Solved: The lifetime of a typical excited state in an atom

The neutrons in a parallel beam, each having kinetic energy 0.025 eV, are directed through two slits 0.40 mm apart. How far apart will the interference peaks be on a screen 1.0 m away? [Hint: First find the wavelength of the neutron.]

Pellets of mass 2.0 g are fired in parallel paths with speeds of through a hole 3.0 mm in diameter. How far from the hole must you be to detect a 1.0-cmdiameter spread in the beam of pellets?

A proton is traveling with a speed of With what maximum precision can its position be ascertained?

If an electrons position can be measured to a precision of how precisely can its speed be known?

An electron remains in an excited state of an atom for typically What is the minimum uncertainty in the energy of the state (in eV)?

The boson, discovered in 1985, is the mediator of the weak nuclear force, and it typically decays very quickly. Its average rest energy is 91.19 GeV, but its short lifetime shows up as an intrinsic width of 2.5 GeV. What is the lifetime of this particle? [Hint: See Example 283.

What is the uncertainty in the mass of a muon specified in given its lifetime of 2.20 ms?

A free neutron has a mean life of 880 s. What is the uncertainty in its mass (in kg)?

An electron and a 140-g baseball are each traveling measured to a precision of 0.065%. Calculate and compare the uncertainty in position of each.

A radioactive element undergoes an alpha decay with a lifetime of If alpha particles are emitted with 5.5-MeV kinetic energy, find the percent uncertainty in the particle energy.

(II) If an electron’s position can be measured to a precision of 15 nm, what is the uncertainty in its speed? Assuming the minimum speed must be at least equal to its uncertainty, what is the electron’s minimum kinetic energy?

Estimate the lowest possible energy of a neutron contained in a typical nucleus of radius [Hint: Assume a particle can have an energy as large as its uncertainty.]

How precisely can the position of a 5.00-keV electron be measured assuming its energy is known to 1.00%?

Use the uncertainty principle to show that if an electron were present in the nucleus its kinetic energy (use relativity) would be hundreds of MeV. (Since such electron energies are not observed, we conclude that electrons are not present in the nucleus.) [Hint: Assume a particle can have an energy as large as its uncertainty.]

For what values can have?

For what are the possible values of

How many electrons can be in the subshell?

How many different states are possible for an electron whose principal quantum number is Write down the quantum numbers for each state

List the quantum numbers for each electron in the ground state of (a) carbon (b) aluminum (Z = 13)

List the quantum numbers for each electron in the ground state of oxygen (Z = 8).

Calculate the magnitude of the angular momentum of an electron in the state of hydrogen

(I) If a hydrogen atom has \(\ell = 4\), what are the possible values for n, \(m_{\ell}\), and \(m_s\)?

(II) If a hydrogen atom has \(m_{\ell} = -3\), what are the possible values of n, \(\ell\), and \(m_s\)?

Show that there can be 18 electrons in a g subshell

What is the full electron configuration in the ground state for elements with Z equal to (a) 26, (b) 34, (c) 38? [Hint: See the Periodic Table inside the back cover.]

What is the full electron configuration for (a) silver (Ag), (b) gold (Au), (c) uranium (U)? [Hint: See the Periodic Table inside the back cover.]

A hydrogen atom is in the 5d state. Determine (a) the principal quantum number, (b) the energy of the state, (c) the orbital angular momentum and its quantum number and (d) the possible values for the magnetic quantum number

Estimate the binding energy of the third electron in lithium using the Bohr model. [Hint: This electron has and sees a net charge of approximately ] The measured value is 5.36 eV

Show that the total angular momentum is zero for a filled subshell

For each of the following atomic transitions, state whether the transition is allowed or forbidden, and why: (a) (b) (c)

An electron has and is in its lowest possible energy state. What are the values of n and for this electron?

An excited H atom is in a 6d state. (a) Name all the states to which the atom is allowed to make a transition with the emission of a photon. (b) How many different wavelengths are there (ignoring fine structure)?

What are the shortest-wavelength X-rays emitted by electrons striking the face of a 28.5-kV TV picture tube? What are the longest wavelengths?

(I) If the shortest-wavelength bremsstrahlung X-rays emitted from an X-ray tube have \(\lambda = 0.035\ nm\), what is the voltage across the tube?

Show that the cutoff wavelength in an X-ray spectrum is given by where V is the X-ray tube voltage in volts

For the spectrum of X-rays emitted from a molybdenum target (Fig. 2811), determine the maximum and minimum energy

Use the result of Example 287 to estimate the X-ray wavelength emitted when a cobalt atom makes a transition from n = 2 n = 1.

Estimate the wavelength for an to transition in iron (Z = 26)

Use the Bohr model to estimate the wavelength for an to transition in molybdenum The measured value is 0.063 nm. Why do we not expect perfect agreement

(II) A mixture of iron and an unknown material is bombarded with electrons. The wavelengths of the \(k_{\alpha}\) lines are 194 pm for iron and 229 pm for the unknown. What is the unknown material?

