Physics Friday 69

Consider a particle of mass m moving freely in a rectangular box, and undergoing perfectly elastic collisions with the walls of the box. Let us consider the motion in the x direction. Let the x-component of the particle’s velocity be vx, and the length of the box in the x direction be Lx. The time it takes to travel from one of the x-bounding walls to the other is , and so the number of collisions with a particular x-bounding wall per unit time is . When it collides with the wall, the x-component of the velocity changes from vx to –vx, and so, due to conservation of momentum, it imparts a momentum of 2m|vx|. The product of the momentum imparted per collision and the rate of collisions per unit time gives the (time-)average force exerted on the wall by the particle:

Now, let us replace the particle with N identical non-interacting particles (an ideal gas). Then the force on one of the x-bounding walls is
, where the bar represents the average over the N particles. Dividing this by the area A of the wall gives us pressure:

where is the volume of the box.
If our gas is isometric, then , where is the root-mean-square speed. This means that the pressure is the same on all sides of the box, and

Now, the Maxwell speed distribution tells us that for ideal gas molecule, the average kinetic energy is given by . This allows us to rewrite the above equation for pressure as
, or multiplying both sides by the volume,

This is the ideal gas law. Often, one is introduced to the law in terms of the number of moles, n, instead of the number of molecules N. As , where NA is Avogadro’s number, this gives us the form more commonly seen in introductory courses:
where is the ideal gas constant.


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4 Responses to “Physics Friday 69”

  1. Physics Friday 71: The Isothermal Atmosphere « Twisted One 151’s Weblog Says:

    […] of finding a particle at a given height z above the “floor” of our column? The ideal gas law tells us that for an ideal gas of N particles, the themperature T, pressure P, and volume V obey […]

  2. Physics Friday 72 « Twisted One 151’s Weblog Says:

    […] Friday 72 By twistedone151 When we previously derived the ideal gas law PV=NkT=nRT from a particle bouncing in a box, we used the Maxwell speed […]

  3. Physics Friday 73 « Twisted One 151’s Weblog Says:

    […] freedom for each molecule (f≥3), we have equation of state and internal energy , as discussed here and here. Now, we can use this to find the heat capacity for a given quantity of an ideal gas. The […]

  4. Physics Friday 88 « Twisted One 151’s Weblog Says:

    […] for a classical ideal gas whose molecules have only translational degrees of freedom available (see here and here). Next week, I intend to show that this reduces to the classical ideal gas for high […]

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