The wavelengths of the photons emitted by a potassium atom

For the hydrogen atom, as increases, does the spacing of adjacent energy levels on an energy-level diagram increase or decrease?

The energy of the ground state of doubly ionized lithium is _______, where (a) (b) (c) (d)

Bohrs quantum condition on electron orbits requires (a) that the orbital angular momentum of the electron about the hydrogen nucleus equals where is an integer, (b) that no more than one electron occupy a given state, (c) that the electrons spiral into the nucleus while radiating electromagnetic waves, (d) that the energies of an electron in a hydrogen atom be equal to where is a constant and is an integer, (e) none of the above.

According to the Bohr model, if an electron moves to a larger orbit, does the electrons total energy increase or decrease? Does the electrons kinetic energy increase or decrease?

According to the Bohr model, the kinetic energy of the electron in the ground state of hydrogen is where The kinetic energy of the electron in the state is (a) (b) (c) (d)

According to the Bohr model, the radius of the n=1 orbit in the hydrogen atom is \(a_0=0.053 \mathrm{~nm}\). What is the radius of the n=5 orbit? (a) \(25 a_{0^{\prime}}\) (b) \(5 a_{0^{\prime}}\) (c) \(a_{0^{\prime}}\) (d) \(a_0 / 5\), (e) \(a_0 / 25\).

For the principal quantum number how many different values can the orbital quantum number have? (a) 4, (b) 3, (c) 7, (d) 16, (e) 25

For the principal quantum number how many different combinations of and can occur? (a) 4, (b) 3, (c) 7, (d) 16, (e) 25

Why is the energy of the 3s state considerably lower than the energy of the 3p state for sodium, whereas in hydrogen 3s and 3p states have essentially the same energy?

The d state of an electron configuration corresponds to (a) (b) (c) (d) (e)

Why are three quantum numbers inadequate to describe the states of the electrons in atoms that have more than one electron?

Group the following six atomspotassium, calcium, titanium, chromium, manganese, and copperaccording to their ground-state electron configurations for the n _ 4 shell.

What element has the electron configuration (a) and (b)

For the principal quantum number what are the possible combinations of the quantum numbers and

An electron in the shell means that the electron is represented by (a) (b) (c) (d) or (e)

The Bohr model and the quantum-mechanical model of the hydrogen atom give the same results for the energy levels. Discuss the advantages and disadvantages of each model.

The Sommerfeld–Hosser displacement theorem states that the optical spectrum of any atom is very similar to the spectrum of the singly charged positive ion of the element immediately following it in the periodic table. Discuss why this theorem is accurate.

Using the triplet of numbers to represent an electron that has the principal quantum number orbital quantum number and magnetic quantum number which of the following transitions is allowed? (a) (b) (c) (d) (e)

The Ritz combination principle states that for any atom, one can find different spectral lines and so that Show why this is true using an energylevel diagram.

(a) We can define a thermal de Broglie wavelength for an atom in a gas at temperature as being the de Broglie wavelength for an atom moving at the rms speed appropriate to that temperature. (The average kinetic energy of an atom is equal to where is the Boltzmann constant. Use this value to calculate the rms speed of the atoms.) Show that where is the mass of the atom. (b) Cooled atoms can form a Bose condensate (a new state of matter) when their thermal de Broglie wavelength becomes larger than the average interatomic spacing. From this criterion, estimate the temperature needed to create a Bose condensate in a gas of whose number density is

