Question

Rank the following according to the wavelength of radiant energy each emits from the shortest wavelength to the longest. a. A light bulb with a filament glowing at \(4000^{\circ} \mathrm{C}\) b. A rock at room temperature c. A car engine at \(140^{\circ} \mathrm{C}\)

Short answer

1. Light bulb, 2. Car engine, 3. Rock

Step-by-step solution

Understand Blackbody Radiation

Objects emit radiation depending on their temperature according to the blackbody radiation principle. Higher temperature objects emit radiation with shorter wavelengths, while cooler objects emit radiation with longer wavelengths.

Convert Temperatures to Kelvin

First, convert the given temperatures from Celsius to Kelvin. The formula is: \[T(K) = T(°C) + 273.15\]Light bulb: \[4000^{\circ}\text{C} + 273.15 = 4273.15\text{K}\]Car engine: \[140^{\circ}\text{C} + 273.15 = 413.15\text{K}\]Room Temperature: The average room temperature is about \[25^{\circ}\text{C}\] which converts to \[25 + 273.15 = 298.15\text{K}\].

Use Wien's Displacement Law

Wien's Displacement Law helps to calculate the peak wavelength \( \lambda_{max} \) of the emitted radiation:\[\lambda_{max} = \frac{b}{T}\]where \( b = 2.897 \times 10^{-3} \text{ m K} \) is the Wien's displacement constant and \( T \) is the temperature in Kelvin.

Calculate Wavelength for Each Object

Substitute the temperatures into Wien's Displacement Law to find their respective peak wavelengths:- Light bulb: \[\lambda_{max} = \frac{2.897 \times 10^{-3}}{4273.15} \approx 6.78 \times 10^{-7} \text{ m}\]- Car engine: \[\lambda_{max} = \frac{2.897 \times 10^{-3}}{413.15} \approx 7.01 \times 10^{-6} \text{ m}\]- Room temperature rock:\[\lambda_{max} = \frac{2.897 \times 10^{-3}}{298.15} \approx 9.72 \times 10^{-6} \text{ m}\]

Rank by Wavelength

Based on the calculated wavelengths, the ranking from shortest to longest wavelength is:1. Light bulb at \(4000^{\circ}\text{C}\)2. Car engine at \(140^{\circ}\text{C}\)3. Rock at room temperature

Key concepts

Blackbody Radiation

Blackbody radiation is a concept that describes how objects emit electromagnetic radiation based on their temperatures. Any object with a temperature above absolute zero emits radiation in some form, and this radiation is referred to as blackbody radiation. The principle is rooted in the fact that the intensity and color of the emitted light depend solely on the temperature of the object, making it an "ideal" emitter.

• Higher temperatures produce more energetic radiation emissions, resulting in shorter wavelengths such as ultraviolet or visible light.
• Conversely, lower temperatures result in less energetic emissions, leading to longer wavelengths like infrared radiation.

Understanding blackbody radiation is essential for studying thermal emissions from various objects, natural or artificial, and is instrumental in fields such as astrophysics, climate science, and thermal imaging. It forms the basis for analyzing how objects like the sun, light bulbs, and even the human body lose heat to their surroundings.

Temperature Conversion

Temperature conversion is a necessary step when working with scientific formulas that involve temperature. Since different scales of temperature measurement are used, conversions ensure uniformity and accuracy in calculations. The most common conversion is from Celsius to Kelvin.

• The Kelvin scale is the SI unit of temperature and starts at absolute zero, which is <(-273.15)^{ ext{°C}} .
• To convert Celsius to Kelvin, you use the straightforward formula: .

In our given problem, converting the temperatures of various objects (like a light bulb filament and a car engine) to Kelvin is essential. This allows us to apply Wien's Displacement Law effectively, as the calculations require temperature in Kelvin.

Emission Wavelength

The emission wavelength is a key concept when analyzing how objects emit radiation. It represents the peak wavelength at which the maximum amount of radiation energy is emitted by an object. Wien's Displacement Law provides the relationship between the emission wavelength and temperature.Wien's Displacement Law states:\[ \lambda_{max} = \frac{b}{T} \]Where:

• \( \lambda_{max} \) is the peak emission wavelength.
• \( b = 2.897 \times 10^{-3} \text{ m K} \) is a constant specific to this formula.
• \( T \) refers to the temperature in Kelvin.

This law illustrates that the hotter an object is, the shorter the wavelength of the emission peak will be. For instance, a filament in a light bulb with a high temperature will emit light in the visible spectrum, while cooler objects like a rock at room temperature will emit primarily infrared radiation.

Kelvin Scale

The Kelvin scale is an absolute temperature scale and a fundamental unit of measurement in thermodynamics and many physics-related fields. Unlike the Celsius or Fahrenheit scales, it's not based on arbitrary reference points such as the freezing point of water.

• The Kelvin scale begins at absolute zero, the theoretical point where all molecular motion ceases, which is equivalent to <0K> or <(-273.15)^{ ext{°C}}> .
• It's utilized extensively in scientific research due to its direct correlation with the energy content of a system.

In the context of calculations involving Wien’s Displacement Law or other temperature-dependent equations, using Kelvin ensures precision and consistency. It helps scientists and engineers to predict how changes in temperature affect physical properties like radiation emission, making it crucial for disciplines ranging from astrophysics to chemical thermodynamics.