Chapter 42: Problem 7
The water molecule has an \(l\) = 1 rotational level 1.01 \(\times\) 10\(^-$$^5\) eV above the \(l\) = 0 ground level. Calculate the wavelength and frequency of the photon absorbed by water when it undergoes a rotational-level transition from \(l\) = 0 to \(l\) = 1. The magnetron oscillator in a microwave oven generates microwaves with a frequency of 2450 MHz. Does this make sense, in view of the frequency you calculated in this problem? Explain.
Short Answer
Step by step solution
Calculate the Energy Difference
Convert Energy to Joules
Relate Energy to Wavelength
Calculate Frequency of the Photon
Compare to Microwave Frequency
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Energy Conversion
For example, if we have an energy difference between rotational levels of \( 1.01 \times 10^{-5} \) eV, this can be converted to joules by multiplying with the conversion factor, resulting in \( 1.618 \times 10^{-24} \) J. This conversion is crucial because it allows us to make use of joules in calculations involving Planck's constant and other fundamental constants, which are generally expressed in joules.
Understanding how to make these conversions enables the use of different scientific principles to solve problems involving energy transitions, such as those encountered in rotational spectra.
Frequency Calculation
Using Planck's relation \( E = \frac{hc}{\lambda} \) where \( E \) is energy, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \, \text{J} \cdot \text{s}) \), and \( c \) is the speed of light \( (3 \times 10^8 \, \text{m/s}) \), we can rearrange this equation to solve for frequency \( u \) once we know the wavelength \( \lambda \). We find that \( u = \frac{c}{\lambda} \).
For this particular problem, if we calculate \( \lambda \) to be approximately \( 1.23 \times 10^{-2} \, \text{m} \), the frequency \( u \) of the photon is about \( 2.44 \times 10^{10} \) Hz. Frequency calculation gives us deeper insight into the properties of light and the types of transitions it can induce.
Wavelength Determination
Using the constants \( h = 6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \) and \( c = 3 \times 10^8 \, \text{m/s} \), we can calculate the wavelength given an energy change of \( 1.618 \times 10^{-24} \) J. This results in a wavelength of approximately \( 1.23 \times 10^{-2} \, \text{m} \).
Wavelength determines the type of electromagnetic radiation; in this case, it falls in the microwave region of the spectrum. Knowing the wavelength helps in identifying where these transitions occur and can be essential in applications such as spectroscopy and communication technologies.
Microwave Frequency Comparison
This comparison highlights that the photon frequency is about 10 times higher than the frequency found in a typical microwave oven.
- The microwave oven's frequency is used for heating food through microwave radiation, which causes polar molecules to rotate, generating heat.
- Our calculated frequency corresponds to a different range of the electromagnetic spectrum, implicating different uses and mechanisms.