A laser used to weld detached retinas puts out 25-ms-long pulses of 640-nm light which average 0.68-W output during a pulse. How much energy can be deposited per pulse and how many photons does each pulse contain? [Hint: See Example 274.]

A low-power laser used in a physics lab might have a power of 0.50 mW and a beam diameter of 3.0 mm. Calculate (a) the average light intensity of the laser beam, and (b) compare it to the intensity of a lightbulb producing 100-W light viewed from 2.0 m.

Calculate the wavelength of the HeNe laser (see Fig. 2820).

Estimate the angular spread of a laser beam due to diffraction if the beam emerges through a 3.0- mm-diameter mirror. Assume that What would be the diameter of this beam if it struck (a) a satellite 340 km above the Earth, or (b) the Moon? [Hint: See Sections 245 and 257.]

The magnitude of the orbital angular momentum in an excited state of hydrogen is and the z component is What are all the possible values of n, and for this state?

An electron in the n = 2 state of hydrogen remains there on average about \(10^{-8}\ s\) before jumping to the n = 1 state. (a) Estimate the uncertainty in the energy of the n = 2 state. (b) What fraction of the transition energy is this? (c)What is the wavelength, and width (in nm), of this line in the spectrum of hydrogen?

What are the largest and smallest possible values for the angular momentum L of an electron in the shell?

A 12-g bullet leaves a rifle at a speed of (a) What is the wavelength of this bullet? (b) If the position of the bullet is known to a precision of 0.60 cm (radius of the barrel), what is the minimum uncertainty in its momentum?

If an electrons position can be measured to a precision of what is the uncertainty in its momentum? Assuming its momentum must be at least equal to its uncertainty, estimate the electrons wavelength.

The ionization (binding) energy of the outermost electron in boron is 8.26 eV. (a) Use the Bohr model to estimate the effective charge, seen by this electron. (b) Estimate the average orbital radius.

Using the Bohr formula for the radius of an electron orbit, estimate the average distance from the nucleus for an electron in the innermost orbit of a uranium atom Approximately how much energy would be required to remove this innermost electron?

Protons are accelerated from rest across 480 V. They are then directed at two slits 0.70 mm apart. How far apart will the interference peaks be on a screen 28 m away?

How many electrons can there be in an h subshell?

(a) Show that the number of different states possible for a given value of is equal to (b) What is this number for 1, 2, 3, 4, 5, and 6?

Show that the number of different electron states possible for a given value of n is (See Problem 54.)

A beam of electrons with kinetic energy 45 keV is shot through two narrow slits in a barrier. The slits are a distance apart. If a screen is placed 45.0 cm behind the barrier, calculate the spacing between the bright fringes of the interference pattern produced on the screen

The angular momentum in the hydrogen atom is given both by the Bohr model and by quantum mechanics. Compare the results for n = 2.

The lifetime of a typical excited state in an atom is about 10 ns. Suppose an atom falls from one such excited state to a lower one, and emits a photon of wavelength about 500 nm. Find the fractional energy uncertainty and wavelength uncertainty of this photon.

A 1300-kg car is traveling with a speed of 2260.22) ms. With what maximum precision can its position be determined?

An atomic spectrum contains a line with a wavelength centered at 488 nm. Careful measurements show the line is really spread out between 487 and 489 nm. Estimate the lifetime of the excited state that produced this line.

An electron and a proton, each initially at rest, are accelerated across the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, find the ratio of the uncertainty in their momentum.

If the principal quantum number n were limited to the range from 1 to 6, how many elements would we find in nature?

If your de Broglie wavelength were 0.50 m, how fast would you be moving if your mass is 68.0 kg? Would you notice diffraction effects as you walk through a doorway? Approximately how long would it take you to walk through the doorway?

Suppose that the spectrum of an unknown element shows a series of lines with one out of every four matching a line from the Lyman series of hydrogen. Assuming that the unknown element is an ion with Z protons and one electron, determine Z and the element in question.

Photons of wavelength 0.154 nm are emitted from the surface of a certain metal when it is bombarded with highenergy radiation. If this photon wavelength corresponds to the line, what is the element?

Problem 1COQ The uncertainty principle states that (a) no measurement can be perfect because it is technologically impossible to make perfect measuring instruments. (b) it is impossible to measure exactly where a particle is, unless it is at rest. (c) it is impossible to simultaneously know both the position and the momentum of a particle with complete certainty. (d) a particle cannot actually have a completely certain value of momentum.

Problem 1MCQ An atom has the electron configuration 1s2 2s22P63s23P64s1 How many electrons does this atom have? (a) 15. (b) 19. (c) 30. (d) 46.

Problem 1P (II) The neutrons in a parallel beam, each having kinetic energy 0.025 eV, are directed through two slits 0.40 mm apart. How far apart will the interference peaks be on a screen 1.0 m away? [Hint: First find the wavelength of the neutron.]

