James Richard Fromm
In the kinetic-molecular theory of gases, pressure is the force exerted against the wall of a container by the continual collision of molecules against it. From Newton's second law of motion, the force exerted on a wall by a single gas molecule of mass m and velocity v colliding with it is:
F = ma = m(Dv/Dt)
In the above equation, the change in a quantity is indicated by the symbol D; a more common symbol is the Greek upper-case delta. It is assumed that the molecule rebounds elastically and no kinetic energy is lost in a perpendicular collision, so Dv = v - (-v) = 2v. If the molecule is moving perpendicular to the wall it will strike the opposite parallel wall, rebound, and return to strike the original wall again. If the length of the container or distance between the two walls is the path length l, then the time between two successive collisions on the same wall is Dt = 2l/v. The continuous force which the molecule moving perpendicular to the wall exerts is therefore
F = m(2v)/(2l/v) = mv2/l
The molecules in a sample of gas are not, of course, all moving perpendicularly to a wall, but the components of their actual movement can be considered to be along the three mutually perpendicular x, y, and z axes. If the number of molecules moving randomly, N, is large, then on the average one-third of them can be considered as exerting their force along each of the three perpendicular axes. The square of the average velocity along each axis, v2(x), v2(y), or v2(z), will be one-third of the square of the average total velocity v2:
v2(x) = v2(y) = v2(z) = v2/3
The average or mean of the square of the total velocity can replace the square of the perpendicular velocity, and so for a large number of molecules N,
F = (N/3)(mv2/l)
Since pressure is force per unit area, and the area of one side of a cubic container must be l2, the pressure p will be given by F/l2 as:
p = (N/3)(mv2/l3)
This equation rearranges to pV = Nmv2/3 because volume V is the cube of the length l.
The form of the ideal gas law given above shows the pressure-volume product is directly proportional to the mean-square velocity of the gas molecules. If the velocity of the molecules is a function only of the temperature, and we shall see in the next section that this is so, the kinetic-molecular theory gives a quantitative explanation of Boyle's law.
The square of the velocity is sometimes difficult to conceive, but an alternative statement can be given in terms of kinetic energy. The kinetic energy Ek of a single particle of mass m moving at velocity v is mv2/2. For a large number of molecules N, the total kinetic energy Ek will depend on the mean-square velocity in the same way:
Ek = Nmv2/2 = nMv2/2
The second form is on a molar basis, since n = N/NA and the molar mass M = mNA where NA is Avogadro's number. The ideal gas law then appears in the form:
pV = 2Ek/3 (compare pV = nMv2/2)
This statement that the pressure-volume product of an ideal gas is directly proportional to the total kinetic energy of the gas is also a statement of Boyle's Law, since the total kinetic energy of an ideal gas depends only upon the temperature.