same velocity; there is a distribution in velocities. The above equation, therefore, stands for pressure due to the group of molecules with speed v x in the x -direction and n stands for the number density of that group of molecules. The total pressure is obtained by summing over the contribution due to all groups: P = n m ( . ) where is the average of v x .
Now the gas is isotropic, i.e. there is no preferred direction of velocity of the molecules in the vessel. Therefore, by symmetry, = ( / ) [ ] = ( / ) ( . ) where v is the speed and denotes the mean of the squared speed.
Thus P = ( / ) n m ( . ) Some remarks on this derivation. First, though we choose the container to be a cube, the shape of the vessel really is immaterial. For a vessel of arbitrary shape, we can always choose a small infinitesimal (planar) area and carry through the steps above.
Notice that both A and ∆ t do not appear in the final result. By Pascal’s law, given in Ch. , pressure in one portion of the gas in equilibrium is the same as anywhere else. Second, we have ignored any collisions in the derivation.
Though this assumption is difficult to justify rigorously, we can qualitatively see that it will not lead to erroneous results. The number of molecules hitting the wall in time ∆ t was found to be ½ n Av x ∆ t . Now the collisions are random and the gas is in a steady state. Thus, if a molecule with velocity ( v x , v y , v z ) acquires a different velocity due to collision with some molecule, there will always be some other Fig.
. Elastic collision of a gas molecule with the wall of the container. To calculate the force (and pressure) on the wall, we need to calculate momentum imparted to the wall per unit time. In a small time interval ∆ t , a molecule with x -component of velocity v x will hit the wall