Problem 56 - Chemistry and Chemical Reactivity

Question

Assume your eyes receive a signal consisting of blue light, \(\lambda=470 \mathrm{nm} .\) The energy of the signal is 2.50 \(\times 10^{-14} \mathrm{J} .\) How many photons reach your eyes?

Step-by-Step Solution

Answer

Approximately 5.91 x 10^4 photons reach your eyes.

Step 1: Identify Known Values

We are given the wavelength of the blue light (\( \lambda = 470 \ \mathrm{nm} \)) and the energy of the signal (\( E = 2.50 \times 10^{-14} \ \mathrm{J} \)).

Step 2: Convert Wavelength to Meters

Convert the wavelength from nanometers to meters by using the conversion \( 1 \ \mathrm{nm} = 10^{-9} \ \mathrm{m} \). This gives \( \lambda = 470 \times 10^{-9} \ \mathrm{m} \).

Step 3: Use Energy Formula for a Photon

The energy of a single photon is given by the formula \( E_{\text{photon}} = \frac{hc}{\lambda} \), where \( h = 6.626 \times 10^{-34} \ \mathrm{Js} \) (Planck's constant) and \( c = 3 \times 10^8 \ \mathrm{m/s} \) (speed of light).

Step 4: Calculate Energy of One Photon

Substitute the known values into the formula:\[E_{\text{photon}} = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{470 \times 10^{-9}}\]This calculates to \( E_{\text{photon}} = 4.23 \times 10^{-19} \ \mathrm{J} \).

Step 5: Determine Number of Photons

To find the number of photons, divide the total energy by the energy of one photon:\[N = \frac{E}{E_{\text{photon}}} = \frac{2.50 \times 10^{-14}}{4.23 \times 10^{-19}}\]Calculating this gives \( N \approx 5.91 \times 10^4 \).

Key Concepts

Wavelength Conversion

When discussing light, we often encounter the term "wavelength," which is crucial in understanding the properties of the light we observe. In the context of this exercise, we are dealing with the wavelength of blue light, specified as 470 nanometers (nm).
To convert this wavelength from nanometers to meters, a common SI unit, we multiply by the conversion factor where 1 nanometer equals \( 10^{-9} \) meters. So, 470 nm becomes \( 470 \times 10^{-9} \) meters.

This conversion is essential because it allows us to use the standard units in calculations related to light, such as finding the energy of photons, which will be discussed in later sections. Always remember:

• 1 nm = \( 10^{-9} \) m
• Converting units is often the first step in scientific problem-solving

By making sure the units are consistent, we can ensure accurate and meaningful calculations.

Planck's Constant

Planck's constant is a fundamental value in physics that plays a key role in quantum mechanics and the study of electromagnetic waves like light.
It is represented by the symbol \( h \) and its value is approximately \( 6.626 \times 10^{-34} \) Js.

This constant is crucial because it relates the energy of a photon to its frequency (or inversely, to its wavelength). The relationship is given by the formula:
\[ E = \frac{hc}{\lambda} \]
where:

• \( E \) is the energy of the photon
• \( h \) is Planck's constant
• \( c \) is the speed of light \((3 \times 10^8 \ m/s)\)
• \( \lambda \) is the wavelength of the light

This formula shows that the energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength. In simpler terms, higher frequency (or shorter wavelength) light carries more energy.
Remembering the role of Planck's constant not only helps in solving physics problems but also deepens the understanding of how light behaves as both a wave and a particle.

Photon Calculation

Photon calculation ties together the concepts of wavelength, energy, and quantum theory to figure out important characteristics of light.
In our exercise, we want to determine how many photons make up a particular energy signal.First, we calculate the energy of an individual photon with the formula \( E_{\text{photon}} = \frac{hc}{\lambda} \).
Using the exercise's values: \( h = 6.626 \times 10^{-34} \) Js, \( c = 3 \times 10^8 \ m/s \), and \( \lambda = 470 \times 10^{-9} \) m, we find
\[ E_{\text{photon}} = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{470 \times 10^{-9}} \approx 4.23 \times 10^{-19} \] J.
Finally, by dividing the total energy received \((2.50 \times 10^{-14} \) J) by the energy of one photon \((4.23 \times 10^{-19} \) J), we get the number of photons.The formula to solve for the number of photons \( N \) is:
\[ N = \frac{E}{E_{\text{photon}}} = \frac{2.50 \times 10^{-14}}{4.23 \times 10^{-19}} \approx 5.91 \times 10^4 \]This shows us that approximately 59,100 photons reach your eyes, demonstrating how even minuscule energies can imply vast numbers of photons, given their incredibly small size and energy.