### Calculate the frequency when given the wavelength

To solve these problems, you usually need to convert the wavelenth to meters before using λν = c. The reason for meters is that, to solve these type of problems, using 3.00 x 108 m s¯1 for the speed of light is usually the best choice.

That being said, there are problems worded in such a way for which 3.00 x 1010 cm s¯1 (for the speed of light) is the better-suited value. Some examples are below. Sometimes a teacher might supply the centimeter value in the problem, but the meter value would be better suited for the solution. Be careful on your units!

Often, these problems require metric conversions. If you wish to review metric conversions, click here.

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.

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 10141

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 10141

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.
Problem #2b: What is the frequency of infrared radiation of wavelength 67.5 μm?

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 10141

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.
Problem #3b: Calculate the frequency of radiation with a wavelength of 4.55 x 10¯9 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 1091

Solution to 3b:

(4.55 x 10¯9 cm) (x) = 3.00 x 1010 cm s¯1

x = 6.59 x 10181

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 10141

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 10151

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.