Calculate the wavenumber of a beam of infrared radiation with a wavelength of $5.00 \mu m$.
Answer (1)
Convert the wavelength from micrometers to meters: λ = 5.00 μ m = 5.00 × 1 0 − 6 m .
Calculate the wavenumber using the formula: ν ~ = λ 1 .
Substitute the value of λ to find the wavenumber: ν ~ = 5.00 × 1 0 − 6 m 1 = 200000 m − 1 .
The wavenumber of the infrared radiation is: 200000 m − 1 .
Explanation
Problem Analysis We are given the wavelength of infrared radiation, λ = 5.00 μ m , and we need to calculate the wavenumber, ν ~ .
Wavelength Conversion First, we need to convert the wavelength from micrometers ( μ m ) to meters ( m ). We know that 1 μ m = 1 0 − 6 m . Therefore, we have: λ = 5.00 μ m = 5.00 × 1 0 − 6 m
Wavenumber Calculation The wavenumber, ν ~ , is the reciprocal of the wavelength, λ , expressed in meters. The formula is: ν ~ = λ 1 Substituting the value of λ in meters, we get: ν ~ = 5.00 × 1 0 − 6 m 1 = 200000 m − 1
Final Answer Therefore, the wavenumber of the infrared radiation is 200000 m − 1 .
Examples
Wavenumber is used in spectroscopy to identify the vibrational modes of molecules. For example, an infrared spectrometer measures the absorption of infrared radiation by a sample. The wavenumber at which absorption occurs corresponds to the frequency of a particular molecular vibration. By analyzing the absorption spectrum, we can identify the molecules present in the sample and determine their structure. This technique is widely used in chemistry, materials science, and environmental science.
