Heat and Thermodynamics


   
 
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 a 
molecule moving with velocity . The components of the velocity along the axes are vx, vy and vz.
 
 
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 -vx. Any collision of the molecule with any other face (except for Ax) does not change the value of vx. Therefore, it travels between A1 and A2 with a constant X-component of the velocity, which is equal to -vx.
 
 
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 as   the 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 vrms. Thus,
 
 
 
Equation (iv) may be written as
 
 
 
  
     
   
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