Calculate Wavelength
Problem #1: Light with a frequency of 7.26 x 1014 Hz lies in the violet region of the
visible spectrum. What is the wavelength of this frequency of light? Answer in units
of nm.
Problem #2: When an electron beam strikes a block of copper, x-rays of frequency
1.07 x 1019 Hz are emitted. What is the wavelength of these x-rays? Answer in units
of pm.
Problem #3: Calculate the wavelength (in meters) of radiation a frequency of 5.00 x
1014 s¯1.
Problem #4: The radio station KUSC (in Southern California) broadcasts at 91.5
MHz. Calculate its wavelength in meters.
Problem #5: Calculate the wavelength of radiation emitted from radioactive cobalt
with a frequency of 2.80 x 1020 s¯1. What region of the eletromagnetic spectrum does
this lie in?
Problem #6: 1.50 x 1013 Hz? Does this radiation have a longer or shorter wavelength
than red light?
Problem #7: What is the wavelength of sound waves having a frequency of 256.0
sec¯1 at 20 °C? Speed of sound = 340.0 m/sec. (The problem is about sound, but this
does not change the basic idea of the equation "wavelength times frequency = speed."
Problem # 8: A photon has a frequency of 6.0 x 104 Hz. Convert this frequency into
wavelength (nm). Does this frequency fall in the visible region?
Calculate frequency
Problem #1a: Calculate the frequency of radiation with a wavelength of 442 nm.
Example #1b: The wavelength of an argon laser's output is 488.0 nm. Calculate the
frequency of this wavelength of electromagnetic radiation.
Problem #2a: Calculate the frequency of electromagnetic radiation that has a
wavelength of 1.315 micrometers.
Problem #2b: What is the frequency of infrared radiation of wavelength 67.5 μm?
Problem #3a: Calculate the frequency of radiation with a wavelength of 4.92 cm.
Problem #3b: Calculate the frequency of radiation with a wavelength of 4.55 x
10¯9 cm.
Problem #4: Calculate the frequency of radiation with a wavelength of 8973 Å.
Example #5: What is the frequency of radiation with a wavelength of 5.00 x 10¯ 8 m?
In what region of the electromagnetic spectrum is this radiation?
Calculate Bhor energy
Problem #1: What is the energy change when an electron drops from the n=3 energy
state to the 𝑛=1 energy state in a hydrogen atom?
Problem #2: Calculate the energy of a photon emitted by a hydrogen atom when its
electron drops from the n = 5 state to the n = 3 state.
Calculate Wavelength
Problem #1: Light with a frequency of 7.26 x 1014 Hz lies in the violet region of the
visible spectrum. What is the wavelength of this frequency of light? Answer in units
of nm.
Solution:
1) Substitute into λν = c:
(x) (7.26 x 1014 s¯1) = 3.00 x 108 m s¯1
x = 4.13 x 10¯7 m
2) Convert from m to nm:
4.13 x 10¯7 m x (109 nm / 1 m) = 413 nm
Problem #2: When an electron beam strikes a block of copper, x-rays of frequency
1.07 x 1019 Hz are emitted. What is the wavelength of these x-rays? Answer in units
of pm.
Solution:
1) Substitute into λν = c:
(x) (1.07 x 1019 s¯1) = 3.00 x 108 m s¯1
x = 2.80 x 10¯11 m
2) Convert from m to pm:
2.80 x 10¯11 m x (1012 pm / 1 m) = 28 pm
By the way, if the question had asked for the answer in nm, it would have been 0.0280
nm. Notice that the wavelength unit in the above questions was deliberately picked to
give a whole number. Generally speaking, the wavelength unit is picked to give a
whole number, be it tens, hundreds or thousands.
Problem #3: Calculate the wavelength (in meters) of radiation a frequency of 5.00 x
1014 s¯1.
Solution:
1) Substitute into λν = c:
(x) (5.00 x 1014 s¯1) = 3.00 x 108 m s¯1
x = 6.00 x 10¯7 m
2) By the way, the unit typically used for wavelengths of visible light is nanometers:
6.00 x 10¯7 m times (109 nm / 1 m) = 600 nm.
Visible light has a wavelegth somewhere between 400 and 700 nm.
Sometimes you might be asked what color a radiation might be. Look at problems
below for examples of this. What color might 600 nm be?
Problem #4: The radio station KUSC (in Southern California) broadcasts at 91.5
MHz. Calculate its wavelength in meters.
Solution:
1) Convert MHz to s¯1:
91.5 MHz = 91.5 x 106 s¯1 = 9.15 x 107 s¯1
1) Use λν = c:
(x) (9.15 x 107 s¯1) = 3.00 x 108 m/s
x = 3.28 m
Problem #5: Calculate the wavelength of radiation emitted from radioactive cobalt
with a frequency of 2.80 x 1020 s¯1. What region of the eletromagnetic spectrum does
this lie in?
Solution:
1) Calculate the wavelength:
λν = c
(x) (2.80 x 1020 s¯1) = 3.00 x 108 m/s
x = 1.07 x 10¯12 m
2) Determine the region of the EM spectrum:
Consult a convenient reference source.
Gamma rays.
Problem #6: 1.50 x 1013 Hz? Does this radiation have a longer or shorter wavelength
than red light?