Problem 1Q Compare a matter wave to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Problem 1SL Use the uncertainty principle to estimate the position uncertainty for the electron in the ground state of the hydrogen atom. [Hint: Determine the momentum using the Bohr model of Section 27–12 and assume the momentum can be anywhere between this value and zero.] How does this result compare to the Bohr radius?

Problem 2MCQ For the electron configuration of MisConceptual Question 1, what orbital quantum numbers do the electrons have? (a) 0. (d) 0 and 1 and 2 and 3. (b) 0 and 1. (e) 0 and 1 and 2 and 3 and 4. (c) 0 and 1 and 2

(II) Pellets of mass 2.0 g are fired in parallel paths with speeds of 120 m/s through a hole 3.0 mm in diameter. How far from the hole must you be to detect a 1.0-cm-diameter spread in the beam of pellets?

Problem 2Q Explain why Bohr’s theory of the atom is not compatible with quantum mechanics, particularly the uncertainty principle.

On what factors does the periodicity of the Periodic Table depend? Consider the exclusion principle, quantization of angular momentum, spin, and any others you can think of.

Problem 3MCQ If a beam of electrons is fired through a slit, (a) the electrons can be deflected because of their wave properties. (b) only electrons that hit the edge of the slit are deflected. (c) electrons can interact with electromagnetic waves in the slit, forming a diffraction pattern. (d) the probability of an electron making it through the slit depends on the uncertainty principle.

Problem 3P (I) A proton is traveling with a speed of (8.660 ± 0.012) X 105 m/s With what maximum precision can its position be ascertained? [Hint: ?P =m ? v]

Problem 3Q Explain why it is that the more massive an object is, the easier it becomes to predict its future position.

Problem 4MCQ What is meant by the ground state of an atom? (a) All of the quantum numbers have their lowest values (n =1, l =ml =0.) (b) The principal quantum number of the electrons in the outer shell is 1. (c) All of the electrons are in the lowest energy state, consistent with the exclusion principle. (d) The electrons are in the lowest state allowed by the uncertainty principle.

Problem 4Q In view of the uncertainty principle, why does a baseball seem to have a well-defined position and speed, whereas an electron does not?

Estimate (a) the quantum number \(\ell\) for the orbital angular momentum of the Earth about the Sun, and (b) the number of possible orientations for the plane of Earth’s orbit.

Problem 5MCQ The Pauli exclusion principle applies to all electrons (a) in the same shell, but not electrons in different shells. (b) in the same container of atoms. (c) in the same column of the Periodic Table. (d) in incomplete shells. (e) in the same atom.

Problem 5P (I) An electron remains in an excited state of an atom for Typically 10-8 What is the minimum uncertainty in the energy of the state (in eV)?

Problem 5Q Would it ever be possible to balance a very sharp needle precisely on its point? Explain.

Show that the diffraction spread of a laser beam, \(\approx \lambda /D\) (Section 28-11), is precisely what you might expect from the uncertainty principle. See also Chapters 24 and 25. [Hint: Since the beam’s width is constrained by the dimension of the aperture D, the component of the light’s momentum perpendicular to the laser axis is uncertain.]

Problem 6MCQ Which of the following is the best paraphrasing of the Heisenberg uncertainty principle? (a) Only if you know the exact position of a particle can you know the exact momentum of the particle. (b) The larger the momentum of a particle, the smaller the position of the particle. (c) The more precisely you know the position of a particle, the less well you can know the momentum of the particle. (d) The better you know the position of a particle, the better you can know the momentum of the particle. (e) How well you can determine the position and momentum of a particle depends on the particle’s quantum numbers.

(II) The \(Z^0\) boson, discovered in 1985, is the mediator of the weak nuclear force, and it typically decays very quickly. Its average rest energy is 91.19 GeV, but its short lifetime shows up as an intrinsic width of 2.5 GeV. What is the lifetime of this particle? [Hint: See Example 28–3.]

Problem 6Q A cold thermometer is placed in a hot bowl of soup. Will the temperature reading of the thermometer be the same as the temperature of the hot soup before the measurement was made? Explain.

Problem 6SL For noble gases, the halogens, and the alkali metals, explain the atomic structure that is common to each group and how that structure explains a common property of the group. (See Section 28–8.)

Problem 7P (II) What is the uncertainty in the mass of a muon (m =105.7 Me V/c2) , Specified eV/c2 in given its lifetime of 2.20 µs?

Problem 7Q Does the uncertainty principle set a limit to how well you can make any single measurement of position? Explain.

Imagine a line whose length equals the diameter of the smallest orbit in the Bohr model. If we are told only that an electron is located somewhere on this line, then the electron’s position can be specified as \(x=0 \pm r_1\), where the origin is at the line’s center and \(r_1\) is the Bohr radius. Such an electron can never be observed at rest, but instead at a minimum will have a speed somewhere in the range from \(v=0-\Delta v\) to \(0+\Delta v\). Determine \(\Delta v\).

