Problem 5 - College Physics

Question

\(\bullet\) The predominant wavelength emitted by an ultraviolet lamp is 248 nm. If the total power emitted at this wavelength is \(12.0 \mathrm{W},\) how many photons are emitted per second?

Step-by-Step Solution

Answer

Approximately \(1.50 \times 10^{19}\) photons per second are emitted.

Step 1: Understand the Parameters

We are given the wavelength of light emitted by the ultraviolet lamp, which is \[\lambda = 248 \text{ nm} = 248 \times 10^{-9} \text{ meters}\]and the total power emitted, which is\[P = 12.0 \text{ watts} = 12.0 \text{ joules per second}.\] We need to find out how many photons are emitted per second.

Step 2: Calculate the Energy of a Single Photon

The energy of a single photon can be found using the formula:\[E = \frac{hc}{\lambda},\]where \(h\) is Planck's constant \(6.626 \times 10^{-34} \text{ J s}\) and \(c\) is the speed of light \(3.00 \times 10^8 \text{ m/s}\). Substituting the given values, we get:\[E = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{248 \times 10^{-9}} \approx 8.02 \times 10^{-19} \text{ J}.\]This is the energy of one photon.

Step 3: Calculate the Number of Photons Emitted per Second

The number of photons emitted per second can be calculated by dividing the total power by the energy of a single photon. Hence, the number of photons \(N\) is given by:\[N = \frac{P}{E} = \frac{12.0}{8.02 \times 10^{-19}} \approx 1.50 \times 10^{19} \text{ photons per second}.\]

Step 4: Conclusion

Thus, using the total emitted power and the energy of a single photon, we conclude that approximately \(1.50 \times 10^{19}\) photons are emitted by the ultraviolet lamp per second.

Key Concepts

Understanding Ultraviolet Light

Ultraviolet (UV) light is a type of electromagnetic radiation with a wavelength that is shorter than visible light but longer than X-rays. This makes it invisible to the human eye. UV light is found in sunlight and contributes to sunburns. It's divided into categories based on wavelength, such as UVA, UVB, and UVC, each with different effects and applications. UV light is commonly used for sterilization and disinfection because it can kill or inactivate microorganisms by destroying nucleic acids and disrupting their DNA.

• UV Light Wavelength: 100-400 nm
• Can cause skin damage and is used in tanning beds
• Used in medical and disinfection applications

Calculating the number of photons emitted from UV light requires understanding the wavelength of the UV light involved.

Energy of a Photon Explained

Photons are particles of light that carry energy. The energy of a particular photon is directly related to its wavelength. The shorter the wavelength, the higher the energy of the photon. This energy is crucial in processes such as photosynthesis, vision, and photovoltaic energy conversion.

• Photon energy is related to frequency and wavelength
• Can influence chemical reactions when absorbed

This can be calculated using the formula: \[E = \frac{hc}{\lambda}\]where:

• \(E\) is the energy of the photon
• \(h\) is Planck's constant, \(6.626 \times 10^{-34} \, \mathrm{J\cdot s}\)
• \(c\) is the speed of light, \(3.00 \times 10^8 \, \mathrm{m/s}\)
• \(\lambda\) is the wavelength of the light

In our calculation for UV light, knowing the energy per photon helps determine how many photons are emitted overall.

The Role of Planck's Constant

Planck's constant is a fundamental quantity in quantum mechanics and plays a crucial role in determining the energy of photons. It establishes the scale at which quantum thermal action occurs, and without it, the calculations of quantum properties like photon energy would not be possible.
This constant, denoted as \(h\), is valued at approximately:\[h = 6.626 \times 10^{-34} \, \mathrm{J\cdot s}\]

• Key to quantum mechanics
• Relates energy to frequency for particles like photons

It is fundamental in the equation for photon energy \(E = \frac{hc}{\lambda}\), allowing for precise calculations of how energy varies with wavelength. Understanding Planck's constant is vital in all of physics and chemistry, especially when working with processes involving light and energy calculations.

Calculating the Number of Photons

Determining the number of photons emitted by a light source in a given time is important for both scientific research and practical applications. The number is derived by dividing the total power by the energy per photon.

• Total Power: Amount of energy emitted per unit time, measured in watts
• Photon Energy: Energy carried by each photon

The formula used is:\[N = \frac{P}{E}\]where:

• \(N\) is the number of photons emitted per second
• \(P\) is the total power of the light in watts
• \(E\) is the energy of one photon

In practice, this means using known values like the total power output of the lamp and the calculated energy of a single photon to find the photon emission rate. This method of calculation is also useful in fields such as lighting design, laser technology, and quantum computing. Understanding how to calculate the number of photons can provide insights into the efficiency and intensity of light-emitting devices.