Hey guys! Ever wondered about the tiny packets of energy that make up light, especially infrared radiation? Let's dive into the fascinating world of photons and their energy levels. We're going to tackle a specific problem today: calculating the energy of a photon of infrared radiation given its frequency. So, buckle up, and let's get started!
Understanding the Energy of a Photon
When delving into the realm of photon energy, it's essential to grasp the fundamental principles that govern this phenomenon. At its core, the energy of a photon is directly proportional to its frequency. This relationship is elegantly encapsulated by the renowned equation: E = hν, where E represents the energy of the photon, h denotes Planck's constant, and ν symbolizes the frequency of the radiation. This equation serves as a cornerstone in the field of quantum mechanics, offering profound insights into the behavior of light and matter at the atomic and subatomic levels.
Now, let's break down each component of this equation to gain a clearer understanding. Firstly, E, the energy of the photon, is typically measured in joules (J), which is the standard unit of energy in the International System of Units (SI). This value quantifies the amount of energy carried by a single photon of electromagnetic radiation. Secondly, h, Planck's constant, is a fundamental physical constant that plays a pivotal role in quantum mechanics. Its value is approximately 6.63 × 10⁻³⁴ joule-seconds (J⋅s). This constant serves as a bridge between the classical world and the quantum world, dictating the scale at which quantum effects become significant. Lastly, ν, the frequency of the radiation, represents the number of oscillations or cycles of the electromagnetic wave per unit of time. It is typically measured in hertz (Hz), where one hertz corresponds to one cycle per second. The frequency of radiation determines its position within the electromagnetic spectrum, ranging from low-frequency radio waves to high-frequency gamma rays. Understanding these components and their interplay is crucial for unraveling the mysteries of photon energy and its implications in various scientific and technological applications.
Why is this important?
Understanding photon energy is pivotal in various fields. In medical applications, it helps us understand how infrared radiation interacts with our bodies in thermal imaging. In telecommunications, it's crucial for designing efficient fiber optic communication systems. Moreover, in astronomy, analyzing the energy of photons from distant stars and galaxies helps us understand the composition and evolution of the universe. So, yeah, it's pretty important stuff!
Problem Statement: Calculating Infrared Photon Energy
Okay, let's get to the problem at hand. We're tasked with finding the energy of a photon of infrared radiation. We know the frequency is 2.53 × 10¹² Hz, and we're given Planck's constant (h) as 6.63 × 10⁻³⁴ J⋅s. The question essentially boils down to applying the formula we discussed earlier: E = hν.
This type of problem is a classic example of how physics principles are applied to real-world scenarios. By understanding the relationship between frequency and energy, we can calculate the energy carried by a single photon, which is a fundamental concept in quantum mechanics. This calculation not only reinforces our understanding of the formula E = hν but also highlights the practical applications of physics in understanding the electromagnetic spectrum and its various forms of radiation. Moreover, solving this problem requires careful attention to units and significant figures, further honing our problem-solving skills in physics. By mastering such calculations, we gain a deeper appreciation for the intricate workings of the universe and the fundamental laws that govern it.
Breaking Down the Given Information
Before we jump into the calculation, let's clearly identify what we know:
- Frequency (ν): 2.53 × 10¹² Hz
- Planck's constant (h): 6.63 × 10⁻³⁴ J⋅s
Our goal is to find the energy (E) in joules (J). Now that we have all the pieces, let's put them together!
Step-by-Step Solution
Alright, let's walk through the solution step-by-step so you can see exactly how it's done. This is where the magic happens!
1. Recall the Formula
The first thing we need to do is remember the formula that links energy, frequency, and Planck's constant. As we discussed, it's:
E = hν
This simple yet powerful equation is the key to unlocking the energy of our infrared photon.
2. Plug in the Values
Next, we're going to substitute the given values into the formula. This is a straightforward step, but accuracy is crucial. We don't want to mess up the numbers!
E = (6.63 × 10⁻³⁴ J⋅s) × (2.53 × 10¹² Hz)
Notice how we've replaced h with its value and ν with the given frequency. Now it's just a matter of doing the math.
3. Perform the Calculation
Now comes the fun part – the calculation itself! We'll multiply the two numbers together. When multiplying numbers in scientific notation, we multiply the coefficients and add the exponents.
E = (6.63 × 2.53) × 10⁽⁻³⁴ ⁺ ¹²⁾ J
E ≈ 16.7739 × 10⁻²² J
4. Adjust Scientific Notation
We're almost there, but we need to make sure our answer is in proper scientific notation. That means the coefficient should be between 1 and 10. So, we'll adjust the decimal point:
E ≈ 1.67739 × 10⁻²¹ J
5. Round to Significant Figures
Finally, let's round our answer to the appropriate number of significant figures. In this case, both our given values (Planck's constant and frequency) have three significant figures, so our answer should also have three.
E ≈ 1.68 × 10⁻²¹ J
And there we have it! We've successfully calculated the energy of the infrared photon.
Analyzing the Answer
So, we've calculated the energy of the photon to be approximately 1.68 × 10⁻²¹ J. This is a tiny amount of energy, which is characteristic of individual photons. Remember, photons are the fundamental units of electromagnetic radiation, and their energies are often very small on a macroscopic scale.
Comparing with the Options
Now, let's compare our result with the options given in the problem:
A. 1.68 × 10²³ J B. 1.68 × 10⁴⁷ J C. 1.68 × 10⁻²¹ J D. *1.68 *
Our calculated value matches option C, 1.68 × 10⁻²¹ J. Options A and B are way too large, and option D is missing the exponent. This confirms that our calculation is correct.
The Significance of the Result
The fact that the energy is on the order of 10⁻²¹ joules highlights the quantum nature of light. It emphasizes that energy is quantized, meaning it exists in discrete packets (photons) rather than a continuous stream. This concept is a cornerstone of quantum mechanics and has profound implications in various fields, from laser technology to medical imaging.
Key Takeaways and Conclusion
We've journeyed through the process of calculating the energy of an infrared photon, and hopefully, you now have a solid understanding of the concepts involved. Let's recap the key takeaways:
- The energy of a photon is directly proportional to its frequency, as described by the equation E = hν.
- Planck's constant (h) is a fundamental constant that links energy and frequency.
- Calculating photon energy involves plugging in the given values and performing the calculation carefully, paying attention to units and significant figures.
- The energy of individual photons is typically very small, reflecting the quantum nature of light.
So, the correct answer to our problem is C. 1.68 × 10⁻²¹ J. But more importantly, we've gained a deeper understanding of photon energy and its significance in the world around us. Keep exploring, guys, and keep asking questions! The universe is full of fascinating mysteries waiting to be uncovered.
Practice Problems
To solidify your understanding, try solving these practice problems:
- What is the energy of a photon of visible light with a frequency of 5.00 × 10¹⁴ Hz?
- A photon has an energy of 3.98 × 10⁻¹⁹ J. What is its frequency?
- Calculate the energy of a photon of ultraviolet radiation with a frequency of 1.50 × 10¹⁵ Hz.
Solving these problems will help you become more comfortable with the formula E = hν and the concept of photon energy. Good luck, and happy calculating!
Further Exploration
If you're interested in learning more about photons and their energy, here are some topics you might want to explore:
- The electromagnetic spectrum and the different types of radiation (radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, gamma rays)
- The photoelectric effect and its implications for the quantum nature of light
- Applications of photon energy in various technologies, such as lasers, solar cells, and medical imaging
There's a whole universe of knowledge out there, so keep learning and keep exploring!