Photon Energy Calculation: A Deep Dive

Hey everyone, let's dive into a cool physics problem! We're going to calculate the energy of a mole of photons with a wavelength of 779 nm. This kind of calculation is super important when we're talking about light, its properties, and how it interacts with the world around us. Plus, it's a great example of how to use some fundamental physics equations. To get started, let's break down the concepts and the steps involved. We'll be using some key constants like Planck's constant (h) and the speed of light (c), which are essential for this calculation. So, grab your calculators, and let's get started. We'll be going through the formulas step by step, making sure everything is clear and easy to follow. Remember, the goal is not just to get the answer, but also to understand how we get there.

First off, we need to understand what a photon is. A photon is a tiny packet of light. It's the smallest unit of light, and it carries energy. The energy of a single photon is directly related to its wavelength. That is, the shorter the wavelength, the more energy the photon has. This relationship is crucial for understanding various phenomena, from how the sun's rays warm the Earth to how lasers work. The energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength. We'll use this understanding to guide us through our calculations. So, we'll start with the energy of a single photon and then scale it up to the energy of a mole of photons. This will involve using Avogadro's number, which tells us how many particles are in a mole. It’s like saying, if you have a dozen eggs, you have 12 eggs; a mole of anything has 6.022 x 10^23 units.

In our case, we're going to be talking about a mole of photons, which is a massive number of photons. The concept of a mole is super important in chemistry and physics because it gives us a way to relate the macroscopic world (what we can see and measure) to the microscopic world (atoms and photons). The reason we want to calculate the energy of a mole of photons is that it helps us understand the total energy content of a beam of light. Think about it: a laser pointer emits a lot of photons, and each of those photons carries energy. By knowing the energy of a mole of photons, we can calculate how much energy the laser is putting out. This has implications for a lot of fields like solar energy, medical treatments, and even telecommunications. It’s all about quantifying the energy transfer that happens when light interacts with matter. So, by solving this problem, we gain a deeper insight into the fundamental nature of light.

Step-by-Step Energy Calculation

Alright, let’s get into the step-by-step calculation. We are going to find the energy of a mole of photons with a wavelength of 779 nm. To do this, we need to apply a few physics formulas and constants. It’s super straightforward when you break it down into smaller parts. Let’s start with the basics, we'll begin with the formula for the energy of a single photon. Then, we will expand that to find the energy for a mole of them. This process is important because it shows the relationship between individual photons and bulk properties like the total energy of a light source. So, buckle up, this is where we actually crunch the numbers.

First, we know the wavelength of the photons (λ) is 779 nm. But we need to convert this to meters because all our other units will need to be in the International System of Units (SI). Next, we need to know Planck's constant (h), which is 6.626 x 10^-34 J·s, and the speed of light (c), which is 3.00 x 10^8 m/s. These constants are fundamental to understanding the behavior of light and matter. The energy of a single photon (E) is given by the formula: E = hc/λ. This formula connects the energy of a photon to its wavelength, through Planck's constant and the speed of light. Now we can substitute the values and calculate the energy of a single photon. Once we have the energy of a single photon, we can calculate the energy of a mole of photons, using Avogadro's number (N_A), which is 6.022 x 10^23 mol^-1. So let’s convert the wavelength, plug everything into the formula, and find out the energy of a single photon. This will be our base to find the energy for the whole mole. Then we’ll multiply this by Avogadro's number.

Let’s start with converting the wavelength from nanometers to meters: 779 nm = 779 x 10^-9 m = 7.79 x 10^-7 m. Now, we’ll use the formula for the energy of a single photon, E = hc/λ. Plugging in the values, we get E = (6.626 x 10^-34 J·s * 3.00 x 10^8 m/s) / 7.79 x 10^-7 m. Doing the math, we find the energy of a single photon is approximately 2.55 x 10^-19 J. Now, we’ll move on to calculating the total energy for one mole of photons. This is where we multiply the energy of a single photon by Avogadro's number (N_A). The energy per mole = E_single * N_A. We multiply 2.55 x 10^-19 J by 6.022 x 10^23. This gives us the total energy per mole of photons. This calculation shows how the tiny energy of individual photons adds up to something measurable and significant.

