Calculating Probability for a C O2 Molecule in a Closed Room

In summary, the problem involves finding the probability of finding a C O2 molecule, with a diffusion coefficient of 1.4×10^-5 m^2/s in air at 300K, within 0.5m of the origin after one hour. This is calculated by converting the integral to spherical coordinates and using the Gaussian distribution integral, possibly involving the error function erf(x). The proper limits must be used when integrating and <r^2> is not relevant to the problem.
  • #1
v_pino
169
0

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?
 
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  • #2
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
[tex]\int_V f(\vec{x})\,d^3\vec{x}[/tex]You need to express this integral in spherical coordinates and then integrate over the proper limits (not -∞ to +∞).
 
  • #3
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?
 

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  • #4
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.
 
  • #5
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
 
  • #6
Yes, and what I'm saying is that you're not evaluating the integral correctly if you're getting an answer of 0.
 
  • #7
Doesn't seem that way. In the exponential, how'd you get the 3/2 power and why isn't r squared?
 

1. What is probability density?

Probability density is a concept in statistics that measures the likelihood of a continuous random variable taking on a specific value within a given range. It is represented by a probability density function (PDF) and is often used to analyze and make predictions about uncertain events.

2. How is probability density different from probability?

While probability measures the likelihood of a specific outcome occurring, probability density measures the likelihood of a continuous random variable taking on a specific value within a given range. In other words, probability density is a more precise and detailed way of expressing probability.

3. What is the relationship between probability density and standard deviation?

Probability density and standard deviation are closely related. The standard deviation of a probability density function represents the spread or variability of the data, and it can be calculated by taking the square root of the variance of the function. Essentially, the higher the standard deviation, the more spread out the data is and the lower the probability density at each point.

4. How do you calculate probability density?

To calculate probability density, you need to first have a probability density function (PDF) for the specific distribution you are analyzing. Then, you can plug in a specific value into the PDF to get the corresponding probability density. Alternatively, you can use calculus to integrate the PDF over a given range to find the probability density within that range.

5. Why is probability density important in statistics?

Probability density is important in statistics because it allows us to analyze and make predictions about uncertain events. It is used in a variety of fields, including finance, engineering, and science, to model and understand real-world phenomena. Probability density also helps us to measure and compare the likelihood of different outcomes, which is essential for making informed decisions.

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