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Why a Gas Exerts a Pressure
We will use the kinetic theory of matter to attempt to explain some aspects of  the behaviour of gases.
The theory suggests that the molecules of a gas move at random, occupying all the space available to them.  
 
The molecules are continually colliding with each other and with the walls of the container.  
On the diagram, only a few paths of molecules have been drawn because...itís a bit tedious drawing them!  
Remember that, in the real case, the numbers of particles are "ginormous", for example, of the order of 1027 particles in a few litres of air (at normal atmospheric pressure).  
It is assumed that all collisions are perfectly elastic.  
To convince yourself that this must be the case, consider what would happen if the collisions were not elastic.  
At each collision, the colliding particles would lose some kinetic energy (slow down).
Collisions between molecules in the air which you are breathing now occur thousand of times per second.
Therefore, even if only a small percentage of the K.E. was lost at each collision, in a very short time, the air would have turned to a liquid... this does not happen.
Conclusion: collisions elastic.
 
   
When a molecule collides with the wall of the container, a change of momentum occurs.  
The change in momentum is caused by the force exerted by the wall on the molecule (see here for the relation between force and momentum)  
The molecule exerts an equal but opposite force on the wall (Newton's third law).  
The pressure exerted by the gas is due to the sum of all these collision forces.  
   
Why the Pressure Exerted by a Gas Increases as the Temperature Increases  
If the temperature of the gas is increased, the average kinetic energy of its molecules increases.  
Therefore, the molecules hit the wall "harder" and also more frequently.  
The total force due to the collisions is greater.  
Therefore the pressure increases.  
   
Why the Temperature of a Gas Increases When the Gas is Compressed  
The diagram below represents a quantity of gas in a cylinder with a moveable piston.  
 
Molecules experiencing elastic collisions with stationary piston will rebound at the same speed.  
However, molecules colliding with the surface of the piston as it is moving so as to compress the gas into a smaller volume will rebound moving faster than before the collision.  
Thus the average speed of the molecules will increase.  
This means the temperature will increase.  
   
Similarly, of course, if the piston is moving the opposite way, the average speed (and therefore the temperature) will decrease.  
   
Why the Pressure Decreases When The Volume Increases  
Consider some gas in a cylinder with a movable piston.  
 
Imagine that the piston is displaced so as to increase the volume occupied by the gas.  
 
The gas is now less dense.  
We will also imagine that the gas is at the same temperature in the two cases (this can be arranged by heating after the expansion).  
   
Now, after the increase in volume:  
1. The molecules have, on average, a shorter distance to move between collisions and therefore they will experience a greater number of collisions per unit time and
2. The total surface area of the cylinder/piston with which the molecules are interacting will increase.
 
Both these changes will have the effect of decreasing the pressure exerted by the gas.  
This logic can, of course, be reversed to consider a decrease in volume.  
   
Thus, the kinetic theory predicts that the pressure of a gas will decrease if its volume increases.  
This is confirmed by experiment.  
See also the Boyle-Mariotte law.  
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