Problem 8P (II) A free neutron (m =1.67 X 10-27 kg) has a mean life of 880 s. What is the uncertainty in its mass (in kg)?

Problem 8SL What is uncertain in the Heisenberg uncertainty principle? Explain. (See Section 28–3.)

Problem 9P (II) An electron and a 140-g baseball are each traveling 120 m/s measured to a precision of 0.065%. Calculate and compare the uncertainty in position of each.

Problem 9Q When you check the pressure in a tire, doesn’t some air inevitably escape? Is it possible to avoid this escape of air altogether?What is the relation to the uncertainty principle?

Problem 10MCQ Which of the following is not always a property of lasers? (a) All of the photons in laser light have the same phase. (b) All laser photons have nearly identical frequencies. (c) Laser light moves as a beam, spreading out very slowly. (d) Laser light is always brighter than other sources of light. (e) Lasers depend on an inverted population of atoms where more atoms occupy a higher energy state than some lower energy state.

Problem 10P (II) A radioactive element undergoes an alpha decay with a lifetime of 12 µs. If alpha particles are emitted with 5.5-MeV kinetic energy, find the percent uncertainty ?E/E in the particle energy.

Problem 10Q It has been said that the ground-state energy in the hydrogen atom can be precisely known but the excited states have some uncertainty in their values (an “energy width”). Is this consistent with the uncertainty principle in its energy form? Explain.

Problem 11P (II) If an electron’s position can be measured to a precision of 15 nm, what is the uncertainty in its speed? Assuming the minimum speed must be at least equal to its uncertainty, what is the electron’s minimum kinetic energy?

Problem 11Q Which model of the hydrogen atom, the Bohr model or the quantum-mechanical model, predicts that the electron spends more time near the nucleus? Explain.

Problem 12P (II) Estimate the lowest possible energy of a neutron contained in a typical nucleus of radius 1.2 X 10-15 m [Hint: Assume a particle can have an energy as large as its uncertainty.]

Problem 12Q The size of atoms varies by only a factor of three or so, from largest to smallest, yet the number of electrons varies from one to over 100. Explain.

Problem 13P (III) How precisely can the position of a 5.00-keV electron be measured assuming its energy is known to 1.00%?

Problem 13Q Excited hydrogen and excited helium atoms both radiate light as they jump down to n = 1, l = 0, ml = 0 the state. Why do the two elements have very different emission spectra?

Problem 14P (III) Use the uncertainty principle to show that if an electron were present in the nucleus (r =10-15 m) its kinetic energy (use relativity) would be hundreds of MeV. (Since such electron energies are not observed, we conclude that electrons are not present in the nucleus.) [Hint: Assume a particle can have an energy as large as its uncertainty.]

Problem 14Q How would the Periodic Table look if there were no electron spin but otherwise quantum mechanics were valid? Consider the first 20 elements or so.

Which of the following electron configurations are not allowed: () \(1 s^{2} 2 s^{2} 2 p^{4} 3 s^{2} 4 p^{2}\) () \(1 s^{2} 2 s^{2} 2 p^{8} 3 s^{1}\) () \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{5} 4 s^{2} 4 d^{5} 4 f^{1}\) If not allowed, explain why. Equation Transcription: Text Transcription: 1s^2 2s^2 2p^4 3s^2 4p^2 1s^2 2s^2 2p^8 3s^1 1s^2 2s^2 2p^6 3s^2 3p^5 4s^2 4d^5 4f^1

Problem 16P (I) For n=6, l =3, what are the possible values of ml and ms?

In what column of the Periodic Table would you expect to find the atom with each of the following configurations: () \(1 s^{2} 2 s^{2} 2 p^{5}\) () \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2}\) () \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6}\) () \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{1}\)? Equation Transcription: Text Transcription: 1s^2 2s^2 2p^5 1s^2 2s^2 2p^6 3s 1s^2 2s^2 2p^6 3s^2 3p^6 1s^2 2s^2 2p^6 3s^2 3p^6 4s^1

Problem 17P (I) How many electrons can be in the n = 5,l =3 subshell?

Problem 17Q Why do chlorine and iodine exhibit similar properties?

Problem 18P (I) How many different states are possible for an electron whose principal quantum number is n =4? Write down the quantum numbers for each state.

Problem 18Q Explain why potassium and sodium exhibit similar properties.

Problem 19P (I) List the quantum numbers for each electron in the ground state of (a) carbon (Z =6), (b) aluminum (Z =13).

Problem 19Q Why are the chemical properties of the rare earths so similar? [Hint: Examine the Periodic Table.]

Problem 20P (I) List the quantum numbers for each electron in the ground state of oxygen (Z =8).

Problem 20Q The ionization energy for neon(Z = 10) is 21.6 eV, and that for sodium(Z = 11) is 5.1 eV. Explain the large difference.

(I) Calculate the magnitude of the angular momentum of an electron in the n = 5, \(\ell = 3\) state of hydrogen.

