Question

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

Short answer

Answer: The peak wavelength for a blackbody at 3000 K is 966 nm, which falls into the Infrared portion of the electromagnetic spectrum.

Step-by-step solution

Use Wien's Displacement Law formula

Wien's Displacement Law states that the peak wavelength \(\lambda_{max}\) of a blackbody's radiation is inversely proportional to its temperature, or \(\lambda_{max} T = b\), where \(b\) is Wien's constant and has a value of \(2.898 \times 10^{-3} m K\). In our case, we have the temperature \(T = 3000 K\). To find the peak wavelength \(\lambda_{max}\), we can rearrange Wien's Displacement Law formula as follows: \(\lambda_{max} = \frac{b}{T}\) Now, we can plug in the values of \(b\) and \(T\) to calculate \(\lambda_{max}\).

Calculate the peak wavelength

Plugging in the values of Wien's constant \(b\) and the temperature \(T\), we find the peak wavelength: \(\lambda_{max} = \frac{2.898 \times 10^{-3} m K}{3000 K} = 9.66 \times 10^{-7} m\) Thus, the peak wavelength is \(9.66 \times 10^{-7} m\) or \(966 \ nm\). This is the answer for part (a).

Identify the Electromagnetic Spectrum portion

Now, we have the peak wavelength, and we need to identify which portion of the electromagnetic spectrum this wavelength falls into. The electromagnetic spectrum can be roughly divided as follows: - Radio waves: \(\lambda > 1 m\) - Microwaves: \(1 m > \lambda > 1 mm\) - Infrared: \(1 mm > \lambda > 700 nm\) - Visible light: \(700 nm > \lambda > 400 nm\) - Ultraviolet: \(400 nm > \lambda > 10 nm\) - X-rays: \(10 nm > \lambda > 1 pm\) - Gamma rays: \(\lambda < 1 pm\) Our calculated peak wavelength of \(966 nm\) falls into the Infrared portion of the electromagnetic spectrum. This is the answer to part (b).

Key concepts

Blackbody Radiation

Blackbody radiation refers to the electromagnetic radiation emitted by an idealized object known as a blackbody. A blackbody is a theoretical object that absorbs all radiation that falls onto it and reflects none. It is a perfect emitter and absorber of radiation. The radiation emitted is solely a function of the blackbody's temperature, which means the energy distribution of the radiation depends only on the temperature of the object and not on its shape or composition.

At any given temperature, a blackbody emits a characteristic spectrum of electromagnetic radiation. This spectrum is continuous and covers a range of wavelengths. Interestingly, the distribution of energy across different wavelengths follows a predictable pattern described by various laws of physics, one of which is Wien's Displacement Law. This law allows us to determine the wavelength at which the emission of radiation is strongest, or the 'peak' wavelength, for a given temperature. Understanding blackbody radiation is fundamental in fields such as astronomy, climate science, and even thermal imaging technology.

Electromagnetic Spectrum

The electromagnetic spectrum is a term used to describe the entire range of electromagnetic waves organized by their wavelengths or frequencies. These waves travel at the speed of light but vary in size, from the very long wavelengths of radio waves to the extremely short wavelengths of gamma rays.

The spectrum is traditionally divided into regions based on wavelength or frequency. These regions include, from longest to shortest wavelength: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Each region of the electromagnetic spectrum has unique properties and uses. For instance, radio waves are used for communication, microwaves for cooking and satellite transmissions, and X-rays for medical imaging. The visible light portion is the narrow band that can be detected by the human eye, encompassing all the colors that we can see from red to violet.

Infrared Wavelength

Infrared wavelength corresponds to a specific region of the electromagnetic spectrum, situated between microwaves and visible light. It covers wavelengths from roughly 1 millimeter to 700 nanometers (nm). The term 'infrared' means 'below red', referencing the fact that these wavelengths are just below the red end of the visible light spectrum that the human eye can perceive.

Objects at various temperatures emit infrared radiation. For instance, the human body radiates infrared, which is why night-vision goggles can detect people by their heat signature. Infrared is categorized into three types: near-infrared (closest to visible light), mid-infrared, and far-infrared (closest to microwave radiation). Each type is used in different technologies, such as remote controls, thermal imaging cameras, and telecommunications. Understanding infrared wavelengths is crucial for interpreting phenomena like the heat produced by stars, planets, and other celestial bodies in space.