Ch 20: The Micro/Macro Connection

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 20: The Micro/Macro Connection

Problem 49b

Chapter 20, Problem 49b

Photons of light scatter off molecules, and the distance you can see through a gas is proportional to the mean free path of photons through the gas. Photons are not gas molecules, so the mean free path of a photon is not given by Equation 20.3, but its dependence on the number density of the gas and on the molecular radius is the same. Suppose you are in a smoggy city and can barely see buildings 500 m away. How far would you be able to see if the temperature suddenly rose from 20°C to a blazing hot 1500°C with the pressure unchanged?

Verified step by step guidance

Step 1: Understand the relationship between the mean free path of photons and the number density of gas molecules. The mean free path (λ) is inversely proportional to the number density (n) of the gas molecules. Mathematically, λ ∝ 1/n.

Step 2: Recall the ideal gas law, which relates the number density (n) to the pressure (P), temperature (T), and Boltzmann constant (k). The number density is given by n = P / (kT). Since the pressure (P) is constant, the number density is inversely proportional to the temperature: n ∝ 1/T.

Step 3: Combine the relationships from Steps 1 and 2. Since λ ∝ 1/n and n ∝ 1/T, it follows that λ ∝ T. This means the mean free path of photons (and thus the distance you can see) is directly proportional to the temperature.

Step 4: Calculate the ratio of the new mean free path to the original mean free path using the proportionality λ_new / λ_original = T_new / T_original. Here, T_original = 20°C = 293 K (convert to Kelvin by adding 273), and T_new = 1500°C = 1773 K.

Step 5: Multiply the original visibility distance (500 m) by the ratio of temperatures to find the new visibility distance. The new visibility distance is given by λ_new = λ_original × (T_new / T_original).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Mean Free Path

The mean free path is the average distance a particle travels between collisions with other particles. In the context of photons scattering off gas molecules, it describes how far a photon can travel before interacting with a molecule. This distance is influenced by the number density of the gas and the size of the molecules, which affects the likelihood of scattering events.

Temperature and Gas Density

Temperature affects the kinetic energy of gas molecules, which in turn influences their density. As temperature increases, gas molecules move faster and can spread out, potentially reducing the number density of the gas. In the scenario of rising temperature from 20°C to 1500°C, the gas may become less dense, impacting the mean free path of photons and thus visibility.

Scattering of Photons

Scattering occurs when photons interact with particles in a medium, such as gas molecules. This interaction can change the direction of the photons, affecting how far they can travel before reaching the observer. The extent of scattering is influenced by factors like the size of the particles and the wavelength of the light, which is crucial for understanding visibility in different atmospheric conditions.

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Textbook Question

Interstellar space, far from any stars, is filled with a very low density of hydrogen atoms (H, not H₂). The number density is about 1 atom/cm³ and the temperature is about 3 K. Estimate the pressure in interstellar space. Give your answer in Pa and in atm.

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Textbook Question

A gas cylinder has a piston at one end that is moving outward at speed v_piston during an isobaric expansion of the gas. Find an expression for the rate at which v_rms is changing in terms of v_piston, the instantaneous value of v_rms, and the instantaneous value L of the length of the cylinder.

Textbook Question

Interstellar space, far from any stars, is filled with a very low density of hydrogen atoms (H, not H₂). The number density is about 1 atom/cm³ and the temperature is about 3 K. What is the edge length L of an L ✕ L ✕ L cube of gas with 1.0 J of thermal energy?

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Textbook Question

Equation 20.3 is the mean free path of a particle through a gas of identical particles of equal radius. An electron can be thought of as a point particle with zero radius. Electrons travel 3.0 km through the Stanford Linear Accelerator. In order for scattering losses to be negligible, the pressure inside the accelerator tube must be reduced to the point where the mean free path is at least 50 km. What is the maximum possible pressure inside the accelerator tube, assuming T = 20℃? Give your answer in both P_a and atm.

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Textbook Question

Photons of light scatter off molecules, and the distance you can see through a gas is proportional to the mean free path of photons through the gas. Photons are not gas molecules, so the mean free path of a photon is not given by Equation 20.3, but its dependence on the number density of the gas and on the molecular radius is the same. Suppose you are in a smoggy city and can barely see buildings 500 m away. How far would you be able to see if all the molecules around you suddenly doubled in volume?

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Textbook Question

You are watching a science fiction movie in which the hero shrinks down to the size of an atom and fights villains while jumping from air molecule to air molecule. In one scene, the hero's molecule is about to crash head-on into the molecule on which a villain is riding. The villain's molecule is initially 50 molecular radii away and, in the movie, it takes 3.5 s for the molecules to collide. Estimate the air temperature required for this to be possible. Assume the molecules are nitrogen molecules, each traveling at the rms speed. Is this a plausible temperature for air?