Problem 21Q Why do we expect electron transitions deep within an atom to produce shorter wavelengths than transitions by outer electrons?

Problem 22P (I) If a hydrogen atom has l=4, what are the possible values for n,ml, and ms?

Problem 22Q Does the Bohr model of the atom violate the uncertainty principle? Explain.

Problem 23P (II) If a hydrogen atom has ml =-3, what are the possible values of n, l and ms?

Problem 23Q Briefly explain why noble gases are nonreactive and why alkali metals are highly reactive. (See Section 28–8.)

Problem 24P (II) Show that there can be 18 electrons in a “g” subshell

Problem 24Q Compare spontaneous emission to stimulated emission.

Problem 25P (II) What is the full electron configuration in the ground state for elements with Z equal to (a) 26, (b) 34, (c) 38? [Hint: See the Periodic Table inside the back cover.]

How does laser light differ from ordinary light? How is it the same?

Explain how a 0.0005-W laser beam, photographed at a distance, can seem much stronger than a 1000-W street lamp at the same distance.

Problem 26P (II) What is the full electron configuration for (a) silver (Ag), (b) gold (Au), (c) uranium (U)? [Hint: See the Periodic Table inside the back cover.]

Problem 27P (II) A hydrogen atom is in the 5d state. Determine (a) the principal quantum number, (b) the energy of the state, (c) the orbital angular momentum and its quantum number l, and (d) the possible values for the magnetic quantum number.

Problem 27Q Does the intensity of light from a laser fall off as the inverse square of the distance? Explain.

Problem 28EA Return to the Chapter-Opening Question, page 803, and answer it again now. Try to explain why you may have answered differently the first time. CHAPTER-OPENING QUESTION---Guess now! The uncertainty principle states that 1. No measurement can be perfect because it is technologically impossible to make perfect measuring instruments. 2. It is impossible to measure exactly where a particle is, unless it is at rest. 3. It is impossible to simultaneously know both the position and the momentum of pa particle with complete certainty. 4. A particle cannot actually have a completely certain value of momentum.

An electron’s position is measured with a precision of \(0.50 \times 10^{-10}\ m\). Find the minimum uncertainty in its momentum and velocity.

An electron has \(n=4, \ell=2\). Which of the following values of \(m_{\ell}\) are possible: \(4,3,2,1,0,-1,-2,-3,-4\)? Equation Transcription: Text Transcription: n=4,\ell=2 m\ell 4,3,2,1,0,-1,-2,-3,-4

Construct a Table of the ground-state quantum numbers for beryllium, \(z=4\) (like those in Table 28–2). Equation Transcription: Text Transcription: z=4

What are the energy and angular momentum of the electron in a hydrogen atom with \(n=6, \ell=4\)? Equation Transcription: Text Transcription: n=6, \ell=4

Problem 28EF Write the complete ground-state configuration for gallium, with its 31 electrons.

(II) Estimate the binding energy of the third electron in lithium using the Bohr model. [Hint: This electron has n = 2 and “sees” a net charge of approximately +1e.] The measured value is 5.36 eV.

Why does the cutoff wavelength in Fig. 28–11 imply a photon nature for light?

Problem 29P (II) Show that the total angular momentum is zero for a filled subshell.

Problem 29Q Why do we not expect perfect agreement between measured values of characteristic X-ray line wavelengths and those calculated using the Bohr model, as in Example 28–7?

\(\text { (II) }\) For each of the following atomic transitions, state whether the transition is allowed or forbidden, and why: () \(4 p \rightarrow 3 p\) () \(3 p \rightarrow 1 s\) () \(4 d \rightarrow 2 d\) (d) \(5 d \rightarrow 3 s\) () \(4 s \rightarrow 2 p\) Equation Transcription: Text Transcription: (II) 4p \rightarrow 3p 5p \rightarrow 1s 4d \rightarrow 2d 5d \rightarrow 3s 4s \rightarrow 2p

Problem 31P (II) An electron has ml =2 and is in its lowest possible energy state. What are the values of nand l for this electron?

Problem 32P (II) An excited H atom is in a 6d state. (a) Name all the States (n,l) to which the atom is “allowed” to make a transition with the emission of a photon. (b) How many different wavelengths are there (ignoring fine structure)?

Problem 33P (I) What are the shortest-wavelength X-rays emitted by electrons striking the face of a 28.5-kV TV picture tube? What are the longest wavelengths?

Problem 34P (I) If the shortest-wavelength bremsstrahlung X-rays emitted from an X-ray tube have ? = 0.035 nm, what is the voltage across the tube?

