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DERIVATION FOR "PRESSURE OF A GAS" ?

According to the kinetic theory of gas,

- Gases are composed of very small molecules and their number of molecules is very large.
- These molecules are elastic.
- They are negligible size compare to their container.
- Their thermal motions are random.



The time between collisions with the wall is the distance of travel between wall collisions divided by the speed.

 t=\frac{2L}{v_x}

The frequency of collisions with the wall in collisions per second is

 f=\frac{1}{t}=\frac{1}{2L/v_x}=\frac{v_x}{2L}

According to Newton, force is the time rate of change of the momentum

 F=\frac{dp}{dt}=ma

The momentum change is equal to the momentum after collision minus the momentum before collision. Since we consider the momentum after collision to be mv, the momentum before collision should be in opposite direction and therefore equal to -mv.

 \Delta{p}=mv_x-(-mv_x)=2mv_x

According to equation #3, force is the change in momentum \Delta{p} divided by change in time \Delta {t}.  To get an equation of average force \overline{F} in term of particle velocity v_x, we take change in momemtum \Delta{p} multiply by the frequency f from equation #2. 

 \overline{F}=\Delta{p}(f)=2mv(\frac{v_x}{2L})=\frac{mv_x^2}{L}

The pressure, P, exerted by a single molecule is the average force per unit area, A. Also V=AL which is the volume of the rectangular box.

 P_{1\:Molecule}=\frac{\overline{F}}{A}=(\frac{mv_x^2}{L})/A=\frac{mv_x^2}{LA}=\frac{mv_x^2}{V}

Let's say that we have N molecules of gas traveling on the x-axis. The pressure will be

 P_{N\:Molecules}=\frac{m}{V}(v_{x_1}^2+v_{x_2}^2+v_{x_3}^2....+v_{x_N}^2)=\sum_{a=0}^{N}\frac{mv_{x_a}^2}{V}

To simplify the situation we will take the mean square speed of N number of molecules instead of summing up individual molecules. Therefore, equation #7 will become

 P_{N\:Particles}=\frac{Nm\overline{v_x^2}}{V}

Earlier we are trying to simplify the situation by only considering that a molecule with mass m is traveling on the x axis.  However, the real world is much more complicated than that. To make a more accurate derivation we need to account all 3 possible components of the particle's speed, vx, vy and vz.

 \overline{v^2}=\overline{v^2_x}+\overline{v^2_y}+\overline{v^2_z}

Since there are a large number of molecules we can assume that there are equal numbers of molecules moving in each of co-ordinate directions.

 \overline{v^2_x}=\overline{v^2_y}=\overline{v^2_z}

Because the molecules are free too move in three dimensions, they will hit the walls in one of the three dimensions one third as often. Our final pressure equation becomes 

 P=\frac{Nm\overline{v^2}}{3V}

However to simplify the equation further, we define the temperature, T, as a measure of thermal motion of gas particles because temperature is much easier to measure than the speed of the particle. The only energy involve in this model is kinetic energy and this kinetic enery is proportional to the temperature T. 

 E_{kinetic}=\frac{mv^2}{2}\propto{T}

To combine the equation ,. we solve kinetic energy equation  for mv2. 

 mv^2=2E_{kinetic}\Rightarrow\frac{mv^2}{3}=\frac{2E_{kinetic}}{3}

Since the temperature can be obtained easily with simple daily measurement like a thermometer, we will now replace the result of kinetic equation #13 with with a constant R times the temperature, T. Again, since T is proportional to the kinentic energy it is logical to say that T times k is equal to the kinetic energy E. k, however, will currently remains unknown.

 kT=\frac{mv^2}{3}=\frac{2E_{kinetic}}{3}

Combining equations, we get:

 P=\frac{N}{V}\frac{m\overline{v^2}}{3}=\frac{N}{V}\frac{2E_{kinetic}}{3}=\frac{N}{V}kT=\frac{NkT}{V}

Because a molecule is too small and therefore impractical we will take the number of molecules, N and divide it by the Avogadro's number, NA= 6.0221 x 1023/mol to get n (the number of moles)

 n=\frac{N}{N_a}

Since N is divided by Na, k must be multiply by Na to preserve the original equation. Therefore, the constant R is created.

R=N_ak

Now we can achieve the final equation by replacing N (number of melecules) with n (number of moles) and k with R. 

 P=\frac{nRT}{V}\Rightarrow{PV=nRT}


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