41 - Iowa State University
1. An electron has a de Broglie wavelength [pic] (a) What is its momentum?
(b) What is its speed? (c) What voltage was needed to accelerate it to this speed?
Solution
(a) We find the momentum from
[pic]
(b) We find the speed from
[pic], [pic], [pic]
(c) With [pic] we can calculate kinetic energy classically:
[pic][pic]
which converted to electron-volts equals [pic]
This is the energy gained by an electron as it is accelerated through a potential difference of [pic]
2. Electrons are accelerated by 2450 V in an electron microscope. What is the maximum possible resolution?
Solution
The wavelength of the electron is
[pic]
[pic]
Method 2
[pic]; [pic]
This is the maximum possible resolution.
3. The neutrons in a parallel beam, each having kinetic energy [pic] are directed through two slits 0.50 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.]
Solution
We find the wavelength of the neutron from
[pic][pic]
[pic]
The peaks of the interference pattern are given by [pic]
and the positions on the screen are [pic]
For small angles, [pic] so we have [pic]
Thus the separation is [pic]
4. An electron remains in an excited state of an atom for typically [pic] What is the minimum uncertainty in the energy of the state (in eV)?
Solution
We find the minimum uncertainty in the energy of the state from
[pic]
5. An electron and a 140-g baseball are each traveling [pic] measured to an accuracy of 0.055%. Calculate and compare the uncertainty in position of each.
Solution
The uncertainty in the velocity is
[pic]
For the electron, we have
[pic]
For the baseball, we have
[pic]
The uncertainty for the electron is greater by a factor of [pic]
6. For [pic] what values can l have?
Solution
The value of [pic] can range from 0 to [pic] Thus for n = 6, we have [pic]
7. How many electrons can be in the [pic] subshell?
Solution
The number of electrons in the subshell is determined by the value of [pic] For each [pic] the
value of [pic] can range from [pic] to [pic] or [pic] values. For each of these there are two values of [pic] Thus the total number for [pic] is
[pic] 14 electrons.
8. If a hydrogen atom has [pic] what are the possible values of n, l, and [pic]
Solution
The value of [pic] can range from [pic] to [pic] so we have [pic]
The value of [pic] can range from 0 to [pic] Thus we have [pic]
There are two values of [pic]
9. What is the full electron configuration in the ground state for elements with Z equal to (a) 27, (b) 36, (c) 38? [Hint: See the periodic table inside the back cover.]
Solution
(a) We start with hydrogen and fill the levels as indicated in the periodic table:
[pic]
Note that the [pic] level is filled before the 3d level is started.
(b) For [pic] we have
[pic]
(c) For [pic] we have
[pic]
Note that the [pic] level is filled before the 4d level is started.
10. A hydrogen atom is in the 6s 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.
Solution
(a) The principal quantum number is [pic]
(b) The energy of the state is
[pic]
(c) The “s” subshell has [pic] The magnitude of the angular momentum depends on [pic] only:
[pic]
(d) For each [pic] the value of [pic] can range from [pic] to [pic] [pic]
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