Modern Physics for Scientists and Engineers
4th Edition
ISBN: 9781133103721
Author: Stephen T. Thornton, Andrew Rex
Publisher: Cengage Learning
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Chapter 4, Problem 55P
To determine
Show that the quantization of
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A sodium atom (Z = 11) contains 11 protons in its nucleus. Strictly speaking, the Bohr model does not apply, because the neutral atom
contains 11 electrons instead of a single electron. However, we can apply the model to the outermost electron as an approximation,
provided that we use an effective value Zeffective rather than 11 for the number of protons in the nucleus. (a) The ionization energy
for the outermost electron in a sodium atom is 5.1 eV. Use the Bohr model with Z = Zeffective to calculate a value for Zeffective. (b) Using
Z = 11, determine the corresponding value for the radius r of the outermost Bohr orbit. (c) Using the value calculated for Zeffective in
part (a), determine the corresponding radius r of the outermost Bohr orbit.
(a) Zeffective
(b) r =
(c) r=
Number i
Number i
Number i
Units
Units
Units
A sodium atom (Z = 11) contains 11 protons in its nucleus. Strictly speaking, the Bohr model does not apply,
because the neutral atom contains 11 electrons instead of a single electron. However, we can apply the model to
the outermost electron as an approximation, provided that we use an effective value Zeffective rather than 11 for
the number of protons in the nucleus. (a) The ionization energy for the outermost electron in a sodium atom is
5.1 eV. Use the Bohr model with Z = Zeffective to calculate a value for Zeffective. (b) Using Z = 11, determine the
corresponding value for the radius r of the outermost Bohr orbit. (c) Using the value calculated for Zeffective in part
(a), determine the corresponding radius r of the outermost Bohr orbit.
(a) Zeffective = Number i 2.04
(b) _r=
(c)_r=
Number i
5.29E-11
Number i 2.12E-11
Units No units
Units
m
Units m
♥
A sodium atom (Z = 11) contains 11 protons in its nucleus. Strictly speaking, the Bohr model does not apply,
because the neutral atom contains 11 electrons instead of a single electron. However, we can apply the model to
the outermost electron as an approximation, provided that we use an effective value Zeffective rather than 11 for
the number of protons in the nucleus. (a) The ionization energy for the outermost electron in a sodium atom is
5.1 eV. Use the Bohr model with Z = Zeffective to calculate a value for Zeffective. (b) Using Z = 11, determine the
corresponding value for the radius r of the outermost Bohr orbit. (c) Using the value calculated for Zeffective in part
(a), determine the corresponding radius r of the outermost Bohr orbit.
(a) Zeffective = Number i
(b)_r=
(c)_r=
Number i
Number i
Units
Units
Units
Chapter 4 Solutions
Modern Physics for Scientists and Engineers
Ch. 4 - Prob. 1QCh. 4 - Prob. 2QCh. 4 - Prob. 3QCh. 4 - Prob. 4QCh. 4 - Prob. 5QCh. 4 - Prob. 6QCh. 4 - Prob. 7QCh. 4 - Prob. 8QCh. 4 - Prob. 9QCh. 4 - Prob. 10Q
Ch. 4 - Prob. 11QCh. 4 - Prob. 12QCh. 4 - Prob. 13QCh. 4 - Prob. 2PCh. 4 - Prob. 3PCh. 4 - Prob. 4PCh. 4 - Prob. 5PCh. 4 - Prob. 6PCh. 4 - Prob. 7PCh. 4 - What fraction of 5-MeV α particles will be...Ch. 4 - Prob. 9PCh. 4 - Prob. 10PCh. 4 - Prob. 11PCh. 4 - Prob. 12PCh. 4 - Prob. 13PCh. 4 - Prob. 14PCh. 4 - Prob. 15PCh. 4 - Prob. 16PCh. 4 - Prob. 17PCh. 4 - Prob. 18PCh. 4 - Prob. 19PCh. 4 - Prob. 20PCh. 4 - Prob. 21PCh. 4 - Prob. 22PCh. 4 - Prob. 23PCh. 4 - Prob. 24PCh. 4 - Prob. 25PCh. 4 - Prob. 26PCh. 4 - Prob. 27PCh. 4 - Prob. 28PCh. 4 - Prob. 29PCh. 4 - Prob. 30PCh. 4 - Prob. 31PCh. 4 - Prob. 32PCh. 4 - Prob. 33PCh. 4 - Prob. 34PCh. 4 - Prob. 35PCh. 4 - Prob. 36PCh. 4 - Prob. 37PCh. 4 - Prob. 38PCh. 4 - Prob. 39PCh. 4 - Prob. 40PCh. 4 - Prob. 41PCh. 4 - Prob. 42PCh. 4 - Prob. 43PCh. 4 - Prob. 44PCh. 4 - Prob. 45PCh. 4 - Prob. 46PCh. 4 - Prob. 47PCh. 4 - Prob. 48PCh. 4 - Prob. 49PCh. 4 - Prob. 50PCh. 4 - Prob. 51PCh. 4 - Prob. 52PCh. 4 - Prob. 54PCh. 4 - Prob. 55PCh. 4 - Prob. 56PCh. 4 - Prob. 57PCh. 4 - Prob. 59PCh. 4 - Prob. 60PCh. 4 - Prob. 61P
