### Wavelength-Frequency Conversions Problems #1 - 10

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.

Given Wavelength, Calculate Frequency (five problems)

Problem #1: Calculate the frequency of radiation with a wavelength of 442 nm.

Solution:

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

Example #2: The wavelength of an argon laser's output is 488.0 nm. Calculate the frequency of this wavelength of electromagnetic radiation.

Solution:

1) Convert nm to m:

488 nm x (1 m / 109 nm) = 4.88 x 10¯7 m

2) 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 very common in science, almost standard usage. However, because of this, many teachers will be tempted to ask these types of problems using different units for the wavelength. Be prepared!

Problem #3: Calculate the frequency of electromagnetic radiation that has a wavelength of 1.315 micrometers.

Solution:

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

Problem #4: What is the frequency of infrared radiation of wavelength 67.5 μm?

Solution:

1) Convert μm to m:

67.5 μm times (1 m / 106 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

I didn't bother to put the wavelength in scientific notation.

Problem #5: What is the frequency of red light having a wavelength of 7000 Å?

The solution below depends on converting Å into cm. This means you must remember that the conversion is 1 Å = 10¯8 cm. The solution:

(7000 x 10¯8 cm) (x) = 3.00 x 1010 cm/sec

Notice how I did not bother to convert 7000 x 10¯8 into scientific notation. If I had done so, the value would have been 7.000 x 10¯5.

Note also that I effectively consider 7000 to be 4 significant figures. I choose to do this because I know wavelength measurements are very accurate and that 6, 7, or even 8 sig figs are possible. At an introductory level, you will not know this, so that is why I am telling you here. Also, the value for the speed of light is known to nine significant figures, as in 299,792,458 m s¯1. However, 3.00 is good enough for introductory work.

The answer is 4.29 x 10141

Given Frequency, Calculate Wavelength (five problems)

Problem #6: 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 10141) = 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 #7: Calculate the wavelength (in meters) of radiation a frequency of 5.00 x 10141.

Solution:

1) Substitute into λν = c:

(x) (5.00 x 10141) = 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. What color might 600 nm be? I will address that in some problems to follow.

Problem #8: Calculate the wavelength given a frequency of 6.80 x 1015 Hz. Answer in units of pm.

Solution:

1) Solve for the wavelength in cm (just to be different):

λν = c

(λ) (6.80 x 10151) = 3.00 x 10 cm/s

λ = 4.41 x 10¯6 cm

2) Convert cm to pm:

4.41 x 10¯6 cm) (1010 pm / 1 cm) = 4.41 x 104 pm

Problem #9: 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 1061 = 9.15 x 1071

1) Use λν = c:

(x) (9.15 x 1071) = 3.00 x 108 m/s

x = 3.28 m

KUSC is a classical radio station. You can find it at KUSC.org. Check it out.

Problem #10: 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 10191) = 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.