In laser cooling and trapping, a beam of atoms traveling in one direction is slowed by interaction with an intense laser beam in the opposite direction. The photons scatter off the atoms by resonance absorption, a process by which the incident photon is absorbed by the atom, and a short time later a photon of equal energy is emitted in a random direction. The net result of a single such scattering event is a transfer of momentum to the atom in a direction opposite to the motion of the atom, followed by a second transfer of momentum to the atom in a random direction. Thus, during photon absorption the atom loses speed, but during photon emission the change in speed of the atom is, on average, zero (because the directions of the emitted photons are random). An analogy often made to this process is that of slowing down a bowling ball by bouncing ping-pong balls off of it. (a) Given that the typical photon energy used in these experiments is about 1 eV, and that the typical kinetic energy of an atom in the beam is the typical kinetic energy of the atoms in a gas that has a temperature of about 500 K (a typical temperature for an oven that produces an atomic beam), estimate the number of photonatom collisions that are required to bring an atom to rest. (The average kinetic energy of an atom is equal to where is the Boltzmann constant and T is the temperature. Use this to estimate the speed of the atoms.) (b) Compare the Part (a) result with the number of ping-pong ballbowling ball collisions that are required to bring the bowling ball to rest. (Assume the typical speed of the incident ping-pong balls are all equal to the initial speed of the bowling ball.) (c) is a type of atom often used during cooling experiments. The wavelength of the light resonant with the cooling transition of the atoms is Estimate the number of photons needed to slow down an atom from a typical thermal velocity of to a stop.

The first Bohr radius is given by (Equation 36-12). Use the known values of the constants in the equation to show that is equal to

The longest wavelength in the Lyman series for the hydrogen atom was calculated in Example 36-2. Find the wavelengths for the transitions (a) and (b)

Find the photon energies for the three longest wavelengths in the Balmer series for the hydrogen atom, and calculate the three wavelengths.

Find the photon energy and wavelength for the series limit (shortest wavelength) in the Paschen series for the hydrogen atom. (b) Calculate the wavelength for the three longest wavelengths in Paschen series.

(a) Find the photon energy and wavelength for the series limit (shortest wavelength) in the Brackett series for the hydrogen atom. (b) Calculate the wavelength for the three longest wavelengths in Brackett series.

In the center-of-mass reference frame of a hydrogen atom, the electron and nucleus have momenta that have equal magnitudes and opposite directions. (a) Using the Bohr model, show that the total kinetic energy of the electron and nucleus can be written where is called the reduced mass, is the mass of the electron, and M is the mass of the nucleus. (b) For the equations for the Bohr model of the atom, the motion of the nucleus can be taken into account by replacing the mass of the electron with the reduced mass. Use Equation 36-14 to calculate the Rydberg constant for a hydrogen atom that has a nucleus of mass Find the approximate value of the Rydberg constant by letting go to infinity in the reduced mass formula. To how many figures does this approximate value agree with the actual value? (c) Find the percentage correction for the ground-state energy of the hydrogen atom by using the reduced mass in Equation 36-16. Note: In general, the reduced mass for a two-body problem with masses and is given by

The Pickering series of the spectrum of (singly ionized helium) consists of spectral lines due to transitions to the state of Every other line of the Pickering series is very close to a spectral line in the Balmer series for hydrogen transitions to (a) Show that this statement is accurate. (b) Calculate the wavelength of the photon during a transition from the level to the level of and show that it corresponds to one of the Balmer series lines.

For an electron in an atom that has an orbital quantum number find (a) the magnitude of the angular momentum and (b) the possible values of the magnetic quantum number (c) Draw a vector diagram to scale showing the possible orientations of relative to the direction.

For an electron in an atom that has an orbital quantum number find (a) the magnitude of the angular momentum and (b) the possible values of (c) Draw a vector diagram to scale showing the possible orientations of relative to the direction.

An electron in an atom has principal quantum number (a) What are the possible values of (b) What are the possible combinations of and (c) Using the fact that there are two quantum states for each combination of and because of electron spin, find the total number of electron states for

In an atom, find the total number of electron states that have (a) and (b) (See Problem 31.)