(I) Show that the cutoff wavelength \(\lambda_{0}\) in an X-ray spectrum is given by \(\lambda_{0}=\frac{1240 \mathrm{~nm}}{V}\) where \(V\) is the X-ray tube voltage in volts. Equation Transcription: Text Transcription: \lambda_{0} \lambda_0=\frac 1240 ~nm V V

\(\text { (I) }\) For the spectrum of \(\text { X-rays }\) emitted from a molybdenum target (Fig. 28–11), determine the maximum and minimum energy. Equation Transcription: Text Transcription: (I) X-rays

Problem 37P (II) Use the result of Example 28–7 (Z =42) to estimate the X-ray wavelength emitted when a cobalt atom (Z =27) makes a transition from n =2 to n =1.

Problem 38P (II) Estimate the wavelength for an n =3 to n =2 transition in iron (Z =26)

Problem 40P (II) A mixture of iron and an unknown material is bombarded with electrons. The wavelengths of the Ka lines are 194 pm for iron and 229 pm for the unknown. What is the unknown material?

\(\text { (II) }\) A laser used to weld detached retinas puts out \(25 \text { - ms - long }\) pulses of \(640-n m\) light which average \(0.68-\mathrm{W}\) output during a pulse. How much energy can be deposited per pulse and how many photons does each pulse contain? [Hint: See Example 27–4.] Equation Transcription: Text Transcription: (II) 25-ms-long 640-nm 0.68-W

Problem 42P (II) A low-power laser used in a physics lab might have a power of 0.50 mW and a beam diameter of 3.0 mm. Calculate (a) the average light intensity of the laser beam, and (b) compare it to the intensity of a lightbulb producing 100-W light viewed from 2.0 m.

Problem 44P (II) Estimate the angular spread of a laser beam due to diffraction if the beam emerges through a 3.0-mm-diameter mirror. Assume that ? =694 nm. What would be the diameter of this beam if it struck (a) a satellite 340 km above the Earth, or (b) the Moon? [Hint: See Sections 24–5 and 25–7.]

\(\text { (II) }\) Calculate the wavelength of the \(H e-N e\) laser (see Fig. 28–20). Equation Transcription: Text Transcription: (II) He-Ne

Problem 46GP An electron in the n = 2 state of hydrogen remains thereon average about 10-8 s before jumping to the n = 1 state.(a) Estimate the uncertainty in the energy of the n = 2state. (b) What fraction of the transition energy is this?(c) What is the wavelength, and width (in nm), of this linein the spectrum of hydrogen?

Problem 47GP What are the largest and smallest possible values for theangular momentum L of an electron in the n = 6 shell?

Problem 48GP A 12-g bullet leaves a rifle at a speed of 150m/s. (a) Whatis the wavelength of this bullet? (b) If the position of thebullet is known to a precision of 0.60 cm (radius of thebarrel), what is the minimum uncertainty in its momentum?

Problem 49GP If an electron’s position can be measured to a precision of2.0 X 10–8 m, what is the uncertainty in its momentum?Assuming its momentum must be at least equal to itsuncertainty, estimate the electron’s wavelength.

The ionization (binding) energy of the outermost electron in boron is 8.26 eV. (a) Use the Bohr model to estimate the “effective charge,” \(Z_{eff}\), seen by this electron. (b) Estimate the average orbital radius.

Problem 51GP Using the Bohr formula for the radius of an electron orbit,estimate the average distance from the nucleus for anelectron in the innermost (n = 1) orbit of a uranium atom(Z = 92). Approximately how much energy would berequired to remove this innermost electron?

Protons are accelerated from rest across 480 V. They are then directed at two slits 0.70 mm apart. How far apart will the interference peaks be on a screen 28 m away?

Problem 53GP How many electrons can there be in an “h” subshell?

Problem 54GP (a) Show that the number of different states possible for agiven value of l is equal to 2(2l + 1). (b) What is thisnumber for l = 0, 1, 2, 3, 4, 5, and 6?

Show that the number of different electron states possible for a given value of \(n \text { is } 2 n^{2}\). (See Problem 54.) Equation Transcription: Text Transcription: n is 2n2

Problem 56GP A beam of electrons with kinetic energy 45 keV is shotthrough two narrow slits in a barrier. The slits are a distance 2.0 X 10–6 m apart. If a screen is placed 45.0 cm behindthe barrier, calculate the spacing between the “bright”fringes of the interference pattern produced on the screen.

Problem 57GP The angular momentum in the hydrogen atom is givenboth by the Bohr model and by quantum mechanics.Compare the results for n = 2.

Problem 58GP The lifetime of a typical excited state in an atom is about10 ns. Suppose an atom falls from one such excited stateto a lower one, and emits a photon of wavelength about500 nm. Find the fractional energy uncertainty ?E/E andwavelength uncertainty ??/? of this photon.

Problem 59GP A 1300-kg car is traveling with a speed of (22 ± 0.22) m/s.With what maximum precision can its position be determined?

Problem 60GP An atomic spectrum contains a line with a wavelengthcentered at 488 nm. Careful measurements show the line isreally spread out between 487 and 489 nm. Estimate thelifetime of the excited state that produced this line.

