Show that, for an ideal gas, P=(1)/(3) ρvrms^2 where P is the pressure, ρis the mass density, an

Show that, for an ideal gas, P=(1)/(3) ρvrms^2 where P is...

Key Concepts

Kinetic Theory of Gases

This concept links the microscopic motion of gas molecules to the macroscopic properties observed in a gas, such as pressure, temperature, and volume. The theory assumes that gas molecules move in random directions and their collisions with each other and the walls of a container are perfectly elastic, allowing one to derive expressions that connect these molecular motions to observable quantities.

Momentum Transfer

The idea of momentum transfer is central to understanding pressure in gases. Gas molecules constantly collide with the walls of their container, and each collision imparts a change in momentum. The cumulative effect of these momentum changes, averaged over many collisions, results in the macroscopic pressure that we measure in a gas.

Isotropic Velocity Distribution

This concept refers to the uniform distribution of molecular velocities in all directions. In three dimensions, the equal partitioning of kinetic energy among the three spatial directions leads to a factor of 1/3 in the pressure equation. It reflects the idea that, on average, only one third of the total kinetic energy of the gas contributes to the momentum transfer in any one specific direction, such as perpendicular to the container wall.

Root Mean Square Speed

Root mean square (rms) speed is a statistical measure of the speed of particles in a gas. It is computed as the square root of the average of the squared speeds of the molecules. This measure is directly related to the kinetic energy of the gas molecules and plays a critical role in linking microscopic motion to macroscopic observables like pressure.

Mass Density

Mass density in the context of gases is defined as the total mass of the gas per unit volume. It provides a measure of how much matter is present in a given space and is crucial in the derivation of relationships between microscopic kinetic properties and macroscopic pressure. The pressure expression P = (1/3)?v_rms² shows that the pressure depends on both the mass density and the average molecular kinetic energy.

Transcript

00:01 In this question we have to show that for an ideal gas the pressure is given by one -third of the mass density times the r -emass velocity squared.

00:11 So how can we show that? i will begin by using this equation.

00:16 So the ideal gas law.

00:17 The ideal gas law tells us that the pressure times the volume is equal to the number of molecules or atoms times boltzman's constant times the temperature.

00:28 Then we can solve this equation.

00:30 For the pressure to get that the pressure is given by n times k times t divided by the volume.

00:38 Now take a look at this factor of k times t right here.

00:43 So this factor of k times t also appear in the expression for the average kinetic energy of a single molecule or atom of ideal gas.

00:53 So we can actually serve this equation for k times t.

00:58 By doing that, you get the following.

01:01 So the average kinetic energy is 3 divided by 2 times k times t.

01:08 It means that k times t is equal to 2 divided by 3 times the average kinetic energy.

01:17 Then, plugging in this result into this equation results in the following.

01:22 The pressure is given by n divided by v times 2, divided by 3 times the average kinetic energy.

01:34 Now, we must remember from classical mechanics that the kinetic energy is the mass times the velocity squared divided by 2...

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