When a small particle is suspended in a fluid, bombardment by molecules makes th
ID: 1450285 • Letter: W
Question
When a small particle is suspended in a fluid, bombardment by molecules makes the particle jitter about at random. Robert Brown discovered this motion in 1827 while studying plant fertilization, and the motion has become known as Brownian motion. The particle's average kinetic energy can be taken as 3 2 kBT, the same as that of a molecule in an ideal gas. Consider a spherical particle of density 1.04 103 kg/m3 in water at 20°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.)
(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 units of t be s and d be m.)
(c) Evaluate the rms speed and the time interval for a particle of diameter 3.05 µm.
(d) Evaluate the rms speed and the time interval for a sphere of mass 61.0 kg, modeling your own body.
Explanation / Answer
a) average kinetic energy = 3/2 * Kb * T
= 3/2 * 1.380 × 10-23 * 293.15
= 6.068 * 10-21 J
mass of particle = 4/3 * 3.14 * d3/8 * 1040
= 544.26 d3
=> 1/2 * 544.26 d3 * V2 = 6.068 * 10-21
=> V = sqrt(2.229* 10-23/d3) m/sec
b)
time taken = d/sqrt(2.229* 10-23/d3) m
c) rms speed = sqrt(2.229* 10-23/(3.05 * 10-6)3)
= 0.0008865 m/sec
time taken = (3.05 * 10-6)/(0.0008865)
= 0.00344 sec
d) rms speed = sqrt(2 * 6.068 * 10-21 /61)
= 1.4104 * 10-11 m/sec
diameter of sphere = 0.5 m This is a particular data because it depends on you own body**
time taken = 0.5/(1.4104 * 10-11)
= 3.544 * 1010 sec
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