# Chem 154 - 2012 Winter - handout 03 - NOTE: (Three sigma...

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NOTE:(Three sigma rule)’For a Normal distribution, almost all points lie within threestandard deviations from the average’(x-μ)σerfx-μσ21.00.682692.00.954503.00.9973000.10.20.30.40.50.6-4-3-2-101234σ=1.0, µ=0.0Gaussian P.D.FNOTE:The Normal Distribution can be generalised into higherdimensions:2–19
QUESTION:Where does the Normal distribution play a rôle?
Here, just two examples in the context of chemistry:The Maxwell-Boltzmann Velocity distribution of particlesREMINDER:At the microscopic level, particles in a gasare constantly changing their velocity. However, it can beshown that the p.d.f. of particle velocity~vis a Gaussian of theform:f(~v) =m2πkBT32e-βm2~v2,(2.31)wherekBis Boltzmann’s constant,mis the particle mass andβ=1kBT.This is simply a Gaussian distribution with an average(mean) ofμ=h~vi=~0.The width of the curve is related to the temperature and2–20
particle mass:σ2= (Δ~v)2=kBTm3.(2.32)For example, this can be used to calculate the probability

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