linear scale. Indicate the series limit (shortest wave- length) on this scale. Are any of these lines in the visible spectrum? 4-16. If the angular momentum of Earth in its motion around the Sun were quantized like a hydrogen electron according to Equation 4-17, what would Earth's quantum number be? How much energy would be released in a transition to the next lowest level? Would that energy release (presumably as a gravity wave) be detectable? What would be the radius of that orbit? (The radius of Earth's orbit is 1.50 × 10¹¹ m.) 4-17. On the average, a hydrogen atom will exist in an excited state for about 10-8 s before making a transition to a lower energy state. About how many revolutions does an electron in the n = 2 state make in 10-8 s?
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Section 4-3 Ine Bohr Model of the Hyarogen Atom
4-13. The radius of the n 1 orbit in the hydrogen atom is a = 0.053 nm. (a) Compute
the radius of the n = 6 orbit. (b) Compute the radius of the n = 6 orbit in singly ionized
helium (He¹), which is hydrogenlike, that is, it has only a single electron outside the
nucleus.
Q
ao
hc
^c
amc² 2πα
4-14. Show that Equation 4-19 for the radius of the first Bohr orbit and Equation 4-20 for
the magnitude of the lowest energy for the hydrogen atom can be written as
-α²mc²
-
E₁
T Add text
=
H
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where λ = h/mc is the Compton wavelength of the electron and a = ke²/ħc is the fine-
structure constant. Use these expressions to check the numerical values of the constants
ao and E₁.
=
4-15. Calculate the three longest wavelengths in the Lyman series (n 1) in nm, and
indicate their position on a horizontal linear scale. Indicate the series limit (shortest wave-
length) on this scale. Are any of these lines in the visible spectrum?
4-16. If the angular momentum of Earth in its motion around the Sun were quantized like
a hydrogen electron according to Equation 4-17, what would Earth's quantum number be?
How much energy would be released in a transition to the next lowest level? Would that
energy release (presumably as a gravity wave) be detectable? What would be the radius of
that orbit? (The radius of Earth's orbit is 1.50 × 10¹¹ m.)
4-17. On the average, a hydrogen atom will exist in an excited state for about 10-8 s
before making a transition to a lower energy state. About how many revolutions does an
electron in the n = 2 state make in 10-8 s?
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