Problem 34 - Chemistry: An Atoms-Focused Approach

Question

Incandescent Lightbulbs A variable power supply is connected to an incandescent lightbulb. At the lowest power setting, the bulb feels warm to the touch but produces no light. At medium power, the lightbulb filament emits a red glow. At the highest power, the lightbulb emits white light. Explain this emission pattern.

Step-by-Step Solution

Answer

Question: Explain the emission pattern of an incandescent light bulb at three different power settings: lowest, medium, and highest. Answer: The emission pattern of an incandescent light bulb at different power settings is determined by the filament's temperature and the wavelength of light emitted, which is governed by the concept of blackbody radiation and Wien's displacement law. At the lowest power setting, the temperature is not high enough to emit visible light, and the bulb only emits infrared radiation. At the medium power setting, the temperature increases, and the filament emits red light due to a shift in the peak wavelength. At the highest power setting, the filament temperature is further increased, emitting a larger portion of the visible spectrum, which is perceived as white light.

Step 1: Understanding Blackbody Radiation

Blackbody radiation is the phenomenon where an object emits electromagnetic radiation due to its temperature. The hotter the object, the higher the frequency (shorter wavelength) of the emitted radiation. This is described by Planck's law, which gives the spectral distribution of radiation emitted by a black body at a given temperature.

Step 2: Relationship between Temperature and Wavelength

According to Wien's displacement law, the wavelength of light emitted at the peak of the blackbody radiation curve is inversely proportionalt to the temperature of the object. This law can be expressed as follows: λ_max (T) = (b) / T where λ_max is the peak wavelength, T is the temperature in Kelvin, and b is Wien's constant (2.9 x 10^{-3} m*K).

Step 3: Power Settings and Filament's Temperature

At different power settings, the filament of the light bulb will reach different temperatures. Let's consider the temperature and the color of light emitted at each power setting:

Step 4: a) Lowest power setting:

At the lowest power setting, the temperature of the filament is not high enough to emit visible light. The radiation emitted is mainly in the infrared range, and as a result, the bulb feels warm to the touch but produces no visible light.

Step 5: b) Medium power setting:

At the medium power setting, the temperature of the filament is increased. According to Wien's displacement law, the peak wavelength of the emitted radiation shifts to shorter wavelengths, with a larger fraction now in the visible range. The lowest energy visible light is red, so at this stage, the filament emits a red glow.

Step 6: c) Highest power setting:

At the highest power setting, the filament temperature is further increased, causing a shift in the peak wavelength to even shorter values. Now, a larger portion of the emitted radiation is within the visible spectrum, covering all colors and resulting in a combination that is perceived as white light. In conclusion, the emission pattern of the incandescent lightbulb at different power settings can be explained by the relationship between the filament's temperature and the wavelength of light emitted, which is described by blackbody radiation and Wien's displacement law.

Key Concepts

Wien's Displacement Law

As an object becomes hotter, the color of the light it emits changes. This change can be explained through Wien's Displacement Law, which states that hotter objects will emit radiation with shorter wavelengths. The peak wavelength \(\lambda_{\text{max}}\)as described in this law is inversely proportional to the temperature of the object. This means that as the temperature increases, the peak wavelength of the emitted radiation decreases.

Mathematically, this relationship is expressed using the formula:
\[ \lambda_{\text{max}} = \frac{b}{T}\]where \(\lambda_{\text{max}}\) is the peak wavelength, \(T\) is the temperature in Kelvin, and \(b\) is Wien's constant (approximately \(2.9 \times 10^{-3} \text{ m*K}\)).

• Low Temperature: Emits longer wavelengths (infrared).
• Medium Temperature: Shifts to shorter wavelengths (red).
• High Temperature: Emits even shorter wavelengths (white).

Understanding this law helps us interpret the behavior of objects such as lightbulbs, which shift colors from infrared to red to white as their temperature increases.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation. Radiation is classified by wavelength or frequency, and the spectrum is typically divided into several regions such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Each type of radiation in the electromagnetic spectrum has a different wavelength and frequency.

When considering the light emitted by a hot filament, like in a lightbulb, the focus is typically on the visible part of the spectrum, which ranges from approximately 400 nm (violet) to 700 nm (red). As the filament heats:

• Initial warmth primarily emits infrared, which we feel as heat.
• As it heats, it emits visible red light at the lower end of visible spectrum.
• At the highest temperatures, it emits light across the visible spectrum, perceived as white light.

The change in color with temperature illuminates the relationship between wavelength and energy across the electromagnetic spectrum.

Planck's Law

Planck's Law is a pivotal concept in understanding blackbody radiation. It describes how the intensity of electromagnetic radiation emitted by a blackbody at a certain temperature is distributed across various wavelengths. This law provides a model for the spectral distribution of radiation that helps explain the interplay of light with temperature.

The Planck's law formula is expressed as:\[I(\lambda, T) = \frac{8\pi h c}{\lambda^5} \left( \frac{1}{e^{\frac{hc}{\lambda k T}} - 1} \right)\]Where:

• \(I(\lambda, T)\) is the intensity at wavelength \(\lambda\) and temperature \(T\).
• \(h\) is Planck's constant.
• \(c\) is the speed of light.
• \(k\) is the Boltzmann constant.

Planck's law accounts for the observed shift in peak emission wavelength with temperature, thus expanding on the understanding provided by Wien’s Displacement Law. In light bulbs, it explains why the spectrum of emitted light changes as the filament temperature increases, manifesting as the transition from infrared, to red, and ultimately to white light.