Question

The wavelength 10.0 $\mu$m is in the infrared region of the electromagnetic spectrum, whereas 600 nm is in the visible region and 100 nm is in the ultraviolet. What is the temperature of an ideal blackbody for which the peak wavelength ${\lambda }_m$ is equal to each of these wavelengths?

Short answer

289.8 K, 4830 K, and 28980 K for 10.0 µm, 600 nm, and 100 nm, respectively.

Step-by-step solution

Understand the Problem

We need to calculate the temperature of an ideal blackbody for which its peak wavelength ${\lambda }_m$ corresponds to specific given wavelengths: 10.0 $\mu$m, 600 nm, and 100 nm. These values indicate the type of electromagnetic region based on wavelength.

Recall Wien's Displacement Law

Wien's Displacement Law relates the temperature $T$ of a blackbody to its peak wavelength ${\lambda }_m$ using the formula: ${\lambda }_m\cdot T=2.898\times {10}^{-3}\text{ m K}$. We will use this formula to find $T$ for each given wavelength.

Convert Units for Consistency

Before substituting the wavelength values into Wien's displacement law, convert them to meters: $10.0\mu m=10.0\times {10}^{-6}\text{ m},600\text{ nm}=600\times {10}^{-9}\text{ m},100\text{ nm}=100\times {10}^{-9}\text{ m}.$

Calculate Temperature for 10.0 µm

Using Wien's law, set ${\lambda }_m=10.0\times {10}^{-6}\text{ m}$: $$T=\frac{2.898\times {10}^{-3}}{10.0\times {10}^{-6}}=289.8\text{ K}.$$

Calculate Temperature for 600 nm

With ${\lambda }_m=600\times {10}^{-9}\text{ m}$: $$T=\frac{2.898\times {10}^{-3}}{600\times {10}^{-9}}=4830\text{ K}.$$

Calculate Temperature for 100 nm

For ${\lambda }_m=100\times {10}^{-9}\text{ m}$: $$T=\frac{2.898\times {10}^{-3}}{100\times {10}^{-9}}=28980\text{ K}.$$

Conclusion

The corresponding temperatures are 289.8 K for 10.0 $\mu$m, 4830 K for 600 nm, and 28980 K for 100 nm.

Key concepts

blackbody radiation

Blackbody radiation refers to the electromagnetic radiation emitted by an object that absorbs all radiation incident upon it, making it a perfect absorber. This type of emitter is also known as an ideal blackbody. Such objects do not reflect or transmit any radiation and instead, hold all energy within them, emitting a continuous spectrum that depends solely on their temperature.
Understanding blackbody radiation is crucial as it serves as a fundamental model for describing how objects emit thermal radiation. The energy emitted by a blackbody is distributed across various wavelengths, and the intensity depends on the temperature of the blackbody.
Scientists use blackbody radiation concepts to understand real-world objects like stars and planets, which approximate ideal blackbody behavior by emitting radiation based on their temperature. Even though real-world objects are not perfect blackbodies, they often exhibit behavior similar enough to make these concepts very useful in astrophysics and thermal physics.

electromagnetic spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, ranging from very low-energy radio waves to extremely high-energy gamma rays. Light, as we see it, is just a tiny part of this expansive spectrum.
Key regions of the electromagnetic spectrum include:

• Radio waves
• Microwaves
• Infrared
• Visible light
• Ultraviolet
• X-rays
• Gamma rays

Each category within the spectrum is distinguished by its wavelength range. For instance, visible light comprises colors from violet (shortest wavelength) to red (longest wavelength).
Researchers across various fields, including astronomy and medicine, utilize different parts of the electromagnetic spectrum to explore the universe and diagnose health conditions, respectively. Understanding how each region of the spectrum behaves helps scientists unlock numerous practical applications.

peak wavelength

The peak wavelength is a critical concept in understanding blackbody radiation. It refers to the specific wavelength at which the emission of radiation from a blackbody is at its maximum.
According to Wien's Displacement Law, there's an inverse relationship between the temperature of a blackbody and its peak wavelength. This means that as the temperature of a blackbody increases, the peak wavelength, where the radiation is most intense, shifts to shorter values. This principle allows us to draw conclusions about the temperature of stars and other celestial bodies, simply by determining their peak wavelengths. Wien's Displacement Law is expressed mathematically as: $${\lambda }_m\cdot T=2.898\times {10}^{-3}\text{ m K}$$ where ${\lambda }_m$ is the peak wavelength and $T$ is the temperature of the blackbody.

infrared

Infrared radiation is a part of the electromagnetic spectrum with wavelengths longer than visible light but shorter than radio waves. Typically, infrared wavelengths range from about 700 nm to 1 mm.
Infrared radiation is not visible to the human eye but can be felt as heat. Objects emit infrared radiation as a function of their temperature, which makes this region valuable in understanding thermal characteristics of substances.
In everyday life, infrared is commonly used in:

• Remote controls
• Night-vision devices
• Thermal imaging cameras
• Infrared heaters

This radiation also plays an essential role in meteorology, astronomy, and medicine, such as spectroscopy, enhancing the understanding of both the Earth's atmosphere and distant celestial objects.

ultraviolet

Ultraviolet (UV) radiation consists of electromagnetic waves with wavelengths shorter than visible light, typically ranging from about 10 nm to 400 nm. UV radiation falls just beyond the violet end of the visible spectrum.
UV radiation is not visible to the naked eye, but it carries enough energy to cause chemical reactions, which makes it both useful and dangerous. For example, UV rays are crucial for synthesizing vitamin D in the skin, yet overexposure can result in skin damage and increase cancer risk.
Key uses of ultraviolet radiation include:

• Sterilization and disinfection
• Fluorescence studies in laboratory settings
• Water purification
• Studying the composition of stars and galaxies in astronomy

Understanding UV radiation helps scientists address health issues related to sun exposure and enhances our knowledge about interstellar matter.