Find the minimum value of the angle between and the direction for an electron in an atom that has (a) (b) and (c)

What are the possible values of and for an electron in an atom that has (a) (b) and (c)

For an electron in an atom that is in an state, find (a) the magnitude of the angular momentum squared (b) the maximum value of and (c) the smallest value of

For the ground state of the hydrogen atom, find the values of (a) at (b) at and (c) the radial probability density at Give your answers in terms of

(a) If electron spin is not included, how many different wave functions are there corresponding to the first excited energy level for a hydrogen atom? (b) Specify the quantum numbers for each of these wave functions.

For the ground state of the hydrogen atom, calculate the probability of finding the electron in the region between and where and (a) and (b)

The value of the constant in the equation (Equation 36-35) is given by Find the values of (a) at (b) at and (c) the radial probability density at for the state and of a hydrogen atom. Give your answers in terms of

Show that the radial probability density for the and state of a one-electron atom can be written as where is a constant.

Calculate the probability of finding the electron in the region between and where and (a) and (b) for the state and in hydrogen. (See Problem 39 for the value of )

Show that the ground-state hydrogen atom wave function (Equation 36-32) is a solution to Schrdingers equation in spherical coordinates:

Show by dimensional analysis that the expression for the hydrogen atom ground-state energy given by (Equation 36-27) has the dimensions of energy.

By dimensional analysis, show that the expression for the first Bohr radius given by (Equation 36-12) has the dimensions of length.

The radial probability distribution function for a oneelectron atom in its ground state can be written where is a constant. Show that has its maximum value at

Show that the number of states in the hydrogen atom for a given is

Calculate the probability that the electron in the ground state of a hydrogen atom is in the region

The potential energy of a magnetic moment in an external magnetic field is given by (a) Calculate the difference in energy between the two possible orientations of an electron in a magnetic field (b) If the electrons are bombarded with photons of energy equal to that energy difference, spin flip transitions can be induced. Find the wavelength of the photons needed for such transitions. This phenomenon is called electron spin resonance.

The total angular momentum of a hydrogen atom in a certain excited state has the quantum number What can you say about the value of the orbital angular-momentum quantum number

A hydrogen atom is in the state What are the possible values of

Using a scaled vector diagram, show how the orbital angular momentum combines with the spin angular momentum to produce the two possible values of total angular momentum for the state of the hydrogen atom.

The total number of states of a hydrogen atom that has principal quantum number n _ 4 is (a) 4, (b) 16, (c) 32, (d) 36, (e) 48

How many of the eight electrons in an oxygen atom in the ground state are in a p state? (a) 0, (b) 2, (c) 4, (d) 6, (e) 8

Write the ground-state electron configuration of (a) an atom of carbon and (b) an atom of oxygen.

Give the possible values of the component of the orbital angular momentum of (a) a d electron and (b) an f electron.

The optical spectra of atoms that have two electrons in the same highest energy shell are similar, but they are quite different from the spectra of atoms that have just one electron in the highest energy shell because of the interaction of the two electrons. Group the elements according to similar spectra: lithium, beryllium, sodium, magnesium, potassium, calcium, chromium, nickel, cesium, and barium.

Write down the possible electron configurations for the first excited state of (a) a hydrogen atom, (b) a sodium atom, and (c) a helium atom.

Indicate which of the following atoms should have optical spectra similar to a hydrogen atom and which of the following atoms should have optical spectra similar to a helium atom: Li, Ca, Ti, Rb, Hg, Ag, Cd, Ba, Fr, and Ra.

(a) Calculate the next two longest wavelengths in the series (after the line) of molybdenum. (b) What is the wavelength of the shortest wavelength in this series?

The wavelength of the line for a certain element is 0.3368 nm. What is the element?

Calculate the wavelength of the line in (a) a magnesium atom and (b) a copper atom.

What is the energy of the shortest wavelength photon emitted by the hydrogen atom?

The wavelength of a spectral line of hydrogen is 97.254 nm. Identify the transition that results in this line, assuming that the transition is to the ground state.

The wavelength of a spectral line of hydrogen is 1093.8 nm. Identify the transition that results in this line.

