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MP_Assignment#10 Due: 9:35am on Mond Monday, ay, May 9, 2011 Note: You Note: You will receive no credit for late submissions. To learn more, read your instruc instructor's tor's Gradin Grading g Policy
Electron Transitions and Spectral Lines Conceptual Question Description: Conceptual question on spectral lines corresponding to electron transitions. The spectrum of a hypoth hypothetical etical atom is shown in the figure. Three distinct distinct spectral series are shown, shown, with the center series corresponding to transitions to a final state of . The indica indicated ted spectr spectral al line corresponds to the transition from an initial state of to the final state .
Part A Which of the lettered spectral lines corresponds to the transition from Hint A.1
to
?
Spectral series
In many atoms, the spectral lines corresponding to transitions to the same final state are naturally grouped together into spectral series. series . (In hydrogen, these series are given the names Lyman, Balmer, Paschen, etc.) In the hypothetical atom under investigation, the same phenomenon occurs, where the middle series corresponds to all of the transitions involving the final state .
Hint A.2
Finding Fi nding lines within within a serie series s
Within any series, the lines furthest furthest to the right correspond correspond to the longe longest st wavelength, wavelength, and hence lowest energy, energ y, transitions. transitions. As you move to the left within a serie series, s, the energy involved involved in the transition increases. ANSWER:
D
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Part B Which of the lettered spectral lines corresponds to the transition from Hint B.1
to
?
Identifying different series
Series are arranged such that transitions involving larger amounts of energy are to the left of series involving smaller amounts of energy. Since the energy difference between successive energy levels gets smaller as increases, this allows you to differentiate between different series.
Hint B.2
Identify the
The
series
series involves transitions with a final energy state of
between successive energy levels gets smaller as the
. Since the energy difference
increases, is the
series to the right or left of
series?
ANSWER:
right left
ANSWER:
H
Part C Which of the lettered spectral lines corresponds to the transition from ANSWER:
to
?
A
The Bohr Atom Description: The problem applies quantization of angular momentum to the classical model of an electron orbiting a nucleus. Learning Goal: To understand the Bohr model of the hydrogen atom. In 1913 Niels Bohr formulated a method of calculating the different energy levels of the hydrogen atom. He did this by combining both classical and quantum ideas. In this problem, we go through the steps needed to understand the Bohr model of the atom. Part A Consider an electron with charge single proton) with charge
and mass
orbiting in a circle around a hydrogen nucleus (a
. In the classical model, the electron orbits around the nucleus, being held
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in orbit by the electromagnetic interaction between itself and the protons in the nucleus, much like planets orbit around the sun, being held in orbit by their gravitational interaction. When the electron is in a circular orbit, it must meet the condition for circular motion: The magnitude of the net force toward the center, , is equal to
. Given these two pieces of information, deduce the velocity
of the electron as it orbits
around the nucleus. Hint A.1
Electrostatic force
Recall that the force
between two charged particles is ,
where
and
are the charges on the particles,
is the separation of the particles, and
is the
permittivity of free space. This is the force that keeps the electron in a circular orbit, so Express your answer in terms of
,
,
, and
.
, the permittivity of free space.
ANSWER: =
Part B The key insight that Bohr introduced to his model of the atom was that the angular momentum of the electron orbiting the nucleus was quantized. He introduced the postulate that the angular momentum could only come in quantities of , where is Planck's constant and is a nonnegative integer ( ). Given this postulate, what are the allowable values for the velocity
of the electron in the
Bohr atom? Recall that, in circular motion, angular momentum is given by the formula Express your answer in terms of
, Planck's constant
,
, and
.
.
ANSWER: =
Part C In Parts A and B you found two different expressions to describe the allowed electron velocities these two values (eliminating Express
in terms of
,
,
) and solve for the allowable radii , , and
. Equate
in the Bohr model.
.
