How Does Kinetic Theory Apply to Real Gas Behavior with Few Atoms?

[kelvin490]

I have a question about deriving the equation of kinetic theory of ideal gas - PV=1/3 Nmc_r², where N is number of atoms, c_r is root mean square of atom speed and m is mass of one atom.

In deriving the equation, many textbooks consider the total rate of change of momentum within a certain time and calculate the pressure. However, in actual case the atoms collide on the wall of container separately at different position at different time, how can these changes of momentum be considered as a whole?

 

[sophiecentaur]

If you spray a hose against a door, the door will move. each drop of water has imparted its share of momentum. Likewise with the molecules hitting the sides of the container. It is the normal component of the momentum of each that imparts an impulse onto the side. The effect is to produce a net force that is averaged over the area to give a pressure. Is that enough or is something still not right for you? This has given a number of people trouble.

 

[kelvin490]

If you spray a hose against a door, the door will move. each drop of water has imparted its share of momentum. Likewise with the molecules hitting the sides of the container. It is the normal component of the momentum of each that imparts an impulse onto the side. The effect is to produce a net force that is averaged over the area to give a pressure. Is that enough or is something still not right for you? This has given a number of people trouble.

You answer helps me to understand, but I still have a point needs to clarify. In your example the sprayed water has huge amount of molecules. In kinetic theory, is it also a necessary assumption that huge amount of atoms are present in the container of gas? Imagine in the case there is a few number of atoms, the container wall is just collided occasionally and it doesn't seem make sense to consider the average pressure for a relatively large area of container wall.

 

[klimatos]

However, in actual case the atoms collide on the wall of container separately at different position at different time, how can these changes of momentum be considered as a whole?

If by this you are wondering about molecules impacting on the wall with every possible angle of incidence and a wide distribution of free-path energies, you can relax. Those changes in momentum are defined as impulses.

Only the impulse normal to and toward the wall is transferred at the instant of impact. The mean pressure in Pascals is the simple product of the mean value of those impulses in Newtons and the mean number of impacts per square meter of surface (flux).

Pressure equals flux times the impulse.

 

[ehild]

You answer helps me to understand, but I still have a point needs to clarify. In your example the sprayed water has huge amount of molecules. In kinetic theory, is it also a necessary assumption that huge amount of atoms are present in the container of gas? Imagine in the case there is a few number of atoms, the container wall is just collided occasionally and it doesn't seem make sense to consider the average pressure for a relatively large area of container wall.

Kinetic theory assumes a huge number of identical molecules. As an example, 0.024 m³ volume contains1 mol gas at room temperature and at atmospheric pressure. 1 mol substance contains about 6x10²³ molecules. Is it big enough?



ehild