Problem 61GP An electron and a proton, each initially at rest, are accelerated across the same voltage. Assuming that the uncertaintyin their position is given by their de Broglie wavelength,find the ratio of the uncertainty in their momentum.

Problem 62GP If the principal quantum number n were limited to the rangefrom 1 to 6, how many elements would we find in nature?

Problem 63GP If your de Broglie wavelength were 0.50 m, how fast wouldyou be moving if your mass is 68.0 kg? Would you noticediffraction effects as you walk through a doorway? Approximately how long would it take you to walk through thedoorway?

Problem 64GP Suppose that the spectrum of an unknown element showsa series of lines with one out of every four matching a linefrom the Lyman series of hydrogen. Assuming that theunknown element is an ion with Z protons and oneelectron, determine Z and the element in question.

Problem 65GP Photons of wavelength 0.154 nm are emitted from thesurface of a certain metal when it is bombarded with highenergy radiation. If this photon wavelength corresponds tothe K? line, what is the element?

The neutrons in a parallel beam, each having kinetic energy 0.025 eV, are directed through two slits 0.40 mm apart. How far apart will the interference peaks be on a screen 1.0 m away? [Hint: First find the wavelength of the neutron.]

Pellets of mass 2.0 g are fired in parallel paths with speeds of through a hole 3.0 mm in diameter. How far from the hole must you be to detect a 1.0-cmdiameter spread in the beam of pellets?

A proton is traveling with a speed of With what maximum precision can its position be ascertained?

(I) If an electron’s position can be measured to a precision of \(2.4 \times 10^{-8}\ m\), how precisely can its speed be known?

An electron remains in an excited state of an atom for typically What is the minimum uncertainty in the energy of the state (in eV)?

The boson, discovered in 1985, is the mediator of the weak nuclear force, and it typically decays very quickly. Its average rest energy is 91.19 GeV, but its short lifetime shows up as an intrinsic width of 2.5 GeV. What is the lifetime of this particle? [Hint: See Example 283.

What is the uncertainty in the mass of a muon specified in given its lifetime of 2.20 ms?

A free neutron has a mean life of 880 s. What is the uncertainty in its mass (in kg)?

An electron and a 140-g baseball are each traveling measured to a precision of 0.065%. Calculate and compare the uncertainty in position of each.

A radioactive element undergoes an alpha decay with a lifetime of If alpha particles are emitted with 5.5-MeV kinetic energy, find the percent uncertainty in the particle energy.

If an electrons position can be measured to a precision of 15 nm, what is the uncertainty in its speed? Assuming the minimum speed must be at least equal to its uncertainty, what is the electrons minimum kinetic energy?

Estimate the lowest possible energy of a neutron contained in a typical nucleus of radius [Hint: Assume a particle can have an energy as large as its uncertainty.]

How precisely can the position of a 5.00-keV electron be measured assuming its energy is known to 1.00%?

Use the uncertainty principle to show that if an electron were present in the nucleus its kinetic energy (use relativity) would be hundreds of MeV. (Since such electron energies are not observed, we conclude that electrons are not present in the nucleus.) [Hint: Assume a particle can have an energy as large as its uncertainty.]

(I) For n = 6, what values can \(\ell\) have?

For what are the possible values of

How many electrons can be in the subshell?

How many different states are possible for an electron whose principal quantum number is Write down the quantum numbers for each state

List the quantum numbers for each electron in the ground state of (a) carbon (b) aluminum (Z = 13)

List the quantum numbers for each electron in the ground state of oxygen (Z = 8).

Calculate the magnitude of the angular momentum of an electron in the state of hydrogen

(I) If a hydrogen atom has \(\ell=4\), what are the possible values for \(n, m_{\ell}\), and \(m_s\)?

(II) If a hydrogen atom has \(m_{\ell}=-3\) what are the possible values of \(n, \ell\) and \(m_s\)?

Show that there can be 18 electrons in a g subshell

What is the full electron configuration in the ground state for elements with Z equal to (a) 26, (b) 34, (c) 38? [Hint: See the Periodic Table inside the back cover.]

What is the full electron configuration for (a) silver (Ag), (b) gold (Au), (c) uranium (U)? [Hint: See the Periodic Table inside the back cover.]

A hydrogen atom is in the 5d state. Determine (a) the principal quantum number, (b) the energy of the state, (c) the orbital angular momentum and its quantum number and (d) the possible values for the magnetic quantum number

Estimate the binding energy of the third electron in lithium using the Bohr model. [Hint: This electron has and sees a net charge of approximately ] The measured value is 5.36 eV

Show that the total angular momentum is zero for a filled subshell

For each of the following atomic transitions, state whether the transition is allowed or forbidden, and why: (a) (b) (c)

An electron has and is in its lowest possible energy state. What are the values of n and for this electron?

An excited H atom is in a 6d state. (a) Name all the states to which the atom is allowed to make a transition with the emission of a photon. (b) How many different wavelengths are there (ignoring fine structure)?

