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 1

Chapter 20, Problem 1

The mean free path of a molecule in a gas is 300 nm. What will the mean free path be if the gas temperature is doubled at (a) constant volume and (b) constant pressure?

Verified step by step guidance

The mean free path (λ) of a molecule in a gas is given by the formula:

λ = \frac{1}{√2 π d_{2} n}

, where

d_{2}

is the molecular diameter and

n

is the number density of molecules. The number density depends on the pressure, temperature, and volume of the gas.

For part (a), at constant volume, the number density

n

is proportional to

\frac{P}{T}

, where

P

is pressure and

T

is temperature. Since the volume is constant and the temperature is doubled, the number density

n

will decrease by a factor of 2. Substituting this into the formula for mean free path, the mean free path

λ

will increase by a factor of 2.

For part (b), at constant pressure, the number density

n

is proportional to

\frac{1}{T}

. Since the temperature is doubled, the number density

n

will decrease by a factor of 2. Substituting this into the formula for mean free path, the mean free path

λ

will increase by a factor of 2.

To summarize, in both cases (a) constant volume and (b) constant pressure, the mean free path will increase by a factor of 2 when the temperature is doubled.

Finally, multiply the initial mean free path (300 nm) by the factor of 2 to determine the new mean free path in each case. This step involves simple multiplication.

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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 molecule travels between collisions with other molecules. It is influenced by factors such as the density of the gas and the size of the molecules. In general, a higher density or larger molecules result in a shorter mean free path, while a lower density or smaller molecules lead to a longer mean free path.

Kinetic Theory of Gases

The kinetic theory of gases describes the behavior of gases in terms of the motion of their molecules. It states that gas temperature is proportional to the average kinetic energy of the molecules. When the temperature of a gas increases, the molecules move faster, which can affect the mean free path depending on whether the volume or pressure is held constant.

Ideal Gas Law

The ideal gas law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law helps to understand how changes in temperature affect gas behavior. Under constant volume, increasing temperature leads to increased pressure, while under constant pressure, the volume must increase to accommodate the higher temperature, impacting the mean free path.