ANSWER: =
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As the electron orbits the nucleus with a speed potential energy. The total energy
at a radius
, it has both kinetic energy and electric
of the electron can be expressed as .
Part D In Parts B and C you saw that, according to Bohr's postulate, the electron radius velocity
and the electron
only have certain allowable values. Plug the values obtained for these two quantities into the
energy statement given above to arrive at a new statement for the allowed energy levels in the Bohr atom. Express your answer in terms of
,
,
,
, and
.
ANSWER: =
The standard formula for energy in the Bohr model is . Despite the serious flaws of the Bohr model, such as casually mixing quantum and classical ideas with little if any justification, this formula turns out to be equivalent to the energy formula for hydrogen obtained from quantum mechanics. To adequately deal with atoms other than hydrogen, however, requires the full quantum theory.
The Photoelectric Effect Experiment Description: Three part problem to understand the photoelectric effect experiment. Learning Goal: To understand the experiment that led to the discovery of the photoelectric effect. In 1887, Heinrich Hertz investigated the phenomenon of light striking a metal surface, causing the ejection of electrons from the metal. The classical theory of electromagnetism predicted that the energy of the electrons ejected should have been proportional to the intensity of the light. However, Hertz observed that the energy of the electrons was independent of the intensity of the light. Furthermore, for low enough frequencies, no electrons were ejected, no matter how great the intensity of the light became. The following problem outlines the methods used to investigate this new finding in physics: the photoelectric effect . Suppose there is a potential difference between the metal that ejects the electrons and the detection device, such that the detector is at a lower potential than the metal. The electrons slow down as they go from higher to lower electric potential; since they must overcome this potential difference to reach the detector, this potential is known as the stopping potential . To reach the detector, the initial kinetic energy of an ejected electron must be greater than or equal to the amount of energy it will lose by moving through the potential difference. http://session.masteringphysics.com/myct/assignments
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Part A If there is a potential difference
between the metal and the detector, what is the minimum energy
that an electron must have so that it will reach the detector? Hint A.1
Relating potential difference to energy
Recall that potential difference is defined as the difference in energy for a particle that is moved between two points, divided by the charge of the particle: , where is the potential difference,
is the energy difference, and
Express your answer in terms of
is the charge of the particle.
and the magnitude of the charge on the electron,
.
ANSWER: =
For the incident light to cause the ejection of an electron, the light must impart a certain amount of energy to the electron to overcome the forces that constrain it within the metal. The minimum amount of energy required to overcome these forces is called the work function . Different metals will have different values for
. For an electron to reach the detector, the light must impart enough energy for the electron to
overcome both the work function and the stopping potential. Part B Suppose that the light carries energy
. What is the maximum stopping potential
that can be
applied while still allowing electrons to reach the detector? Hint B.1
Find the energy of the ejected electron
What will be the energy
of the electron immediately after it leaves the metal?
Express your answer in terms of
and
.
ANSWER: =
Now, think about how this answer relates to what you found in Part A. Express your answer in terms ,
, and
.
ANSWER: =
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Part C Classical electromagnetism predicted that
should have increased as the intensity of the incident light
increased. On the contrary, it was found that voltage
increased as the frequency
of the light increased. The
was found to obey the following linear relationship: ,
where
and
are numerical constants (representing the slope and the intercept, respectively). By
comparing this equation to your answer from Part B, find an expression for the intercept in this equation changes with different light but Express your answer in terms of
and
. (Notice that
is a constant of the metal.)
.
ANSWER: =
Part D In a 1905 paper that later won him a Nobel Prize, Albert Einstein postulated that the energy of light was proportional to its frequency. The constant of proportionality turned out to be Planck's constant : . Using your previous results, and the equation given in Part C, find an expression for
in
terms of experimentally determinable quantities. Express your answer in terms of the slope ANSWER:
and
.
