/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Absorption of radiation. The pas... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 4

Absorption of radiation. The passage of radiation through a medium is accompanied by partial absorption of the radiation. In many cases (the linear theory) one can assume that the absorption in passing through a layer two units thick is the same as the absorption in successively passing through two layers, each one unit thick. a) Show that under this condition the absorption of radiation is subject to the law \(I=I_{0} \mathrm{e}^{-k l}\), where \(I_{0}\) is the intensity of the radiation falling on the absorbing substance, \(I\) is the intensity after passing through a layer of thickness \(l\), and \(k\) is a coefficient having the physical dimension inverse to length. b) In the case of absorption of light by water, the coefficient \(k\) depends on the wave length of the incident light, for example as follows: for ultraviolet \(k=1.4 \times\) \(10^{-2} \mathrm{~cm}^{-1} ;\) for blue \(k=4.6 \times 10^{-4} \mathrm{~cm}^{-1} ;\) for green \(k=4.4 \times 10^{-4} \mathrm{~cm}^{-1}\); for red \(k=2.9 \times 10^{-3} \mathrm{~cm}^{-1} .\) Sunlight is falling vertically on the surface of a pure lake 10 meters deep. Compare the intensities of these components of sunlight listed above the surface of the lake and at the bottom.

Short Answer

Expert verified
Answer: Blue and green light penetrate the deepest in the water, as their intensities reduce less compared to ultraviolet and red light at the bottom of the lake.

Step by step solution

01

Deriving the absorption law

To derive the law of absorption of radiation \(I=I_{0} \mathrm{e}^{-k l}\), according to the linear theory, we can assume that the absorption in passing through a layer two units thick is the same as the absorption in successively passing through two layers, each one unit thick. Let \(螖I\) be the amount of radiation absorbed in a small section of thickness \(螖l\). Since absorption is proportional to the intensity \(I\) and thickness \(螖l\), we can write: \(\frac{螖I}{I}=-k螖l\). Now integrate this equation to find the relationship between \(I\) and \(l\): \(\int_{I_{0}}^{I} \frac{dI}{I}=-k \int_{0}^{l} dl\).
02

Solve the integrals

Solving each integral, we have: \(\ln \left(\frac{I}{I_{0}}\right)=-kl\). Now, taking the exponent of both sides to remove \(\ln\): \(\frac{I}{I_{0}}=\mathrm{e}^{-kl}\) So the law of absorption of radiation is: \(I=I_{0} \mathrm{e}^{-k l}\).
03

Calculate the intensities at the bottom of the lake

Given the absorption coefficients for ultraviolet (\(k_{1}\)), blue (\(k_{2}\)), green (\(k_{3}\)), and red (\(k_{4}\)) lights, we need to calculate the intensities of these components at the bottom of the lake, with a depth of 10 meters (1000 cm). The intensities are given by: \(I_{i}=I_{0} \mathrm{e}^{-k_{i} l}\), for \(i=1,2,3,4\). Let \(I_{0}=100 \%\), then we can calculate the intensity ratios for each color component at the bottom of the lake: \(I_{1}=100 \% \times \mathrm{e}^{-(1.4\times10^{-2})(1000)}\) \(I_{2}=100 \% \times \mathrm{e}^{-(4.6\times10^{-4})(1000)}\) \(I_{3}=100 \% \times \mathrm{e}^{-(4.4\times10^{-4})(1000)}\) \(I_{4}=100 \% \times \mathrm{e}^{-(2.9\times10^{-3})(1000)}\)
04

Compare the intensities of the light components at the bottom of the lake

By evaluating the above expressions, we get the intensities at the bottom of the lake: \(I_{1}=0.244 \%\) \(I_{2}=40.8 \%\) \(I_{3}=42.3 \%\) \(I_{4}=5.14 \%\) As we can see, the intensities of the blue and green components of sunlight do not reduce as much as the ultraviolet and red components at the bottom of the lake. This means that blue and green light can penetrate deeper into the water, whereas ultraviolet and red light is mostly absorbed near the surface.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Beer-Lambert Law
The Beer-Lambert Law is a fundamental principle in the study of light absorption by a medium. It describes how the intensity of light decreases as it travels through a substance that absorbs light. In more scientific terms, the law establishes a relation where the absorbance of light is directly proportional to the concentration of the absorbing species and the path length through which the light travels. A mathematical expression for this concept is:
\[ I = I_0 e^{-k l} \]
Here, \(I\) represents the intensity of light after passing through the medium, \(I_0\) is the initial intensity of light before entering the medium, \(k\) is the absorption coefficient, and \(l\) denotes the path length. The negative sign in the exponent indicates that there is a reduction in light intensity as it travels through the absorbing medium. This principle applies widely in spectrophotometry and helps in measuring concentrations of solutions, among other applications.
Exponential Decay
Exponential decay is a process by which an original amount decreases at a rate proportional to its current value. When discussing light intensity in the context of the Beer-Lambert Law, exponential decay describes how the intensity of radiation diminishes as it penetrates a medium. This type of decay can be visualized using a graph showing a rapid decrease in intensity which then levels off as it approaches zero, never actually reaching zero. Its equation is given by the same expression as the Beer-Lambert Law:
\[ I = I_0 e^{-k l} \]
The key characteristic of exponential decay is that the amount of decrease in intensity is always proportional to its current intensity. This concept is exceedingly important, not just in physics but across many other fields including biology, economics, and even population studies.
Absorption Coefficient
The absorption coefficient, symbolized by \(k\), is a crucial parameter in the Beer-Lambert Law that quantifies the absorbing ability of a medium. It is a measure of how strongly a material can absorb light or other types of radiation at a given wavelength. The higher the absorption coefficient, the more the radiation is absorbed, and thus, the intensity is reduced more rapidly as it passes through the material. The unit for the absorption coefficient is inverse length (e.g., \(cm^{-1}\)) indicating it is a ratio of the amount of absorption per unit distance traveled by the light. Factors affecting the absorption coefficient include the properties of the absorbing substance, the temperature, and the wavelength of the incoming light. As demonstrated in our exercise, different wavelengths of light (ultraviolet, blue, green, red) have different absorption coefficients when passing through water, leading to variations in light intensities observed at different depths of the lake.
Intensity of Radiation
The intensity of radiation, represented as \(I\), is the power carried by radiation per unit area. In the context of the Beer-Lambert Law and our exercise, it is essential to understand that the initial intensity \(I_0\) will diminish as radiation passes through an absorbing medium due to the medium's absorption characteristics. The integrity of radiation is affected by many factors including medium composition, distance traveled through the medium, and the environment's physical conditions. In our example with the lake, sunlight (comprising different colors or wavelengths) enters the water surface with a certain intensity, but as it goes deeper, the water absorbs some of the light, causing the intensity to decrease. The different absorption coefficients for each color indicate how well that particular color can penetrate the water. For instance, blue and green light maintain relatively high intensities even at the bottom of a 10-meter deep lake, whereas ultraviolet and red lights get significantly absorbed closer to the surface.

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Most popular questions from this chapter

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