Comment: there are a number of ways to answer this question. I'll do two.
Solution #1:
Calculate the frequency of 7000 Å (the longest wavelength of red light):
7000 Å = 7000 x 10¯8 cm = 7.00 x 10¯5 cm (three sig figs is a reasonable assumption)
λν = c
(7.00 x 10¯5 cm) (x) = 3.00 x 1010 cm/s
x = 4.28 x 10¯14 s¯1
A lower frequency (like the value given in the problem) means a longer wavelength.
The radiation in the problem has a longer wavelength than the red light I used.
Beyond red on the EM spectrum is infrared.
Solution #2:
Consult a convenient reference source.
Compare 1.50 x 1013 Hz to the various values in the frequency column of the above
web site.
Determine that the frequency given in the problem lies in the infrared, a region that
has a longer wavelength than red light does.
Problem #7: What is the wavelength of sound waves having a frequency of 256.0
sec¯1 at 20 °C? Speed of sound = 340.0 m/sec. (The problem is about sound, but this
does not change the basic idea of the equation "wavelength times frequency = speed."
Solution:
a) Use λν = speed:
(x) (256.0 sec¯1) = 340.0 m/sec
The answer, to four sig figs, is 1.328 m.
Just for kicks: in centimeters, 132.8 cm, and in Ångströms, 1.328 x 1010 Å
Problem # 8: A photon has a frequency of 6.0 x 104 Hz. Convert this frequency into wavelength (nm).
Does this frequency fall in the visible region?
Calculate frequency
Problem #1a: Calculate the frequency of radiation with a wavelength of 442 nm.
Solution to 1a:
1) Convert nm to m:
442 nm x (1 m / 109 nm) = 4.42 x 10¯7 m
2) Substitute into λν = c:
(4.42 x 10¯7 m) (x) = 3.00 x 108 m s¯1
x = 6.79 x 1014 s¯1
Example #1b: The wavelength of an argon laser's output is 488.0 nm. Calculate the
frequency of this wavelength of electromagnetic radiation.
Solution to 1b:
1) Convert nm to m:
488 nm x (1 m / 109 nm) = 4.88 x 10¯7 m
Then, substitute into λν = c:
(4.88 x 10¯7 m) (x) = 3.00 x 108 m s¯1
x = 6.15 x 1014 s¯1
The use of nm for wavelength is quite common.
Problem #2a: Calculate the frequency of electromagnetic radiation that has a
wavelength of 1.315 micrometers.
Solution to 2a:
1) Convert μm to m:
1.315 μm x (1 m / 106 μm) = 1.315 x 10¯6 m
2) Substitute into λν = c:
(1.315 x 10¯6 m) (x) = 3.00 x 108 m s¯1
x = 2.28 x 1014 s¯1
Problem #2b: What is the frequency of infrared radiation of wavelength 67.5 μm?
Solution to 2b:
1) Convert μm to m:
67.5 μm = 67.5 x 10-6 m
2) Use λν = c to determine the frequency:
(67.5 x 10-6 m) (x) = 3.00 x 108 m/s
x = 4.44 x 1012 s-1
Problem #3a: Calculate the frequency of radiation with a wavelength of 4.92 cm.
Comment: since the wavelengths are already in cm, we can use c = 3.00 x 1010 cm
s¯1 and not have to do any conversions at all.
Solution to 3a:
(4.92 cm) (x) = 3.00 x 1010 cm s¯1
x = 6.10 x 109 s¯1
Problem #3b: Calculate the frequency of radiation with a wavelength of 4.55 x
10¯9 cm.
Solution to 3b:
(4.55 x 10¯9 cm) (x) = 3.00 x 1010 cm s¯1
x = 6.59 x 1018 s¯1
Problem #4: Calculate the frequency of radiation with a wavelength of 8973 Å.
Comment: since 1 Å = 10¯8 cm, therefore 8973 Å = 8973 x 10¯8 cm. Converting to
scientific notation gives 8.973 x 10¯5 cm. This is another place where the cm
s¯1 value for c can be used, since Å converts to cm very easily.
Solution:
(8.973 x 10¯5 cm) (x) = 3.00 x 1010 cm s¯1
x = 3.34 x 1014 s¯1
Example #5: What is the frequency of radiation with a wavelength of 5.00 x 10¯ 8 m?
In what region of the electromagnetic spectrum is this radiation?
Solution:
1) Use λν = c to determine the frequency:
(5.00 x 10¯8 m) (x) = 3.00 x 108 m/s
x = 6.00 x 1015 s¯1
2) Determine the electromagnetic spectrum region:
Consult a convenient reference source.
This frequency is right in the middle of the ultraviolet region of the spectrum.
Calculate Bhor energy
Problem #1: What is the energy change when an electron drops from the n=3 energy
state to the 𝑛=1 energy state in a hydrogen atom?
-18
E
= 2.18 x 10
J x (1/9 - 1/1)
photon
-18
= E = -1.94 x 10
E
J
photon
Problem #2: Calculate the energy of a photon emitted by a hydrogen atom when its
electron drops from the n = 5 state to the n = 3 state.
-18
E
= 2.18 x 10
J x (1/25 - 1/9)
photon
-19
= E = -1.55 x 10
E
photon
J