Oxygen (O2) has a molar mass of 32.0 32.032.0 g/mol. Suppose an oxygen molecule traveling at this speed bounces back and forth between opposite sides of a cubical vessel 0.100.10 m on a side. What is the average force the molecule exerts on one of the walls of the container? (Assume that the molecule's velocity is perpendicular to the two sides that it strikes.)

Ch 18: Thermal Properties of Matter

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 18: Thermal Properties of Matter

Problem 31e

Chapter 18, Problem 31e

Oxygen (O₂) has a molar mass of $32.0$ g/mol. Suppose an oxygen molecule traveling at this speed bounces back and forth between opposite sides of a cubical vessel $0.10$ m on a side. What is the average force the molecule exerts on one of the walls of the container? (Assume that the molecule's velocity is perpendicular to the two sides that it strikes.)

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First, determine the speed of the oxygen molecule using the root-mean-square speed formula for gases: $ v_{rms} = \sqrt{\frac{3kT}{m}} $, where $ k $ is the Boltzmann constant, $ T $ is the temperature, and $ m $ is the mass of one molecule.

Calculate the time between collisions with the wall. Since the molecule travels back and forth across the cubical vessel, the time $ t $ between collisions is $ t = \frac{2L}{v} $, where $ L $ is the length of the side of the cube (0.10 m) and $ v $ is the speed of the molecule.

Determine the change in momentum $ \Delta p $ for each collision. The momentum change is $ \Delta p = 2mv $, where $ m $ is the mass of the molecule and $ v $ is its velocity. The factor of 2 accounts for the reversal of direction upon collision.

Calculate the average force $ F $ exerted on the wall using the impulse-momentum theorem: $ F = \frac{\Delta p}{\Delta t} $, where $ \Delta t $ is the time between collisions.

Substitute the values obtained from previous steps into the formula for average force to find the force exerted by the molecule on the wall.

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Key Concepts

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

Kinetic Theory of Gases

The kinetic theory of gases explains the behavior of gas molecules in terms of their motion and interactions. It assumes that gas molecules are in constant, random motion and that they collide elastically with the walls of their container. This theory helps in understanding how the speed and mass of molecules relate to the pressure exerted on the container walls.

Momentum and Impulse

Momentum is the product of an object's mass and velocity, representing the quantity of motion it possesses. Impulse is the change in momentum resulting from a force applied over a time interval. In this context, the force exerted by the molecule on the container wall can be calculated by considering the change in momentum as the molecule bounces off the wall.

Newton's Third Law of Motion

Newton's Third Law states that for every action, there is an equal and opposite reaction. When the oxygen molecule strikes the wall of the container, it exerts a force on the wall, and the wall exerts an equal and opposite force on the molecule. This principle is crucial for calculating the average force exerted by the molecule on the container wall.

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