Probability that a molecule will travel a distance at least equal to the mean free pa

[Oojee]

Homework Statement

a) Find the probability that a molecule will travel a distance at least equal to the mean free path before its next collision.
b)After what distance of travel since the last collision is the probability of having suffered the next collision equal to 1/2?

Homework Equations

Exponential probability distribution
f(r) = Ae^{-r/$$\lambda$$}

where A = a constant, $$\lambda$$ = mean free path

The Attempt at a Solution

P = Integral (limits $$\lambda$$ to $$\infty$$)f(r) dr / Integral (limits 0 to $$\infty$$) f(r) dr
I will do the calculations of both parts but not sure about the probability. I have found probability by number of molecules with distance between collision greater than lambda / total number of molecules. Is this probability correct?

 

[vela]

What exactly is f(r) supposed to represent? You say it's a probability distribution but of what exactly?

 

[Oojee]

What exactly is f(r) supposed to represent? You say it's a probability distribution but of what exactly?

f(r) is a continuous function that gives the probability for a molecule to travel a distance r before having a collision.

 

[vela]

OK, your calculation is correct. Typically, though, you normalize f(r). In other words, you set the constant A, which is a normalization constant, so that

$$\int_0^\infty f(r)\,dr = 1$$

You're effectively doing this in your calculation by dividing by the integral from 0 to infinity.

 

[Oojee]

@ vela
thank you.