Small dust particles suspended in air seem to dance randomly about, a phenomenon

Question

Small dust particles suspended in air seem to dance randomly about, a phenomenon called Brownian motion. For this problem you will need to know Boltzmann's constant: Kb=1.38X10^-23J/K (a)What would you expect the mean translational kinetic energy of such particles to be if they are in air at a temperature of 290 K? Express the mean translational kinetic energy numerically, in joules, to two significant figures. Note that has been factored out already to make your answer simpler. (b)Find an expression for the rms (root-mean-square) speed v of these particles, assuming them to be spheres of diameter d and density p Express the rms speed in terms of kb, T, d, p and p

Explanation / Answer

here ,

The mean translational kinetic energy K =(3/2) kT

K is Boltzmann constant k =1.38*10^-23 J/K

Temperature T =290 K

the mean translational kinetic energy

K =(3/2)[ (1.38*10^-23 J/K)(290 K)]

K = 6*10^-21 J (ANSWER)

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B) the root-mean-square speed

v_rms =square root of(3kT/m) .

Volume of the sphere V =4/3 pi r^3

r = d/2   where d = diameter of the sphere

density of the sphere rho = mass/volume

rho =m/(4/3) pi r ^3

m =rho*(4/3) pi r ^3

so thus the rms (root-mean-square) speed is

V_rms = square root of(3kT/rho*(4/3) pi (d/2)^3) (ANSWER)

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