Expression for Pressure

 

 

Consider an ideal gas enclosed in a cubical vessel of edge L. Take a corner of the vessel as the origin O and the X-,Y-, Z- axes along the edges (figure 1). Let A₁ and A₂ be the parallel faces, perpendicular to the X-axis. Consider amolecule moving with velocity . The components of the velocity along the axes are v_x, v_y and v_z.

When the molecule collides with the face A₁, the x-components remain unchanged. This follows from our assumption that the collisions of the molecules with the wall are perfectly elastic. The change in momentum of the molecule is

As the linear momentum remains conserved in a collision, the change in momentum of the wall is

------(i)

After rebound, this molecule travels towards A₂ with the X- component of velocity equal to -v_x. Any collision of the molecule with any other face (except for A_x) does not change the value of v_x. Therefore, it travels between A₁ and A₂ with a constant X-component of the velocity, which is equal to -v_x.

Note :

that we can neglect any collisions with the other molecules in view of the last assumption discussed in the previous section.

The distance traveled parallel to the X-direction between A₁ and A₂ = L. Thus, the time taken by the molecule to go from A₁ to

The molecule rebounds from A₂, travels towards A₁ and collides with it after another time interval Thus, the time between two consecutive collisions of this molecule with A₁ is The number of collisions of this molecule with A₁ in unit time is

The momentum imparted per unit time to the wall by this molecule is, from (i) and (ii),

This is also the force exerted on the wall A₁ due to this molecule. The total force on the wall A₁ due to all the molecules is

----- (iii)

As all directions are equivalent, we have

If N is the total number of molecules in the sample, we can write

The pressure is force per unit area so that

where M is the total mass of the gas taken and r is its density. Also, is the average of the speeds squared. It is written asthe mean square speed. Thus, the pressure is

Root mean square speed

The square root of mean square speed is called root-mean-square speed or rms speed. It is denoted by the symbol v_rms. Thus,

Equation (iv) may be written as