A particular laser consumes 140.0 watts of electrical power and produces a stream of 1.25×10 19 1064-nm photons per second. What is the percent efficiency of the laser in converting electrical power to light?

Hey everyone in this example, we need to calculate percent efficiency for the conversion of power to light for a given LED. That uses 144 watts of energy to emit 1. times 10 to the 20th power photons per second at the wavelength 535 nanometers. And we should recall that we want wavelength to be in units of meters. So we're going to begin by recalling our symbol for wavelength. Lambda given as 535 nanometers should be converted from nanometers. two m. By recalling that our prefix nano tells us we have 10 to the negative ninth power meters. Now we're able to cancel out nanometers, leaving us with meters as our final unit for wavelength. And we're going to get 5.35 times 10 to the negative seventh power meters as our wavelength. Our next step is to call our formula for energy per photon and we should recall that this is going to be equal to Planck's constant. Multiplied by our speed of light divided by our wavelength. So what we're going to have for this is our planks constant, which we should recall is 6.626 times 10 to the negative 34th power jewels. Time seconds. Then multiplied by our speed of light, which we recall is three point oh oh times 10 to the eighth. Power meters per second in our denominator. We're going to plug in our wavelength in meters, which above we set is 5.35 times 10 to the negative seventh power meters. So now we're going to be able to cancel out meters with meters as well as our inverse seconds with seconds. Leaving us with jewels as our final unit for energy. And so what we're going to get is a value of 3. Jewels. Or sorry, 3.7155 Times 10 to the negative 19th power jewels per photon emitted by our led. And so now we want to go ahead and calculate our power per photon. And so this should be in units of Watts. So to do this, we're going to start out with our energy per photon and jewels. So we said that that's 3.7155 times 10 to the negative 19th power joules per photon. And we're going to multiply this by our information. And the problem where we're told that we have our led Emitting 1.27 times 10 to the 20th power photons per second. So we'll plug that in here and we should have Again, times 10 to the 20th power photons per second. And so this allows us to cancel our units of photons leaving us with jewels per second as our final unit for Watts. We're sorry, as our final unit for our power per photon. And we would recall that this makes sense because one watt Is equal to one jewel per second. So we're going to get a value here equal to 4.719 joules per second, which we can also rewrite, I'm sorry. This should be 47.19. So 47.19 joules per second. Which we can rewrite, since we know that one watt is equal to one joule per second as 47. watts. And we can just represent watts with a capital W. So now that we have our power per photon in watts, we're going to take this value to calculate our percent efficiency. And we should recall that to find our percent efficiency. We're going to take our power per photon and watts. So in our room rater that's 47.19 watts and divided by our total number of watts, admitted or used by our LED light. And so that's given in the problem as 144 watts. And then we want to multiply this by 100%. Since our answer should be a percent. So this is going to give us a value and we need to cancel our units of watts. But we're going to get a value equal to 32.8%. And so this is going to complete this example as our final answer for our percent efficiency when we convert our power to light in an LED. So I hope that everything we reviewed was clear. But if you have any questions, please leave them down below and I'll see everyone in the next practice video.

Ch.8 - The Quantum-Mechanical Model of the Atom

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Tro 6th Edition

Ch.8 - The Quantum-Mechanical Model of the Atom

Problem 100

Chapter 8, Problem 100

A particular laser consumes 140.0 watts of electrical power and produces a stream of 1.25×10¹⁹ 1064-nm photons per second. What is the percent efficiency of the laser in converting electrical power to light?

Verified step by step guidance

Calculate the energy of a single photon using the formula: \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ J s} \), \( c \) is the speed of light \( 3.00 \times 10^8 \text{ m/s} \), and \( \lambda \) is the wavelength \( 1064 \times 10^{-9} \text{ m} \).

Determine the total energy of the photons produced per second by multiplying the energy of a single photon by the number of photons: \( E_{\text{total}} = E \times 1.25 \times 10^{19} \).

Convert the electrical power consumed by the laser from watts to joules per second, noting that 1 watt = 1 joule/second.

Calculate the efficiency of the laser by dividing the total energy of the photons produced per second by the electrical power consumed, and then multiply by 100 to express it as a percentage.

Summarize the steps to ensure understanding: calculate photon energy, total photon energy per second, convert power units, and find efficiency percentage.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Photon Energy

The energy of a photon is determined by its wavelength, given by the equation E = hc/λ, where E is energy, h is Planck's constant (6.626 x 10^-34 J·s), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength in meters. For 1064 nm (1.064 x 10^-6 m), this relationship allows us to calculate the energy of each photon emitted by the laser.

Power and Energy Conversion

Power is the rate at which energy is used or produced, measured in watts (W), where 1 W = 1 J/s. In this context, the laser's power consumption (140.0 W) indicates how much electrical energy is being converted into light energy per second. To find efficiency, we compare the useful output energy (light) to the input energy (electricity).

Efficiency Calculation

Efficiency is a measure of how effectively a system converts input energy into useful output energy, expressed as a percentage. It is calculated using the formula: Efficiency (%) = (Useful Output Energy / Input Energy) × 100. In this case, we will calculate the total energy output from the emitted photons and compare it to the electrical power consumed by the laser to determine its efficiency.