A C O2 molecule is released at the center of a closed room where the air is perfectly still. Take the center as the origin of coordinates. After time t has elapsed, the position of the molecule r is uncertain, but is described by the probability distribution function
f(r) = ( 1/ (4pi Dt)^3/2 ) * exp( -r^2 / 4Dt)
The diﬀusion coeﬃcient of C O2 in air at 300◦ K is given by D = 1.4 × 10^−5 m2 /s. Calculate the probability to ﬁnd the molecule within 0.5m of the origin after one hour.
Hint: Use spherical coordinates and convert the integral by integration by parts to one that you can calculate using the Gaussian distribution integral [say, in terms of the error function erf(x)]. Of course, you may need to use a calculator.
Spherical coordinate: dV = r^2 sin (phi) d(phi) d(theta) dr
The Attempt at a Solution
Is it asking me to find <r^2>? If so, do I find it simply by integrating dr r^2 f(r) from infinity to minus infinity?
Why do I need to use spherical coordinates?