Question4. Fast Green Dye is a food dye, which absorbs in the red range. The energy of this photon is $3.18 * 10^{-19} \mathrm{~J}$ . a. Determine the frequency of this photon. (5 points) b. Determine the wavelength of this photon in nanometers. (5 points)

Studdy Solution

STEP 1

1. We are dealing with electromagnetic radiation, where the energy of a photon is related to its frequency and wavelength.
2. The relationship between energy and frequency is given by Planck's equation: $E = h \nu$ .
3. The speed of light equation relates frequency and wavelength: $c = \lambda \nu$ .
4. Planck's constant $h = 6.626 \times 10^{-34} \, \text{J}\cdot\text{s}$ .
5. The speed of light $c = 3.00 \times 10^8 \, \text{m/s}$ .

STEP 2

1. Calculate the frequency of the photon using its energy.
2. Calculate the wavelength of the photon using its frequency.

STEP 3

Use Planck's equation to find the frequency $\nu$ of the photon. The equation is:
$E = h \nu$
Rearrange the equation to solve for $\nu$ :
$\nu = \frac{E}{h}$
Substitute the given energy and Planck's constant into the equation:
$\nu = \frac{3.18 \times 10^{-19} \, \text{J}}{6.626 \times 10^{-34} \, \text{J}\cdot\text{s}}$
Calculate $\nu$ :
$\nu \approx 4.80 \times 10^{14} \, \text{Hz}$

STEP 4

Use the speed of light equation to find the wavelength $\lambda$ of the photon. The equation is:
$c = \lambda \nu$
Rearrange the equation to solve for $\lambda$ :
$\lambda = \frac{c}{\nu}$
Substitute the speed of light and the frequency calculated in Step 1 into the equation:
$\lambda = \frac{3.00 \times 10^8 \, \text{m/s}}{4.80 \times 10^{14} \, \text{Hz}}$
Calculate $\lambda$ :
$\lambda \approx 6.25 \times 10^{-7} \, \text{m}$
Convert the wavelength from meters to nanometers (1 meter = $10^9$ nanometers):
$\lambda \approx 625 \, \text{nm}$
The frequency of the photon is approximately $4.80 \times 10^{14} \, \text{Hz}$ and the wavelength is approximately $625 \, \text{nm}$ .

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