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 45

Chapter 20, Problem 45

The pressure inside a tank of neon is 150 atm. The temperature is 25℃. On average, how many atomic diameters does a neon atom move between collisions?

Verified step by step guidance

Step 1: Convert the temperature from Celsius to Kelvin using the formula:

T = T + 273.15

. This is necessary because the ideal gas law and kinetic theory calculations require temperature in Kelvin.

Step 2: Use the ideal gas law to calculate the number density of neon atoms in the tank. The formula is:

n = \frac{P}{kT}

, where

P

is the pressure,

k

is Boltzmann's constant, and

T

is the temperature in Kelvin.

Step 3: Calculate the mean free path of a neon atom using the formula:

λ = \frac{1}{4 π d^2 n}

, where

d

is the atomic diameter of neon and

n

is the number density calculated in Step 2.

Step 4: Express the mean free path in terms of atomic diameters by dividing the mean free path

λ

by the atomic diameter

d

. This gives the average number of atomic diameters a neon atom moves between collisions.

Step 5: Ensure all units are consistent throughout the calculations (e.g., pressure in atm, temperature in Kelvin, atomic diameter in meters) to avoid errors. Substitute the known values for pressure, temperature, Boltzmann's constant, and atomic diameter to complete the calculation.

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Key Concepts

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

Kinetic Molecular Theory

Kinetic Molecular Theory explains the behavior of gases in terms of particles in constant motion. It posits that gas particles move freely and collide elastically with each other and the walls of their container. This theory helps in understanding how temperature and pressure affect the speed and frequency of collisions between gas particles.

Mean Free Path

Mean Free Path is the average distance a particle travels between collisions in a gas. It is influenced by factors such as the density of the gas and the size of the particles. Understanding mean free path is crucial for calculating how far a neon atom moves before colliding with another atom in the tank.

Ideal Gas Law

The Ideal Gas Law relates pressure, volume, temperature, and the number of moles of a gas through the equation PV = nRT. This law provides a framework for understanding the behavior of gases under various conditions. In this context, it can be used to derive relationships that help estimate the mean free path and the behavior of neon gas in the tank.

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Related Practice

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

What is the entropy change of the nitrogen if 250 mL of liquid nitrogen boils away and then warms to 20℃ at constant pressure? The density of liquid nitrogen is 810 kg/m³.

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A 75 g ice cube at 0℃ is placed on a very large table at 20℃. You can assume that the temperature of the table does not change. As the ice cube melts and then comes to thermal equilibrium, what are the entropy changes of (a) the water, (b) the table, and (c) the universe?

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Textbook Question

Dust particles are ≈ 10 μm in diameter. They are pulverized rock, with ρ ≈ 2500 kg/m³. If you treat dust as an ideal gas, what is the rms speed of a dust particle at 20℃?

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A mad engineer builds a cube, 2.5 m on a side, in which 6.2-cm-diameter rubber balls are constantly sent flying in random directions by vibrating walls. He will award a prize to anyone who can figure out how many balls are in the cube without entering it or taking out any of the balls. You decide to shoot 6.2-cm-diameter plastic balls into the cube, through a small hole, to see how far they get before colliding with a rubber ball. After many shots, you find they travel an average distance of 1.8 m. How many rubber balls do you think are in the cube?

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Textbook Question

2.0 mol of helium at 280℃ undergo an isobaric process in which the helium entropy increases by 35 J/K. What is the final temperature of the gas?

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