28. Heat motion of gas molecules

From the very beginning, we must stop trying to track the motion of all the molecules that make up the gas. There are too many of them, and because of the collisions with each other the movement of molecules is very complicated. Instead, we will make a single, but very important assumption about the general nature of this motion.

Suppose the gas molecules move randomly. The speed of any molecule can be either very high or very low. The direction of movement of molecules changes chaotically when they collide with each other.

However, even though the movement of individual molecules is chaotic, the behavior of all molecules as a whole reveals on average some simple trends. Firstly, if any direction \(x\) is randomly identified in gas, then the average number of molecules moving in this direction should be equal to the average number of molecules moving in the opposite direction. After all, the chaos in the movement of molecules means that none of the directions of motion is dominant. They are all equal.

Similarly, the average number of people walking along a city street in one direction and another, for a fairly long period of time on average is equal. Of course, if we exclude special cases such as holiday demonstrations.

Second, simple laws are valid for the average arithmetic speed of molecules. Let there be \(N\) molecules. Projections of the velocities of these molecules in the x direction can take all sorts of values: \(\upsilon_{x_1}, \upsilon_{x_2}, \upsilon_{x_3}, ..., \upsilon_{x_i}, ..., \upsilon_{x_N}\). And each projection can be either positive or negative. The average arithmetic value of the velocity projection is \(\vec{\upsilon_x}\), by this direction \(x\) is equal to the sum of the velocity projections of all molecules divided by their number

\( \overline{\upsilon_x} {~=~} \frac{\upsilon_{x_1}{~+~}\upsilon_{x_2}{~+~}\upsilon_{x_3}{~+~}...{~+~}\upsilon_{x_i}{~+~}...{~+~}\upsilon_{x_N}}{N} \) (3-1)

Similarly, it is possible to determine the average value of any value taking on different values, for example, the average height of a student in a class. Because of the chaos in the movement of molecules, positive values of velocity projections should occur as often as negative values. Therefore, the average value of the velocity projection in any direction \(x\) is zero

\( \overline{\upsilon_x} {~=~} 0 \) (3-2)

But the average square of the velocity projection (we will see soon that this value is important), of course, is not equal to zero, because all the elements of the sum are positive

\( \overline\upsilon{^2_x} {~=~} \frac{\upsilon{^2_{x_1}} {~+~} \upsilon{^2_{x_2}} {~+~} \upsilon{^2_{x_3}} {~+~} ... {~+~} \upsilon{^2_{x_i}} {~+~} ... {~+~} \upsilon{^2_{x_N}}}{N} \) (3-3)

Due to the fact that the direction of \(x\) does not differ from the directions \(y\) and \(z\) (again, because of the chaos in the movement of molecules), the equations are fair

\(\overline\upsilon{^2_x} {~=~} \overline\upsilon{^2_y} {~=~} \overline\upsilon{^2_z}\) (3-4)

For each molecule, the square speed is equal: \(\upsilon{^2_i} {~=~} \upsilon{^2_{x_i}} {~+~} \upsilon{^2_{y_i}} {~+~} \upsilon{^2_{z_i}}\). The value of the average square of velocity, determined in the same way as the average square of the velocity projection ( equation 3-3), is equal to the sum of the average squares of its projections

\(\overline\upsilon^2 {~=~} \overline\upsilon{^2_x} {~+~} \overline\upsilon{^2_y} {~+~} \overline\upsilon{^2_z}\)(3-5)

From equations \((3-4)\) and \((3-5)\) it follows that

\(\overline\upsilon{^2_x} {~=~} \overline\upsilon{^2_y} {~=~} \overline\upsilon{^2_z} {~=~} \frac{1}{3}\overline\upsilon^2 \)(3-6)

The assumption made by us about the chaotic character of gas molecules motion, from which equations (3-2) and (3-4) follow, is performed better than more collisions between molecules.