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Chapter 12: Problem 20

In the Bohr model of the atom, which of the following statements are false? (More than one response permitted.) (A) The energy levels are spaced apart by \(h v\). (B) The angular momentum of the electron is quantized. (C) The least amount of energy that can be absorbed by the atom is between the \(n=2\) state and the \(n=3\) state. (D) The energy of a photon emitted in the Balmer series is proportional to \(\left(\frac{1}{4}-\frac{1}{n^2}\right)\) where \(n\) is the principal quantum number of the initial energy level. (E) The energy levels characterized by the quantum number \(n\) are stable.

Short Answer

Expert verified
A. The energy levels are spaced apart by quantities such as \(h v\), where \(h\) is Planck's constant and \(v\) is frequency. B. The angular momentum of the electron is quantized, having only certain discrete values. C. The least amount of energy that can be absorbed by the atom occurs between the \(n=2\) state and the \(n=3\) state. Answer: Both statements (A) and (C) are false according to the Bohr model of the atom.

Step by step solution

01

Analyze statement (A)

According to the Bohr model of the atom, the energy levels are not spaced apart by \(h v\), where \(h\) is Planck's constant and \(v\) is the frequency. The energy levels in the hydrogen atom are described by the formula \(E_n = -\frac{13.6 eV}{n^2}\), where \(n\) is the principal quantum number. This statement is false.
02

Analyze statement (B)

The angular momentum of the electron in the Bohr model is quantized, which means it can only have certain discrete values. The quantization of angular momentum is given by \(L = n \hbar\), where \(n\) is a positive integer and \(\hbar\) is the reduced Planck constant. This statement is true.
03

Analyze statement (C)

The least amount of energy that can be absorbed by the atom is the energy difference between two adjacent energy levels. The smallest energy difference occurs between the \(n=1\) state and the \(n=2\) state, not between the \(n=2\) state and the \(n=3\) state as mentioned in the statement. Therefore, this statement is false.
04

Analyze statement (D)

The energy of a photon emitted in the Balmer series is proportional to \(\left(\frac{1}{4} - \frac{1}{n^2}\right)\), where \(n\) is the principal quantum number of the initial energy state. The Balmer series represents the transitions of electrons from energy levels with \(n > 2\) to the \(n=2\) level. This formula is correct according to the Bohr model, so this statement is true.
05

Analyze statement (E)

The energy levels characterized by the quantum number \(n\) in the Bohr model are considered stable because the electron does not radiate energy or lose it as it orbits the nucleus. However, when the electron absorbs or emits energy and changes its energy level, the states are no longer stable. This statement is true. Finally, we can conclude that statements (A) and (C) are false according to the Bohr model of the atom.

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Most popular questions from this chapter

An atomic nucleus is induced to break into two pieces, during which process energy is released. One can say with certainty that (A) the original atomic mass was greater than that of iron. (B) the fragments will each have an atomic mass less than that of iron. (C) the masses of the fragments will add up to be less than the mass of the original nucleus. (D) (A) and (C) (E) None of the above

A photon of energy \(1.75 \mathrm{eV}\) has a wavelength of most nearly (A) \(710 \mathrm{~nm}\) (B) \(700 \mathrm{~nm}\) (C) \(650 \mathrm{~nm}\) (D) \(600 \mathrm{~nm}\) (E) \(550 \mathrm{~nm}\)

Meteorites created in the early solar system contaned aluminum-26, which is a radioactive isotope of aluminum with a half-life of \(7.2 \times 10^5 \mathrm{yrs}\). Aluminum-26 decays first into an excited state of magnesium-26 via the reaction \({ }_{13}^{26} \mathrm{Al} \rightarrow{ }_{12}^{26} \mathrm{Mg}^*+\mathrm{e}^{+}\), where the \(e^{+}\)has energy \(2.99 \mathrm{MeV}\). (The \(e^{+}\)is a positron; see previous problem. The asterisk (*) indicates "excited.") The \({ }_{12}^{26} \mathrm{Mg}^*\) then decays into the stable isotope magnesium- 26 via the reaction \({ }_{12}^{26} \mathrm{Mg}^* \rightarrow{ }_{12}^{26} \mathrm{Mg}+\gamma\). The \(\gamma\) has energy \(1.8 \mathrm{MeV}\). a) If you were asked to calculate the de Broglie wavelength of the positron, would it be permissible to use Newtonian physics? Justify your answer. b) What is the wavelength of the photon emitted when the excited magnesium-26 decays into its ground state? What is its momentum? c) \({ }_{12}^{26} \mathrm{Mg}\) has an atomic mass of \(25.9826 \mathrm{u}\). What is the speed of the recoiling nucleus when the photon is emitted? d) What is the nucleus' kinetic energy in electron volts? e) Precise measurements indicate that for a certain meteorite \(A\) the present ratio \({ }^{26} \mathrm{Mg} /{ }^{27} \mathrm{Al}=5 \times 10^{-5}\), where \({ }^{27} \mathrm{Al}\) is the common, stable isotope of aluminum. For a meteorite \(B\) the ratio is \({ }^{26} \mathrm{Mg} /{ }^{27} \mathrm{Al}=1.55 \times 10^{-7}\). Assuming that the different ratios are due to the difference in the meteorites' times of creation, how much older is meteorite \(B\) than \(A\) ?

Measurements of the spectrum of a certain atom show that one sequence of spectral lines has the identical pattern of the Balmer series except that all the frequencies are exactly four times as high. The most likely explanation for the observation is that (A) one is observing a series of lines in hydrogen that have the ground state \(n=1\) instead of \(n=2\). (B) the atom is helium, with atomic number two. (C) the atom is beryllium with atomic number four. (D) the atom is deuterium with atomic mass two. (E) the atom is hydrogen but excited to energy levels that are four times higher than in the Balmer series.

A virus has a mass of about \(10^{-18} \mathrm{~kg}\). The de Broglie wavelength of a virus being blown in the wind at \(10 \mathrm{~m} / \mathrm{s}\) is nearest to (A) one billionth the size of a hydrogen atom. (B) one millionth the size of a hydrogen atom. (C) one thousandth the size of a hydrogen atom. (D) the size of a hydrogen atom. (E) one thousand times the size of a hydrogen atom.
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