# Probability density

## 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

vela
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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 +∞).

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

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vela
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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.

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

vela
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Yes, and what I'm saying is that you're not evaluating the integral correctly if you're getting an answer of 0.

vela
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Doesn't seem that way. In the exponential, how'd you get the 3/2 power and why isn't r squared?