Problem 1: Red Light Has A Wavelength Of 590 Nm. What Is Its ✓ Solved

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)

Sample Paper For Above instruction

Understanding the properties of light and their implications in chemical bonding and molecular interactions requires an exploration of fundamental concepts of electromagnetic radiation. This paper demonstrates calculations of wavelength, frequency, energy, and wavenumber for red light, assesses the potential of different wavelengths to break chemical bonds, and relates atomic transitions to energy changes, anchoring these discussions within the framework of quantum physics and molecular chemistry.

First, considering the wavelength of red light, given as 590 nanometers (nm), which can be converted to meters for standard SI calculations: 1 nm = 1 × 10⁻⁹ meters, so λ = 590 × 10⁻⁹ m = 5.9 × 10⁻⁷ m. The frequency (ν) of light is related to its wavelength via the speed of light (c), approximately 3.00 × 10⁸ meters per second (m/s), through the equation:

ν = c / λ

Substituting the known values:

ν = (3.00 × 10⁸ m/s) / (5.9 × 10⁻⁷ m) ≈ 5.08 × 10¹⁴ Hz

Hence, the frequency of red light with a wavelength of 590 nm is approximately 5.08 × 10¹⁴ Hz.

Next, the energy (E) of a photon can be calculated using Planck's equation:

E = h × ν

where h is Planck's constant, approximately 6.626 × 10⁻³⁴ Joule seconds (Js). Calculating:

E = (6.626 × 10⁻³⁴ Js) × (5.08 × 10¹⁴ Hz) ≈ 3.37 × 10⁻¹⁹ Joules

Thus, each photon of red light carries approximately 3.37 × 10⁻¹⁹ Joules of energy.

To find the energy in terms of the total energy for 1 mole of photons, we utilize Avogadro's number (6.022 × 10²³ mol⁻¹):

Energy per mole = (3.37 × 10⁻¹⁹ J) × (6.022 × 10²³ mol⁻¹) ≈ 2.03 × 10⁵ Joules per mol

Regarding the conversion to wavenumbers, which is a measure of the number of wave cycles per centimeter, the relation is:

Wavenumber (ṽ) = 1 / λ (in centimeters)

First, convert the wavelength to centimeters: 590 nm = 590 × 10⁻⁷ cm = 5.9 × 10⁻⁵ cm. Therefore:

ṽ = 1 / (5.9 × 10⁻⁵ cm) ≈ 16949.15 cm⁻¹

This value indicates approximately 16,949 wavenumbers for the given red light wavelength.

Now, addressing the capacity of various light wavelengths to break chemical bonds, particularly carbon-carbon (C−C) bonds in DNA, which require around 335 kJ/mol of energy:

First, converting bond energy to Joules per photon:

Energy per bond = 335 kJ/mol / 6.022 × 10²³ mol⁻¹ ≈ 5.56 × 10⁻¹⁹ Joules

Comparing this to the photon energy of UV light at 260 nm, which is more energetic:

Calculate wavelength in meters: λ = 260 nm = 260 × 10⁻⁹ m

Frequency: ν = c / λ = (3.00 × 10⁸ m/s) / (260 × 10⁻⁹ m) ≈ 1.15 × 10¹⁵ Hz

Photon energy: E = h × ν ≈ (6.626 × 10⁻³⁴ Js) × (1.15 × 10¹⁵ Hz) ≈ 7.62 × 10⁻¹⁹ Joules

Per photon energy is sufficient to break the C−C bonds in DNA, as it exceeds the bond energy per photon (~5.56 × 10⁻¹⁹ Joules). Conversely, orange light at 590 nm cannot, due to lower photon energy (~3.37 × 10⁻¹⁹ Joules), which is below the bond energy threshold.

In conclusion, UV light at 260 nm possesses enough energy to break C−C bonds in DNA, emphasizing its potential to cause molecular damage, whereas visible red and orange light are insufficient for such bond cleavage.

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