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Derivation of PV=nRT, The Equation of Ideal 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.

To begin, let's visualize a rectangular box with length L, areas of ends A1 and A2. There is a single molecule with speed vx

traveling left and right to the end of the box by colliding with the end walls.

3D Demonstration of Ideal Gas

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

1.

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

2.

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

3.

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.

4.

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According to equation #3, force is the change in momentum divided by change in time . To get an equation of

average force in term of particle velocity , we take change in momemtum multiply by the frequency from

equation #2.

5.

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

6.

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

7.

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

8.

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.

9.

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.

10.

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

11.

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.

12.

To combine the equation #11 and #12 we solve kinetic energy equation #12 for mv2.

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13.

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.

14.

Combining equation #14 with #11, we get:

15.

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)

16.

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

17.

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

17.