When a small particle is suspended m a fluid, bombardment by molecules makes the
ID: 995961 • Letter: W
Question
When a small particle is suspended m a fluid, bombardment by molecules makes the particle jitter about at random. Robert Brown discovered this motion m 1827 white studying plant fertilization, and the motion has become known as Browrusn motton. The particle's average kinetic energy can be taken as 3/4 the same as that of a molecule in an ideal gas. Consider a spherical particle of density 9 10 times 10^2 kg/m^3 in water at 20degree C. (a) For a particle of diameter d. evaluate the rms speed (Use the following as necessary: d. Let the units of vrms be m/s and d be m.) v_rms = (b) The particle's actual motion is a random walk, but imagine that it moves with constant velocity equal in magnitude to its rms speed. In what time interval would it move by a distance equal to its own diameter' (Use the following as necessary: d. Let the umts of t be s and d be m.) t= (c) Evaluate the rms speed and the time interval for a particle of diameterExplanation / Answer
a.
V(rms) = (3kT/m)1/2,
where m is the mass of particle,
K is the Boltzmann constant = 1.38 x1023m2kg s2K1
T is the temperature = 20o C = 293 K
Density of spherical particle = 9.10 x 102 kg/m3
Volume of spherical particle = 4/3 d3/8 = 1/6 d3 m3
m = volume x density
= 1/6 d3 x 9.10 x 102 = 476.47 d3 kg
V(rms) = (3KT/m)1/2
= (3 x 1.38 x1023 x 293 / 476.47 d3)1/2 = 5.05 x 10-12 x d-3/2 m/s
b.
T = distance / speed = d / (5.05 x 10-12 x d-3/2) = 1.98 x1011 x d5/2
c.
V(rms) = 5.05 x 10-12 x (4 x 10-6)-3/2 = 5.05 x 10-12 x 1.25 x 108 = 6.31 x 10-4 m/s
T = 1.98 x1011 x (4 x 10-6)5/2 = 1.98 x1011 x 3.2 x 10-14 = 6.34 x 10-3 s
d.
V(rms) = (3KT/m)1/2
= (3 x 1.38 x1023 x 293 / 73)1/2 = 1.29 x 10-11 m/s
T = distance / speed
= d / (1.29 x 10-11)
= 1 / (1.29 x 10-11) (Assuming radius of my body is 0.5 m)
= 7.75 x 1010 s
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