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The most widely used wavelength region for infrared analysis is about 2 to \(15 \mu \mathrm{m} .\) Express this range in angstroms and in wavenumbers.

Short Answer

Expert verified
The range is 20,000-150,000 angstroms or 666.67-5000 cm\(^{-1}\).

Step by step solution

01

Convert Micrometers to Angstroms

We know that 1 micrometer (\(\mu\mathrm{m}\)) equals 10,000 angstroms. So, to convert the given range in micrometers to angstroms, we will multiply the lower and upper limits of the range by 10,000. For the lower limit (2 \(\mu\mathrm{m}\)):\[2 \times 10,000 = 20,000 \text{ angstroms}\]For the upper limit (15 \(\mu\mathrm{m}\)):\[15 \times 10,000 = 150,000 \text{ angstroms}\] So, the range in angstroms is from 20,000 to 150,000 angstroms.
02

Understand Wavenumber Concept

Wavenumber is the reciprocal of the wavelength in centimeters and is measured in units of inverse centimeters (cm\(^{-1}\)). The formula to convert wavelength (in centimeters) to wavenumber is:\[\text{Wavenumber} (\bar{u}) = \frac{1}{\text{Wavelength (cm)}}\]Since 1 \(\mu\mathrm{m}\) equals 10\(^{-4}\) cm, we first convert the micrometers into centimeters.
03

Convert Micrometers to Centimeters

Now, let's convert the range from micrometers to centimeters. 1 \(\mu\mathrm{m}\) is equal to 0.0001 cm (or 10\(^{-4}\) cm). Thus, For the lower limit (2 \(\mu\mathrm{m}\)):\[2 \times 0.0001 = 0.0002 \text{ cm}\]For the upper limit (15 \(\mu\mathrm{m}\)):\[15 \times 0.0001 = 0.0015 \text{ cm}\]This means the range is 0.0002 cm to 0.0015 cm.
04

Calculate the Wavenumber Range

Now we calculate the corresponding wavenumbers for our wavelength range in centimeters.For the lower limit (0.0002 cm):\[\bar{u} = \frac{1}{0.0002} = 5000 \text{ cm}^{-1}\]For the upper limit (0.0015 cm):\[\bar{u} = \frac{1}{0.0015} \approx 666.67 \text{ cm}^{-1}\]Thus, the wavenumber range is from approximately 666.67 cm\(^{-1}\) to 5000 cm\(^{-1}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wavelength Conversion
When dealing with infrared spectroscopy, converting between different units of wavelength is crucial. Wavelength determines the characteristics of light, and is therefore fundamental to spectroscopy.
From a practical standpoint, wavelengths can be expressed in various units, such as micrometers, angstroms, and centimeters. This allows scientists and engineers to work with the dimensions relevant to their specific applications.
Converting between these units enables clearer communication and deeper understanding of spectroscopic data. For example, micrometers (渭m) are often used for infrared spectral regions, but sometimes the precision and scale of angstroms are more appropriate. By aligning these units, we can precisely interpret the energy or frequency of the spectroscopic signals.
Wavenumber
In infrared spectroscopy, wavenumber is a concept that needs to be well understood. It is a measure of frequency and is the reciprocal of the wavelength expressed in centimeters.
The unit for wavenumber is inverse centimeters (cm鈦宦), and it is crucial because it provides an alternative way to describe how light interacts with matter.
One of the key benefits of using wavenumbers is that the higher the wavenumber, the higher the energy of the photon. This directly relates to the regions in the electromagnetic spectrum that we are investigating. By understanding wavenumber, we can better appreciate the energy variations within a sample and determine specific properties or changes in the sample's structure.
Micrometers to Angstroms
Converting measures of micrometers to angstroms is often necessary in scientific analysis, especially in infrared spectroscopy.
Micrometers are often a suitable unit to express infrared wavelengths, but when higher precision is required, angstroms are preferred. The conversion factor is simple: 1 micrometer ( 渭m) is equivalent to 10,000 angstroms ( 脜).
  • To convert, multiply the number of micrometers by 10,000 to obtain the equivalent number of angstroms.
  • For example, the conversion of 2 micrometers is accomplished by calculating 2 脳 10,000 = 20,000 angstroms.
  • Similarly, 15 micrometers equals 150,000 angstroms.
This ensures that you work with the most appropriate and convenient units when dealing with infrared data.
Spectroscopic Calculations
Spectroscopy often involves calculations that require precision and attention to detail. These calculations allow us to relate different forms of data and gain insights into the properties of a substance.
For example, when dealing with wavenumbers, we convert the wavelength into centimeters and take the reciprocal to find the corresponding values.
  • First, convert the wavelength from micrometers to centimeters, knowing 1 渭m = 0.0001 cm.
  • Next, calculate the wavenumber using the formula: \(\bar{u} = \frac{1}{\text{Wavelength (cm)}}\).
  • The result provides a range which can help identify specific molecular vibrations or structural changes.
These methods and conversions are vital for interpreting spectroscopic data accurately, leading to understandable and actionable results.

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Most popular questions from this chapter

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