What are the shortest-wavelength X-rays emitted by electrons striking the face of a 28.5-kV TV picture tube? What are the longest wavelengths?

If the shortest-wavelength bremsstrahlung X-rays emitted from an X-ray tube have what is the voltage across the tube?

Show that the cutoff wavelength in an X-ray spectrum is given by where V is the X-ray tube voltage in volts

For the spectrum of X-rays emitted from a molybdenum target (Fig. 2811), determine the maximum and minimum energy

Use the result of Example 287 to estimate the X-ray wavelength emitted when a cobalt atom makes a transition from n = 2 n = 1.

Estimate the wavelength for an to transition in iron (Z = 26)

Use the Bohr model to estimate the wavelength for an to transition in molybdenum The measured value is 0.063 nm. Why do we not expect perfect agreement

A mixture of iron and an unknown material is bombarded with electrons. The wavelengths of the lines are 194 pm for iron and 229 pm for the unknown. What is the unknown material?

A laser used to weld detached retinas puts out 25-ms-long pulses of 640-nm light which average 0.68-W output during a pulse. How much energy can be deposited per pulse and how many photons does each pulse contain? [Hint: See Example 274.]

A low-power laser used in a physics lab might have a power of 0.50 mW and a beam diameter of 3.0 mm. Calculate (a) the average light intensity of the laser beam, and (b) compare it to the intensity of a lightbulb producing 100-W light viewed from 2.0 m.

Calculate the wavelength of the HeNe laser (see Fig. 2820).

Estimate the angular spread of a laser beam due to diffraction if the beam emerges through a 3.0-mm-diameter mirror. Assume that What would be the diameter of this beam if it struck (a) a satellite 340 km above the Earth, or (b) the Moon? [Hint: See Sections 245 and 257.]

The magnitude of the orbital angular momentum in an excited state of hydrogen is and the z component is What are all the possible values of n, and for this state?

An electron in the state of hydrogen remains there on average about before jumping to the state. (a) Estimate the uncertainty in the energy of the state. (b) What fraction of the transition energy is this? (c) What is the wavelength, and width (in nm), of this line in the spectrum of hydrogen?

What are the largest and smallest possible values for the angular momentum L of an electron in the n = 6 shell?

A 12-g bullet leaves a rifle at a speed of (a) What is the wavelength of this bullet? (b) If the position of the bullet is known to a precision of 0.60 cm (radius of the barrel), what is the minimum uncertainty in its momentum?

If an electrons position can be measured to a precision of what is the uncertainty in its momentum? Assuming its momentum must be at least equal to its uncertainty, estimate the electrons wavelength.

The ionization (binding) energy of the outermost electron in boron is 8.26 eV. (a) Use the Bohr model to estimate the effective charge, seen by this electron. (b) Estimate the average orbital radius.

Using the Bohr formula for the radius of an electron orbit, estimate the average distance from the nucleus for an electron in the innermost orbit of a uranium atom Approximately how much energy would be required to remove this innermost electron?

Protons are accelerated from rest across 480 V. They are then directed at two slits 0.70 mm apart. How far apart will the interference peaks be on a screen 28 m away?

How many electrons can there be in an “h” subshell?

(a) Show that the number of different states possible for a given value of is equal to (b) What is this number for 1, 2, 3, 4, 5, and 6?

Show that the number of different electron states possible for a given value of n is (See Problem 54.)

A beam of electrons with kinetic energy 45 keV is shot through two narrow slits in a barrier. The slits are a distance apart. If a screen is placed 45.0 cm behind the barrier, calculate the spacing between the bright fringes of the interference pattern produced on the screen

The angular momentum in the hydrogen atom is given both by the Bohr model and by quantum mechanics. Compare the results for n = 2.

The lifetime of a typical excited state in an atom is about 10 ns. Suppose an atom falls from one such excited state to a lower one, and emits a photon of wavelength about 500 nm. Find the fractional energy uncertainty and wavelength uncertainty of this photon.

A 1300-kg car is traveling with a speed of 2260.22) ms. With what maximum precision can its position be determined?

An atomic spectrum contains a line with a wavelength centered at 488 nm. Careful measurements show the line is really spread out between 487 and 489 nm. Estimate the lifetime of the excited state that produced this line.

An electron and a proton, each initially at rest, are accelerated across the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, find the ratio of the uncertainty in their momentum

If the principal quantum number n were limited to the range from 1 to 6, how many elements would we find in nature?

If your de Broglie wavelength were 0.50 m, how fast would you be moving if your mass is 68.0 kg? Would you notice diffraction effects as you walk through a doorway? Approximately how long would it take you to walk through the doorway?

Suppose that the spectrum of an unknown element shows a series of lines with one out of every four matching a line from the Lyman series of hydrogen. Assuming that the unknown element is an ion with Z protons and one electron, determine Z and the element in question.

Photons of wavelength 0.154 nm are emitted from the surface of a certain metal when it is bombarded with highenergy radiation. If this photon wavelength corresponds to the line, what is the element?