Question

Three hundred thousand years after the Big Bang, the average temperature of the universe was about \(3000 \mathrm{~K}\). a) At what wavelength of radiation would the blackbody spectrum peak for this temperature? b) To what portion of the electromagnetic spectrum does this correspond?

Short answer

Answer: The peak wavelength of radiation for this temperature is 967 nm, and it corresponds to the near-infrared (NIR) portion of the electromagnetic spectrum.

Step-by-step solution

Wien's Law

In order to find the peak wavelength, we will apply Wien's Displacement Law formula, which is given by: $$\lambda_{max} = \dfrac{b}{T}$$ where \(\lambda_{max}\) is the peak wavelength, \(b\) is Wien's constant (approximately \(2.9 \times 10^{-3} \mathrm{m·K}\)), and \(T\) is the temperature in Kelvin.

Calculate the peak wavelength

Using the provided temperature, \(T = 3000 \mathrm{~K}\), we can use Wien's Law to find the peak wavelength: $$\lambda_{max} = \dfrac{2.9 \times 10^{-3} \mathrm{m·K}}{3000 \mathrm{~K}}$$ $$\lambda_{max} = 9.67 \times 10^{-7} \mathrm{m} = 967 \mathrm{~nm}$$ So, the peak wavelength of radiation at this temperature is \(967 \mathrm{~nm}\).

Determine the portion of the electromagnetic spectrum

To determine the corresponding portion of the electromagnetic spectrum, consult the wavelengths of different regions of the spectrum: - Ultraviolet (UV): \(10 \mathrm{~nm}\) to \(400 \mathrm{~nm}\) - Visible light: \(400 \mathrm{~nm}\) to \(700 \mathrm{~nm}\) - Near-infrared (NIR): \(700 \mathrm{~nm}\) to \(2500 \mathrm{~nm}\) The calculated peak wavelength, \(967 \mathrm{~nm}\), falls within the near-infrared (NIR) region of the electromagnetic spectrum. So, the answers are: a) The peak wavelength of radiation for this temperature is \(967 \mathrm{~nm}\). b) This corresponds to the near-infrared (NIR) portion of the electromagnetic spectrum.

Key concepts

Blackbody Radiation

At its core, blackbody radiation is the theoretical thermal emission of electromagnetic radiation from an idealized object which perfectly absorbs all radiation incident upon it. This hypothetical object is called a blackbody. In the realms of physics and astronomy, understanding blackbody radiation is crucial since it provides insight into the thermal properties of objects such as stars, planets, and even the early universe.

A blackbody does not reflect or transmit any light, and therefore, it emits radiation that is solely dependent on its temperature. This emitted spectrum of radiation is continuous and has a characteristic distribution that peaks at a certain wavelength. This is a fundamental concept in thermodynamics and quantum mechanics, as it connects temperature with quantum emissions of electromagnetic waves. The study of blackbody radiation led to the development of quantum mechanics after classical physics could not explain the observed spectra.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, arranged according to their wavelength or frequency. These types of radiation include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays.

Each part of the electromagnetic spectrum has unique properties and interacts differently with matter. For instance, the visible light spectrum is what is detectable by the human eye and is crucial for daily activities like seeing. On the other hand, radio waves are used for communication, and X-rays are employed in medical imaging. The concept of the electromagnetic spectrum is fundamental in understanding the applications and effects of different types of radiation in various technologies and natural phenomena.

Peak Wavelength Calculation

To calculate the peak wavelength of the radiation emitted by a blackbody, scientists use Wien's Displacement Law. This law provides a simple relationship between the temperature of the blackbody and the wavelength at which the emission is strongest.

The formula for Wien's Displacement Law is:
\[\lambda_{max} = \frac{b}{T}\]
Here, \(\lambda_{max}\) is the peak wavelength, \(b\) is Wien's constant (approximately \(2.9 \times 10^{-3} \mathrm{m\cdot K}\)), and \(T\) is the temperature in Kelvin. This calculation aids in understanding how the temperature of an object affects the type of radiation it emits most strongly. For example, hotter objects emit peak radiation at a shorter wavelength, which is why the filament of a light bulb glows with visible light, while cooler objects like humans emit infrared radiation.