Particles in Motion. Particles In Motion2 Velocities and Speeds of Particles ► Consider a Cartesian system with orthogonal axes (x,y,z) ► N(v x )dv x.

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Presentation transcript:

Particles in Motion

Particles In Motion2 Velocities and Speeds of Particles ► Consider a Cartesian system with orthogonal axes (x,y,z) ► N(v x )dv x = number of particles having a velocity component along the x axis between v x and v x +dv x ► N = Total Number of particles ► m = mass of the particle ► Then: The Maxwell-Boltzmann Distribution

Particles In Motion3 Continuing Let Then This is just a gaussian.

Particles In Motion4 Gaussians ► Generally a gaussian is ► The center is at x = 0 and the amplitude is A  To move the center: -a(x-x 0 ) 2 ► FWHM: y = A/2 (but remember to double!) ► This is the Formal Gaussian Probability Distribution where σ is the standard deviation

Particles In Motion5 Continuing some more The y,z contributions are The fraction with components along the respective axes v x +dv x, v y +dv y, v z +dv z

Particles In Motion6 Summation and Normalization Probability Distributions Integrate to 1 The normalization on the gaussians is α Now consider the speeds of the particles: ==> v 2 = v x 2 + v y 2 + v z 2 Let us go to a spherical coordinate system: ==> dv x dv y dv z → 4 π v 2 dv Or

Particles In Motion7 The Speed Distribution NB: Velocities are normally distributed but the speeds are not! Note that α is just the most probable speed

Particles In Motion8 Gas Pressure ► Pressure ≡ rate of momentum transfer normal to surface ► Consider a 3 d 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: N x+ = 1/6 N v and N x- = 1/6 N v.  Therefore the total number of crossings is 1/3 N v Another way to do it!

Particles In Motion9 Gas Pressure II ► Pressure = “N * mv” ► Therefore P x = 1/3 mv 2 N ► Go to a Maxwellian speed distribution: Now use the Maxwell Boltzman Distribution One can either invoke the Perfect Gas Law or one can derive the Perfect Gas Law!