Problem 12 - Chemistry The Molecular Science

Question

Many marine organisms exhibit bioluminescence, which occurs when excited singlet-state molecules return to their lowest energy (ground) state by releasing photons in the visible region (and beyond) of the electromagnetic spectrum. Photostomias guernei, a spiny-rayed oceanic fish, bioluminesces at \(470 . \mathrm{nm}\). (a) What color is the bioluminescence? (b) Calculate the frequency of this bioluminescence.

Step-by-Step Solution

Answer

(a) Blue. (b) Approximately \( 6.38 \times 10^{14} \) Hz.

Step 1: Determine the Color of Bioluminescence

The wavelength given for the bioluminescence of Photostomias guernei is 470 nm. The visible spectrum ranges from about 380 nm (violet) to 750 nm (red). A wavelength of 470 nm falls within the blue region of the visible spectrum. Therefore, the color of the bioluminescence is blue.

Step 2: Use the Speed of Light Formula

Frequency and wavelength are related through the speed of light. The formula to use here is: \[ c = \lambda u \] where \( c \) is the speed of light (\( 3.00 \times 10^8 \) m/s), \( \lambda \) is the wavelength (470 nm), and \( u \) is the frequency. We need to calculate the frequency, so we rearrange the formula to: \[ u = \frac{c}{\lambda} \]

Step 3: Convert Wavelength to Meters

Before substituting the values into the formula, convert the wavelength from nanometers to meters: \[ 470 \text{ nm} = 470 \times 10^{-9} \text{ m} \]

Step 4: Substitute Values and Calculate Frequency

Substitute the known values into the rearranged formula: \[ u = \frac{3.00 \times 10^8 \text{ m/s}}{470 \times 10^{-9} \text{ m}} \] Calculate the frequency: \[ u \approx 6.38 \times 10^{14} \text{ Hz} \]

Step 5: Conclude with Calculated Frequency

The frequency of the bioluminescence of Photostomias guernei is approximately \( 6.38 \times 10^{14} \) Hz. The light emitted corresponds to the blue region of the visible spectrum.

Key Concepts

Electromagnetic Spectrum

The electromagnetic spectrum is a broad range of all the different wavelengths of electromagnetic radiation. Think of it like a rainbow that isn't limited to just the colors we see. This spectrum includes various types such as radio waves, microwaves, infrared light, visible light, ultraviolet light, X-rays, and gamma rays.

The electromagnetic spectrum is categorized based on wavelength or frequency.

• Radio Waves: These have the longest wavelengths and the lowest frequencies.
• Gamma Rays: These have the shortest wavelengths and the highest frequencies.

Bioluminescence, like that exhibited by the fish Photostomias guernei, falls within the visible light portion of this spectrum. This is just a teeny part sandwiched between infrared and ultraviolet light.

Understanding where a particular light falls on the spectrum aids in determining its nature and behavior. This is crucial for tasks like identifying the color of bioluminescent emissions.

Visible Spectrum

The visible spectrum is the part of the electromagnetic spectrum that is visible to the human eye. It ranges from about 380 nm to 750 nm in wavelength. These light waves are what we commonly perceive as colors.

Here’s a quick breakdown of the visible spectrum:

• Violet: 380 - 450 nm
• Blue: 450 - 495 nm
• Green: 495 - 570 nm
• Yellow: 570 - 590 nm
• Orange: 590 - 620 nm
• Red: 620 - 750 nm

Photostomias guernei's bioluminescence at 470 nm lies within the blue portion.

This is why it appears blue to our eyes. The specific wavelength of light determines the unique color we see, due to our eyes having different receptors sensitive to different parts of the visible spectrum.

Wavelength and Frequency Calculation

Calculating wavelength and frequency is crucial in understanding different forms of electromagnetic radiation. The relationship between them is expressed through the formula: \[ c = \lambda u \] Where:

• \( c \): speed of light (\( 3.00 \times 10^8 \) m/s)
• \( \lambda \): wavelength in meters
• \( u \): frequency in Hertz (Hz)

Since Photostomias guernei emits light at a wavelength of 470 nm, we first convert this to meters (as the formula requires standard units).

Now, 470 nm becomes 470 x \( 10^{-9} \) m. With this, we can plug it into the formula, rearranged as: \[ u = \frac{c}{\lambda} \] By substituting the speed of light and the converted wavelength value, we calculate the frequency to be approximately \( 6.38 \times 10^{14} \) Hz.

Such calculations provide a clear understanding of the characteristics of the light emitted by different bioluminescent organisms, helping in fields like marine biology and optical science.