Problem 15 - Chemistry The Molecular Science

Question

Calculate the energy of a photon of blue light that has a wavelength of \(450 \mathrm{nm}\)

Step-by-Step Solution

Answer

The energy of the photon is approximately \( 4.42 \, \times \, 10^{-19} \; \text{J} \).

Step 1: Understand the Equation

To calculate the energy of a photon, we use the equation \( E = \frac{hc}{\lambda} \), where \( E \) is the energy in joules, \( h \) is Planck’s constant \( (6.626 \, \times \, 10^{-34} \, \text{J} \, \cdot \, \text{s}) \), \( c \) is the speed of light \( (3.00 \, \times \, 10^8 \, \text{m/s}) \), and \( \lambda \) is the wavelength in meters.

Step 2: Convert Wavelength to Meters

The given wavelength is \( \lambda = 450 \; \text{nm} \). We need to convert this value into meters because the equation requires standard SI units. Thus, \( 450 \, \text{nm} = 450 \, \times \, 10^{-9} \; \text{m} \).

Step 3: Plug Values into Equation

Now, substitute the values for Planck’s constant \( (6.626 \, \times \, 10^{-34} \, \text{J} \, \cdot \, \text{s}) \), the speed of light \( (3.00 \, \times \, 10^8 \, \text{m/s}) \), and the wavelength \( 450 \, \times \, 10^{-9} \; \text{m} \) into the equation for energy: \[ E = \frac{6.626 \, \times \, 10^{-34} \, \times \, 3.00 \, \times \, 10^8}{450 \, \times \, 10^{-9}} \].

Step 4: Calculate the Energy

Simplify the equation to find \( E \). This calculation gives: \[ E = \frac{6.626 \, \times \, 10^{-34} \, \times \, 3.00 \, \times \, 10^8}{450 \, \times \, 10^{-9}} \approx 4.42 \, \times \, 10^{-19} \; \text{J} \]. Thus, the energy of the photon is approximately \( 4.42 \, \times \, 10^{-19} \; \text{J} \).

Key Concepts

Planck's Constant

Planck's constant is a fundamental constant named after the physicist Max Planck. It is a crucial element of quantum mechanics because it relates the energy of a photon to its frequency. The constant is denoted as \( h \) and equals \( 6.626 \times 10^{-34} \text{J} \cdot \text{s} \).
Planck's constant bridges the gap between the macroscopic and microscopic worlds in physics.
Here’s why it’s significant:

• Planck’s constant is used to calculate the energy of photons, which are particles of light.
• The formula \( E = h u \), where \( u \) is the frequency, shows that energy is proportional to frequency with Planck’s constant as the proportionality constant.
• This constant helps us understand that energy is quantized; it comes in discrete packets.

Understanding Planck’s constant helps us grasp the dual nature of light, acting both as a wave and as a particle (photon).

Wavelength Conversion

In physics, converting units to their proper form is often necessary. For calculations involving the speed of light and Planck’s constant, it is essential to express quantities in standard SI units.
Wavelength is usually given in nanometers (nm) for light.
To convert nanometers to meters, use the conversion factor: \[ 1 \text{ nm} = 1 \times 10^{-9} \text{ m} \].

• Given a wavelength of \( 450 \text{ nm} \), multiplying by this factor yields \( 450 \times 10^{-9} \text{ m} \).
• This conversion ensures compatibility with SI units used in physical constants like Planck's constant and the speed of light.

Therefore, accurately converting wavelength units is critical for reliable calculations in photon energy problems.

Speed of Light

The speed of light is one of the universe’s fundamental constants and plays a vital role in physics.
Symbolized by \( c \), it is the speed at which light travels in a vacuum and is approximately \( 3.00 \times 10^8 \text{ m/s} \).
Understanding the speed of light helps in calculating the energy of photons efficiently.

• It appears in the equation \( E = \frac{hc}{\lambda} \), where \( c \) represents the speed of light, showing its role in determining a photon's energy.
• The constancy of \( c \) ensures the equations used in physics maintain uniformity across calculations.
• It serves as a critical component in various areas of physics, including Einstein's theory of relativity.

The speed of light is not just about the movement of photons, but it also reinforces fundamental physics laws and equations.
Knowing how to incorporate this constant into calculations opens up a clearer understanding of the photonic world.