Spectral lines of the following wavelengths are emitted by a singly ionized helium atom: 164 nm, 230.6 nm, and 541 nm. Identify the transitions that result in those spectral lines.

The combination of physical constants \(\alpha=e^{2} k / \hbar c\) where k is the Coulomb constant, is known as the fine-structure constant. It appears in numerous relations in atomic physics. (a) Show that \(\alpha\) is dimensionless. (b) Show that in the Bohr model of the hydrogen atom \(v_n=c \alpha /n\) where \(v_n\) is the speed of the electron in the state of quantum number n.

The wavelengths of the photons emitted by a potassium atom corresponding to transitions from the and states to the ground state are 766.41 nm and 769.90 nm. (a) Calculate the energies of the photons in electron volts. (b) The difference in the energies of the photons equals the difference in energy between the and states in potassium. Calculate (c) Estimate the magnetic field that the 4p electron in potassium experiences.

To observe the characteristic lines of the X-ray spectrum, one of the electrons must be ejected from the atom. This is generally accomplished by bombarding the target material with electrons of sufficient energy to eject this tightly bound electron. What is the minimum energy required to observe the lines of (a) a tungsten atom, (b) a molybdenum atom, and (c) a copper atom?

We are often interested in finding the quantity in electron volts when is given in nanometers. Show that

The positron is a particle that has the same mass as the electron and carries a charge equal to Positronium is a bound state of an electronpositron combination. (a) Calculate the energies of the five lowest energy states of positronium using the reduced mass, as given in Problem 27. (b) Do transitions between any of the levels found in Part (a) fall in the visible range of wavelengths? If so, which transitions are they?

In 1947, Lamb and Retherford showed that there is a very small energy difference between the and the states of the hydrogen atom. They measured this difference essentially by causing transitions between the two states using very long wavelength electromagnetic radiation. The energy difference (the Lamb shift) is and is explained by quantum electrodynamics as being due to fluctuations in the energy level of the vacuum. (a) What is the frequency of a photon whose energy is equal to the Lamb shift energy? (b) What is the wavelength of that photon? In what spectral region does it belong?

A Rydberg atom is one in which an electron is in a very high excited state Such atoms are useful for experiments that probe the transition from quantum-mechanical behavior to classical. Furthermore, these excited states have extremely long lifetimes (i.e., the electron will stay in this high excited state for (n _ 40 or higher). K a very long time). Ahydrogen atom is in the state. (a) What is the ionization energy of the atom when it is in that state? (b) What is the energy level separation (in electron volts) between that state and the state? (c) What is the wavelength of a photon resonant with the transition between these two states? (d) What is the radius of the atom when it is in the state?

The deuteron, the nucleus of deuterium (heavy hydrogen), was first recognized from the spectrum of hydrogen. The deuteron has a mass that is approximately twice the mass of the proton. (a) Calculate the Rydberg constant for hydrogen and for deuterium using the reduced mass as given in Problem 27. (b) Using the result obtained in Part (a), determine the difference between the longest wavelength Balmer line of hydrogen (protium) and the longest wavelength Balmer line of deuterium.

The muonium atom is a hydrogen atom that has the electron replaced by a particle. The has a mass 207 times as great as the electron. (a) Calculate the energies of the five lowest energy levels of muonium using the reduced mass as given in Problem 27. (b) Do transitions between any of the levels found in Part (a) fall in the visible range of wavelengths (for example, between and If so, which transitions are they?

The triton, a nucleus consisting of a proton and two neutrons, is unstable and has a half-life of approximately 12 years. Tritium is an atom consisting of an electron and a triton. (a) Calculate the Rydberg constant of tritium using the reduced mass as given in Problem 27. (b) Determine the difference between the longest wavelength of the Balmer lines of tritium and the longest wavelength of the Balmer lines of deuterium (see Problem 73). In addition, (c) determine the difference between the longest wavelength of the Balmer lines of tritium and the longest wavelength of the Balmer lines of hydrogen (protium).