=
Part E Suppose that two sets of values were recorded in this experiment: Stopping potential
Frequency
(
(
)
)
Using these data, extrapolate a numerical value for Planck's constant Hint E.1
Finding the value of
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.
from the data
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In Part C, you were told that the voltage
is proportional to the frequency of the light that strikes the
metal: . A numerical value for the slope
can be found from the two data points, using the fact that .
By entering this value for
into the relationship you obtained in Part D, you can obtain a numerical
value for Planck's constant
.
Express your answer in joule-seconds to three significant figures. ANSWER: =
Part F Using the data given, find a numerical value for the work function Hint F.1
of the metal.
How to determine the constant in a linear equation
In Part C, you were told that the voltage
is proportional to the frequency of the light that strikes the
metal: . In Part E, you should have determined the numerical value of the slope into the equation here, along with one of the sets of the data points (for you can determine the numerical value of
. By substituting this value of and
. Finally, using the relationship between
) given in Part E, and
determined
in Part C, you can obtain the numerical value of the work function. Express your answer in joules to two significant figures. ANSWER: =
Photon Density Ranking Task Description: Ranking task on the number of photons in beams of different frequency EM radiation with the same energy. (ranking task) Part A Beams of different frequency electromagnetic radiation are listed below. Each beam has the same total energy. Rank these beams on the number of photons in each beam. http://session.masteringphysics.com/myct/assignments
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Hint A.1
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Photon energy
The energy of a single photon is proportional to its frequency. Hint A.2
Radio waves
By examining a radio dial, you will discover that FM radio stations broadcast with frequencies between 88 and 108 (megahertz, or millions of cycles per second) and AM radio stations broadcast between 520 and 1720
Hint A.3
(kilohertz, or thousands of cycles per second).
Frequency spectrum of electromagnetic waves
Different frequency electromagnetic waves have historically been given different names. The traditional names for the various frequencies are listed below.
Hint A.4
Number of photons in the beam
The number of photons in a beam of electromagnetic energy is inversely proportional to the energy of each individual photon. This means that more low-energy photons are needed than high-energy photons to achieve the same beam energy. Rank from largest to smallest. To rank items as equivalent, overlap them. ANSWER:
View
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Conceptual Question 39.11 Description: The figure shows the energy-level diagram of Element X. (a) What is the ionization energy of Element X? (b) An atom in the ground state absorbs a photon, then emits a photon with a wavelength of 1240 nm. What conclusion can you draw about the... The figure shows the energy-level diagram of Element X.
Part A What is the ionization energy of Element X? Express your answer using two significant figures. ANSWER:
=
Part B An atom in the ground state absorbs a photon, then emits a photon with a wavelength of 1240
. What
conclusion can you draw about the energy of the photon that was absorbed? Express your answer using two significant figures. ANSWER: =
Part C An atom in the ground state has a collision with an electron, then emits a photon with a wavelength of 1240 . What conclusion can you draw about the initial kinetic energy of the electron? Express your answer using two significant figures. ANSWER: http://session.masteringphysics.com/myct/assignments
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=
Problem 39.6 Description: (a) Use Millikan's photoelectric-effect data in Figure 39.10 in the textbook to determine the work function, in eV, of cesium. (b) Use Millikan's photoelectric-effect data in Figure 39.10 in the textbook to determine an experimental value of... Part A Use Millikan's photoelectric-effect data in Figure 39.10 in the textbook to determine the work function, in , of cesium. Express your answer using three significant figures. ANSWER: =
Part B Use Millikan's photoelectric-effect data in Figure 39.10 in the textbook to determine an experimental value of Planck's constant. Express your answer using three significant figures. ANSWER: =
Problem 39.16 Description: The diameter of the nucleus is about 10 fm. (a) What is the kinetic energy, in MeV, of a proton with a de Broglie wavelength of 10 fm? The diameter of the nucleus is about 10 fm. Part A What is the kinetic energy, in MeV, of a proton with a de Broglie wavelength of 10 fm? ANSWER: MeV
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Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 7 points.
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