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Particles in Motion

Particles in Motion. Velocities and Speeds of Particles. The Maxwell-Boltzmann Distribution. 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

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Particles in Motion

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  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 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 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

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