Detailed Calculation

Okay, let's go into more detail on each step to ensure we get to the answer correctly. This section will walk you through the math and the units. This is important to fully grasp how the formula works. Remember that each part of this problem is designed to build our understanding of how light works and interacts with the world. We’ll break down each of these calculations to make sure it's super clear.

First, we need to make sure the units are correct. Wavelength (λ) is given as 779 nm. We convert nanometers to meters by multiplying by 10^-9. Therefore, 779 nm = 779 x 10^-9 m, which simplifies to 7.79 x 10^-7 m. Next, we use the formula E = hc/λ. Where 'h' (Planck's constant) is 6.626 x 10^-34 J·s and 'c' (the speed of light) is 3.00 x 10^8 m/s. Substitute the values: E = (6.626 x 10^-34 J·s * 3.00 x 10^8 m/s) / 7.79 x 10^-7 m. Now, we perform the multiplication in the numerator: 6.626 x 3.00 = 19.878. And the powers of ten will be added too, which will give 10^-34 * 10^8 = 10^-26. So, we'll get 19.878 x 10^-26 / 7.79 x 10^-7. After that, we need to divide by the wavelength: E = 19.878 x 10^-26 J m / 7.79 x 10^-7 m. This calculation gives us approximately 2.55 x 10^-19 J for a single photon. This value tells us how much energy each photon of this particular wavelength carries. We now have to find the energy for a mole of photons. To find the energy for a mole, we multiply the energy of a single photon by Avogadro's number. This gives us the energy for the entire mole, which provides the total energy.

The energy per mole = E_single * N_A = 2.55 x 10^-19 J * 6.022 x 10^23 mol^-1. Multiply the numbers, 2.55 * 6.022 which is approximately 15.35. The powers of ten would be -19 and +23 which will result in 10^4. Therefore, the total energy per mole is 15.35 x 10^4 J, which simplifies to 1.535 x 10^5 J. That’s a significant amount of energy, and it shows the power of many tiny photons working together. The final result is the total energy contained in one mole of photons with the specified wavelength. This calculation connects individual photon characteristics to bulk properties.

Conclusion: Energy of a Mole of Photons

Alright, so after all those calculations, what's the bottom line, guys? The energy of a mole of photons with a wavelength of 779 nm is approximately 1.535 x 10^5 J. We've gone from the very small scale of individual photons to the larger scale of a mole. This kind of problem really highlights the amazing relationship between energy, wavelength, and the number of photons involved. It also illustrates some really important physics concepts. This entire process demonstrates the power of quantum mechanics and how it affects everyday life. We can understand and predict the behavior of light and energy on a macroscopic scale.

To recap, we used the formula E = hc/λ to calculate the energy of a single photon, and then multiplied that energy by Avogadro’s number to find the energy of a mole of photons. This approach is fundamental to many areas of science and technology, like understanding the energy output of light sources or the efficiency of solar panels. So, the next time you see light, remember it's made up of photons with specific energies, and those energies are related to their wavelengths. This knowledge helps us better understand and control light for different applications. Keep exploring, and you'll find there are many more fascinating aspects of the world to discover. This calculation has opened up our understanding of how light interacts with matter and energy. It helps us to see the fundamental links between the very small and the very large.

Now, you should be able to solve similar problems. Keep practicing and exploring, and you’ll get better. This stuff becomes way easier and more intuitive the more you work with it. Remember, understanding the basic formulas and how they apply is key. And always double-check your units! Have fun with it, and keep asking questions. Physics is all about exploring the world around us, and every calculation brings us closer to a deeper understanding. Keep up the good work, and you'll be amazed at how much you can learn. Keep the curiosity alive, and keep calculating!