Problem 1: Red Light Has A Wavelength Of 590 Nm. What 240217

Problem 1red Light Has A Wavelength Of 590 Nma What Is Its Fre

Problem 1red Light Has A Wavelength Of 590 Nma What Is Its Fre

Red light has a wavelength (λ) of 590 nm. (a) What is its frequency? (b) What is its energy? (c) What is this value in wavenumbers (defined as 1/λ typically with units of cm-1 which is the number of waves that fit in a 1 cm length) (d) Can orange light break a carbon-carbon bond in DNA? How about UV light with a wavelength of 260 nm? (carbon-carbon bond energies are typically around 335 kJ/mol)

This collection of questions explores various properties of electromagnetic radiation, emphasizing the relationships between wavelength, frequency, energy, and their biological and chemical implications. Understanding these concepts is vital for applications in spectroscopy, photochemistry, and molecular biology.

Paper For Above instruction

The properties of electromagnetic radiation are fundamental to understanding phenomena in physics, chemistry, and biology. These properties include wavelength, frequency, and energy, which are interconnected by fundamental equations derived from the principles of quantum mechanics and wave theory. This paper aims to comprehensively analyze the properties of red light with a wavelength of 590 nm, calculate its frequency and energy, convert this information into wavenumber units, and explore the biological and chemical implications related to vibrational energies and photon-induced bond rupture.

Calculating the Frequency of Red Light

The frequency (v) of electromagnetic radiation can be calculated using the speed of light equation:

v = c / λ

where c is the speed of light in vacuum, approximately 3.00 × 108 meters per second, and λ is the wavelength in meters. Since the wavelength is given in nanometers (nm), it must be converted to meters:

λ = 590 nm = 590 × 10-9 m = 5.90 × 10-7 m

Thus, the frequency is:

v = (3.00 × 108 m/s) / (5.90 × 10-7 m) ≈ 5.08 × 1014 Hz

This frequency corresponds to the visible red region of the electromagnetic spectrum and defines the rate at which the electric and magnetic fields oscillate.

Calculating the Energy of Red Light

Photon energy (E) is related to frequency by Planck’s equation:

E = h × v

where h is Planck’s constant, approximately 6.626 × 10-34 J·s. Plugging in the frequency:

E = (6.626 × 10-34 J·s) × (5.08 × 1014 Hz) ≈ 3.37 × 10-19 Joules

This energy describes the quantum of energy carried by each photon of red light at 590 nm.

Converting to Wavenumbers

Wavenumber is defined as the reciprocal of wavelength in centimeters:

\[


\tilde{\nu} = \frac{1}{\lambda \text{ (cm)}}


\]

Converting λ from nanometers to centimeters:

λ = 590 nm = 590 × 10-7 cm = 5.9 × 10-5 cm

Therefore,

\[


\tilde{\nu} = \frac{1}{5.9 \times 10^{-5} \text{ cm}} \approx 16949 \text{ cm}^{-1}


\]

This wavenumber indicates the number of wave cycles per centimeter, useful in vibrational spectroscopy.

Biological and Chemical Implications

The energy of a photon at 590 nm, approximately 3.37 × 10-19 Joules, corresponds to the energy involved in electronic transitions within molecules. Comparing this energy to chemical bond energies shows that visible photons generally do not possess enough energy to break covalent bonds, which are typically around 335 kJ/mol (~5.54 × 10-19 Joules per molecule). Consequently, red light at this wavelength can cause electronic excitations but is insufficient to induce bond dissociation directly.

In contrast, ultraviolet (UV) light with shorter wavelengths (around 260 nm) carries much higher energy per photon. Calculating its energy:

λ = 260 nm = 2.60 × 10-7 m


E = h × c / λ = (6.626 × 10-34) × (3.00 × 108) / (2.60 × 10-7) ≈ 7.63 × 10-19 Joules

which exceeds typical bond energies. Hence, UV light at 260 nm can effectively break carbon-carbon bonds, such as those in DNA, which are around 335 kJ/mol (~5.54 × 10-19 Joules per molecule). This explains ultraviolet's role in DNA damage and the initiation of photochemical reactions.

Thus, while visible red light affects electronic states without breaking bonds, UV radiation effectively causes bond cleavage, contributing to biological damage and chemical reactions.

Conclusion

The analysis of light's physical properties and their biological implications demonstrates the important distinctions between different regions of the electromagnetic spectrum. Red light, with its lower energy, influences electronic states without causing bond disruption, whereas UV light carries sufficient energy to induce chemical bond cleavage, underpinning many photochemical processes and biological effects.

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