You are using a microscope to view a dust particle suspended in a drop of water on a microscope slide. As water molecules bombard the particle, it "jitters" about in a random motion $($Brownian motion$).$ The particle's average kinetic energy is the same as that of a molecule in an ideal gas

K $=$

$32$

kBT

$.$

The particle $($assumed to be spherical$)$ has a density of $350$ kg$/$m$3$ in water at $23$C$.$

$($a$)$

If the particle has a diameter d$,$ determine an expression for its rms speed in terms of the diameter d$.($Enter your answer as a multiple of d$3/2.$ Assume vrms is in m$/$s and d$3/2$ is in m$3/2.$ Do not include units in your answer.$)$

vrms $=$ d$3/2$

$($b$)$

Assuming the particle moves at a constant speed equal to the rms speed, determine the time required for it to travel a distance equal to its diameter. $($Enter your answer as a multiple of d$5/2.$ Assume t is in seconds and d$5/2$ is in m$5/2.$ Do not include units in your answer.$)$

t $=$ d$5/2$

$($c$)$

Evaluate the rms speed $($in mm$/$s$)$ and the time $($in ms$)$ required for the particle to travel the distance of one diameter for a particle with a diameter of $4\mathrm{.}00$m$.$

vrms$=$ mm$/$st$=$ ms