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Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Similar questions
- The Balmer series for hydrogen was discovered before either the Lyman or the Paschen series. Why?arrow_forwardThe wave function for an electron in a hydrogen atom is w = Ne. Here, r is the distance from the distance from the nucleus to the electron and a, is Bohr radius. Calculate the normalization factor, N. (A) (7a,)" (B) (1/ 7až) (C) (z/a¿)** (D) (1/ za}) 1/2 1/2 (C) (7 / a)arrow_forwardThe distance r between the electron and the nucleus in a hydrogenatom (in its lowest energy state) is a random variable with probability density p(r) = 4a03r2e- 2r/ao for r > 0, where ao is the Bohr radius (Figure 8). Calculate the probability P that the electron is within one Bohrradius of the nucleus. The value of a0 is approximately 5.29 x 10- 11 m,but this value is not needed to compute P.arrow_forward
- Show that the quantization of angular momentum L= nh follows from Bohr’s Assumption D that the mean value K of the kinetic energy of the electronnucleus system is given by K = nhforb/2. Assume a circular orbit.arrow_forwardIn a one electron system, the probability of finding the electron within a shell of thickness år at a radius of r from the nucleus is given by the radial distribution function, P(r) = ?R? (r). An electron in a 1s hydrogen orbital has the radial wavefunction R (r) given by 3/2 R(r) = 2 ela where a, is the Bohr radius (52.9 pm). Calculate the probability of finding the electron in a sphere of radius 0.80ao centered at the nucleus. 10.99 probability: Incorrectarrow_forwardIf, in 1 1 = Ry - you set ni = 1 and take n2 greater than 1, you generate what is known as the Lyman %3D series. Find the wavelength of the first mem- ber of this series. The value of ħ is 1.05457 × 10¬34 J.s; the Rydberg constant for hydrogen is 1.09735 × 10’ m¬'; the Bohr radius is 5.29177 × 10¬1" m; and the ground state energy for hydrogen is 13.6057 eV. Answer in units of nm. Consider the next three members of this se- ries. The wavelengths of successive members of the Lyman series approach a common limit as n2 → ∞. What is this limit? Answer in units of nm.arrow_forward
- (a) Show that the speed of an electron in the nth Bohr orbit of hydrogen is αc/n, where α is the fine structure constant, equal to e2/4πε0ħc. (b) What would be the speed in a hydrogen like atom with a nuclear charge of Ze?arrow_forwardAssuming that only a single electron is present and a Bohr model, calculate the mean radius, orbital velocity, and energy of a N=1 electron of hydrogen N=4 electron for lead N=1 electron for plutonium N=1 electron for an element with Z = 142arrow_forwardLook up the values of the quantities in aB = h2 / 4π2 me kqe2 ,and verify that the Bohr radius aB is 0.529 x 10-10 m .arrow_forward
- The energy of the n = 2 Bohr orbit is -30.6 eV for an unidentified ionized atom in which only one electron moves about the nucleus. What is the radius of the n = 3 orbit for this species? Number i Units >arrow_forwardIn a one electron system, the probability of finding the electron within a shell of thickness år at a radius of r from the nucleus is given by the radial distribution function, P(r) = = PR²(r). An electron in a 1s hydrogen orbital has the radial wavefunction R(r) given by 3/2 1 R(r) = 2 ao -rlao e where ao is the Bohr radius (52.9 pm). Calculate the probability of finding the electron in a sphere of radius 1.5ao centered at the nucleus. probability: %arrow_forwardThe gravitational attraction between electron and proton in a hydrogen atom is weaker than the coulomb attraction by a factor of about 10-40. An alternative way of looking at this fact is to estimate the radius of the first Bohr orbit of a hydrogen atom if the electron and proton were bound by gravitational attraction. You will find the answer interesting.arrow_forward
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