Ch. 18 - Kinetic Theory of Gases

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 18 - Kinetic Theory of Gases

Problem 22a

Chapter 18, Problem 22a

A gas consisting of 14,500 molecules, each of mass 2.00 x 10⁻²⁶ kg, has the following distribution of speeds, which crudely mimics the Maxwell distribution. Determine vᵣₘₛ for this distribution of speeds.

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Identify the formula for the root-mean-square (rms) speed, which is given by:

\sqrt{\frac{\sum n_{i} {v_{i}}^{2}}{\sum}}

, where ni is the number of molecules at a given speed vi.

Extract the data from the distribution of speeds provided in the problem. For each speed vi, note the corresponding number of molecules ni. This information is necessary to calculate the numerator and denominator of the formula.

Calculate the numerator of the formula, which is the sum of the products of the number of molecules and the square of their respective speeds:

\sum n_{i} {v_{i}}^{2}

. For each speed, square the speed value, multiply it by the corresponding number of molecules, and then sum these products.

Calculate the denominator of the formula, which is the total number of molecules:

\sum n_{i}

. Add up all the values of ni to find the total number of molecules in the distribution.

Divide the numerator by the denominator and take the square root to find the root-mean-square speed:

\sqrt{\frac{\sum n_{i} {v_{i}}^{2}}{\sum}}

. This will give the value of vrms.

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Key Concepts

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

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the distribution of speeds of particles in a gas at thermal equilibrium. It shows that most particles have speeds around a certain value, with fewer particles having very high or very low speeds. This statistical distribution is fundamental in understanding the kinetic theory of gases and how temperature affects molecular motion.

Root Mean Square Speed (vᵣₘₛ)

The root mean square speed (vᵣₘₛ) is a measure of the average speed of particles in a gas, calculated as the square root of the average of the squares of the speeds of all particles. It provides insight into the kinetic energy of the gas molecules and is essential for relating temperature to molecular motion in the context of kinetic theory.

Kinetic Theory of Gases

The kinetic theory of gases explains the behavior of gases in terms of the motion of their molecules. It posits that gas pressure arises from collisions of molecules with the walls of a container and that temperature is a measure of the average kinetic energy of the molecules. This theory underpins many concepts in thermodynamics and statistical mechanics.

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