# Probability density

v_pino

## Homework Statement

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.

## Homework Equations

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?

## Answers and Replies

Staff Emeritus
Homework Helper
No, the problem isn't asking you to find <r2>. It's asking you to find a probability.

The probability of finding the particle in an infinitesimal volume dv at the point x is given by f(x) dv, and the probability to find the particle in a volume V would then be
$$\int_V f(\vec{x})\,d^3\vec{x}$$You need to express this integral in spherical coordinates and then integrate over the proper limits (not -∞ to +∞).

v_pino
Thanks for the reply.

Here's my integral using dV in spherical coordinate system (where R = 0.5m) attached as an image. The integral of d(theta) gives me pi. But doesn't the integral of sin(phi) d(phi) give me zero?

#### Attachments

• eq.pdf
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Last edited:
Staff Emeritus
Homework Helper
Don't see the new image, but I can make out some of the first one. I'm not sure why you have <r2> in there. That has nothing to do with the problem. You're integrating incorrectly if you get 0.

Calculate the probability to ﬁnd the molecule within 0.5m of the origin after one hour.

v_pino
If the integral of sin(phi)d(phi) is zero, doesn't this mean the entire integral becomes zero? And hence, is probability zero?

Thanks

Staff Emeritus