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Q. 23

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
Found in: Page 567

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Short Answer

During a physics experiment, helium gas is cooled to a temperature of at a pressure of What are (a) the mean free path in the gas, (b) the rms speed of the atoms, and (c) the average energy per atom?

(a) The mean free path in gas is

(b) The rms speed of atoms is

(c) The average energy per atom is

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Step by Step Solution

Step : 1 Introduction (part a)

(a) The thermodynamics depicts the link between the four state variables for an ideal gas, which are the container volume, the volume of the gas, the pressure that the gas is exerted, temperature of the gas and the number of moles of the gas in the container. Ideal gas law is given by equation in the form

Where is Boltzmann's constant and in SI unit its value is

The pressure is given in atm, so we need to convert it into Pascal by

We plug the values for and into equation to get number density number by

Number density

Step :2 Explanation 

If a component in a vapour travels a long distance, it collides with a lot of other molecules. The molecules take a long time to disperse to a new place as a result of these collisions.. The average distance that the molecule moves between the collisions is called the mean free path and it is given by equation in the form

Where is the radius of the molecule, is the number of molecules and is the volume. Helium is a monatomic gas, so its radius is . We plug the values for and into equation (2) to get by

Step :3  Kinetic energy (part b)

(b) The molecule with mass and velocity has an average translational kinetic energy and it is given by equation (20.19)in the form

The change in the temperature of the molecule affects its average translational kinetic energy, so it is related to the temperature per molecule in the form

As shown, both equations and have the same left side, so we can use these expressions to get an equation for root mean square velocity

The molecular mass of helium is . Converting this to , we get the mass of one atom of argon by

Now, we plug the values for, and into equation to get

Step :4 Energy for helium (part c)

(c) Plug the values for into equation to get the energy for helium

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