Problem 45 - Chemistry: An Atoms-Focused Approach

Question

Red Lasers The power of a red laser \((\lambda=630 \mathrm{nm})\) is 1.00 watt (abbreviated \(\mathrm{W},\) where \(1 \mathrm{W}=1 \mathrm{J} / \mathrm{s}\) ). How many photons per second does the laser emit?

Step-by-Step Solution

Answer

Answer: Approximately 3.17 x 10^18 photons are emitted per second by a 1.00 W red laser with a wavelength of 630 nm.

Step 1: Convert wavelength to meters

Since the wavelength is given in nanometers, we need to convert it to meters. To do this, simply multiply by 10^(-9): λ = 630 nm * 10^(-9) = 6.30 x 10^(-7) m

Step 2: Calculate the energy of a single photon

To find the energy of a single photon, we use the formula E = h * c / λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength: E = (6.63 x 10^(-34) J s) * (3.00 x 10^8 m/s) / (6.30 x 10^(-7) m) E ≈ 3.15 x 10^(-19) J

Step 3: Calculate the number of photons per second

Now that we have the energy of a single photon, we can find the number of photons emitted per second by dividing the power of the laser by the energy per photon: Number of photons/s = Power / Energy per photon Number of photons/s = 1.00 W / 3.15 x 10^(-19) J Number of photons/s ≈ 3.17 x 10^(18) photons/s Thus, the red laser emits approximately 3.17 x 10^18 photons per second.

Key Concepts

Planck's Constant

Planck's constant, denoted as \(h\), is a fundamental constant in quantum mechanics and has a value of \(6.63 \times 10^{-34} \, \text{J} \cdot \text{s}\). This constant is crucial for understanding the quantum behavior of particles, especially when it comes to energy quantization. It relates the energy of a photon to the frequency of the light through the equation \(E = h \cdot f\), where \(E\) is energy and \(f\) is frequency.
To find the energy of a photon, we often use the modified formula \(E = \frac{h \cdot c}{\lambda}\). Here, \(c\) is the speed of light and \(\lambda\) is wavelength. Planck's constant helps us bridge the concepts of energy, frequency, and wavelength in quantum systems.
In our context, Planck's constant enables us to convert the wavelength of the laser into energy, a necessary step to solve the problem of finding how many photons a laser emits per second.

Speed of Light

The speed of light in a vacuum is a universal constant, symbolized as \(c\), and it equals \(3.00 \times 10^8 \, \text{m/s}\). This immense speed is critical for many fields of physics, establishing the fundamental limit of how fast information can travel.
In calculations related to photons, the speed of light is used to relate energy and wavelength with Planck's constant. The formula \(E = \frac{h \cdot c}{\lambda}\) includes \(c\) to express energy in terms of wavelength. This equation shows us that energy increases with frequency (or decreases with wavelength), highlighting the direct connection between these properties.
Understanding the speed of light and its role in these calculations helps us grasp how photons behave in different scenarios, such as when they are emitted by a laser.

Wavelength Conversion

Wavelength is often measured in nanometers (nm) for visible light; however, scientific calculations frequently require converting this to meters. This conversion is straightforward because 1 nanometer equals \(10^{-9}\) meters.
For example, converting a wavelength of \(630 \, \text{nm}\) into meters involves multiplying by \(10^{-9}\):

• \(630 \, \text{nm} \times 10^{-9} = 6.30 \times 10^{-7} \, \text{m}\)

This step is vital for using the formula \(E = \frac{h \cdot c}{\lambda}\), as it requires the wavelength in meters to calculate photonic energy correctly.
Thus, an understanding of wavelength conversion ensures accurate calculations when dealing with photonic emissions.

Joules to Watts Conversion

To solve problems involving power, like determining photon emission rates, it's essential to understand the relationship between joules and watts. Power, measured in watts, represents energy per time unit. One watt equals one joule per second (\(1 \, \text{W} = 1 \, \text{J/s}\)).
This relationship is crucial for calculating how many photons a laser emits every second. Knowing the energy of one photon (in joules) allows us to divide the laser's power (in watts) by this energy to find the number of photons emitted per second. The equation is:

• \(\text{Number of photons/s} = \frac{\text{Power (W)}}{\text{Energy per photon (J)}}\)

This conversion and understanding ensure accurate results when handling energy and power in photon-related calculations and experiments.