A laser produces 20.0 mW of red light. In 30.0 minutes, the laser emits 1.15×10 20 photons. What is the wavelength of the laser?

Hey everyone in this example we need to calculate the wavelength of the laser, producing an energy value of 35.2 mil a watts, we're told this laser amidst 3.61 times 10 to the 21st power photons in 2.31 hours. We should recognize that our energy value here is in units of mila watts. And we're going to want to convert this into jewels per photon. We also want to go ahead and recall our formula for calculating wavelength which is represented by lambda, which we can find by taking plank's constant, multiplying by the speed of light and dividing by our energy value in joules per photon however were given our energy and units of mila mila watts. So we can take our mila watts 35.2 million watts and focus on converting from mila watts into watts by recalling that are prefix milli tells us we have 10th of the third power mila watts for one watch. Now we're able to cancel out our units of mila watts and focus on going from what to jules by recalling our conversion factor that we have one watt equal to one jewel per second. So we're going to multiply by one over seconds. So this would give us inverse seconds. And so this would allow us to then go ahead and cancel out our units of Watts, leaving us with jewels per second as our final units. And so now that we have these units as our final units, we're going to get the value of all of these products equal to a value of 0.0352 jewels per second. So now we have our energy in joules per second. We're going to take this and convert from jewels per second using the amount of time that they give us as a conversion factor. And so we should recall that we have in one hour 3600 seconds. So now we're able to cancel our units of seconds because they're aligned in the in the numerator and the denominator. And then we're going to use the info and the problem which tells us that 42.31 hours we produce 3.61 times 10 to the 21st power photons. And this conversion works out for us because we want to go ahead and cancel out our units of hours so that we're left with joules per photon for our unit of energy. And so what we're going to have here now that we have jewels per photon left over as our final units, We're going to get an answer equal to 8.11 times 10 to the negative 20th power jewels per photon as our energy value. So now that we have energy in the proper units we can go ahead and find our final answer for our wavelength lambda. So we should have in our numerator plank's constant, which we recall is 6.626 times 10 to the negative 34th power jewels time seconds. And then we're going to plug in our speed of light which we should recall. And let me just make this a bit clear, sorry, is equal to a value of three point oh oh times 10 to the eighth. Power and units of meters per second in our denominator. We're going to plug in that energy in joules per photon as 8.11 times to the negative 20th power jewels per photon. And so now we're going to go ahead and focus on canceling out our units so we can get rid of jewels with jewels. We can get rid of our seconds with our inverse seconds. And then we can also cancel out meters. We're sorry not meters but photons because our units for wavelength should just be in meters. And so this is going to give us a value equal to 2.4151 times 10 to the negative six power meters as our wavelength. However, we want to go ahead and express our answer and units of nanometers. So what we're going to do is take this wavelength 2.4151 times 10 to the negative six power meters. And we're going to place meters in the denominator to get two nanometers in our numerator as our final unit. And so we should recall that our prefix nano tells us that we should have for one m 10 to the ninth Power or tend to the yeah 10 to the ninth power nanometers. And so we can cancel out our units of meters because they're aligned appropriately, leaving us with nanometers as our final unit for our wavelength. And this is going to give us a final answer equal to 2451 nm as our final answer for our wavelength produced by our laser. So I hope that everything I went through was clear. If you have any questions, please leave them down below. Otherwise, I will 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 99

Chapter 8, Problem 99

A laser produces 20.0 mW of red light. In 30.0 minutes, the laser emits 1.15×10²⁰ photons. What is the wavelength of the laser?

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Convert the power of the laser from milliwatts to watts. Since 1 mW = 0.001 W, multiply 20.0 mW by 0.001 to get the power in watts.

Calculate the total energy emitted by the laser in 30.0 minutes. Use the formula: \( \text{Energy} = \text{Power} \times \text{Time} \). Convert 30.0 minutes to seconds by multiplying by 60.

Determine the energy of a single photon. Use the formula: \( E_{\text{photon}} = \frac{\text{Total Energy}}{\text{Number of Photons}} \).

Use the energy of a single photon to find the wavelength. Use the formula: \( E_{\text{photon}} = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ J s} \) and \( c \) is the speed of light \( 3.00 \times 10^8 \text{ m/s} \). Rearrange to solve for \( \lambda \): \( \lambda = \frac{hc}{E_{\text{photon}}} \).

Substitute the values for \( h \), \( c \), and \( E_{\text{photon}} \) into the equation to calculate the wavelength \( \lambda \).

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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 given by the equation E = hν, where E is energy, h is Planck's constant (6.626 × 10^-34 J·s), and ν (nu) is the frequency of the light. This relationship shows that the energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength.

Wavelength and Frequency Relationship

The wavelength (λ) and frequency (ν) of light are related by the equation c = λν, where c is the speed of light (approximately 3.00 × 10^8 m/s). This means that as the wavelength increases, the frequency decreases, and vice versa. Understanding this relationship is crucial for calculating the wavelength from the frequency or energy of the emitted photons.

Power and Photon Emission

Power (P) is defined as the rate at which energy is emitted or transferred. In this context, the power of the laser (20.0 mW) indicates how much energy is emitted per second. By knowing the total number of photons emitted and the time duration, one can calculate the energy per photon, which is essential for determining the wavelength of the laser light.