1. Particles in Motion

2. Velocities and Speeds of Particles
The Maxwell-Boltzmann Distribution
Consider a Cartesian system with orthogonal axes (x,y,z)
N(vx)dvx = number of particles having a velocity component along the x axis between vx and vx+dvx
N = Total Number of particles
m = mass of the particle
Then:
Particles In Motion

3. Continuing
Let
Then
This is just a gaussian.
Particles In Motion

4. Gaussians Generally a gaussian is
The center is at x = 0 and the amplitude is A
To move the center: -a(x-x0)2
FWHM: y = A/2 (but remember to double!)
This is the Formal Gaussian Probability Distribution where σ is the standard deviation
Particles In Motion

5. Continuing some more The y,z contributions are
The fraction with components along the respective axes vx+dvx, vy+dvy, vz+dvz
Particles In Motion

6. Summation and Normalization
Probability Distributions Integrate to 1 Or
The normalization on the gaussians is α
Now consider the speeds of the particles:
==> v2 = vx2 + vy2 + vz2
Let us go to a spherical coordinate system:
==> dvxdvydvz → 4πv2dv
Particles In Motion

7. The Speed Distribution
Note that α is just the most probable speed
NB: Velocities are normally distributed but the speeds are not!
Particles In Motion

8. Gas Pressure Pressure ≡ rate of momentum transfer normal to surface
Another way to do it!
Pressure ≡ rate of momentum transfer normal to surface
Consider a 3d orthogonal axis system
If the particles are confined to move along axes, then 1/6 are moving along any one axis at a single time (on the average)
Let the speed be v
The number crossing a unit area per second on any axis is: Nx+ = 1/6 Nv and Nx- = 1/6 Nv.
Therefore the total number of crossings is 1/3 Nv
Particles In Motion

9. Gas Pressure II Pressure = “N * mv” Therefore Px = 1/3 mv2N
Now use the Maxwell Boltzman Distribution
Pressure = “N * mv”
Therefore Px = 1/3 mv2N
Go to a Maxwellian speed distribution:
One can either invoke the Perfect Gas Law or one can derive the Perfect Gas Law!
Particles In Motion