Stat Mech: Mean, Std Dev & Probability Distrib of n/N as N Increases

[latentcorpse]

An ideal gas of N molecules in a volume V. Write down the mean and standard deviation of the number of molecules n to appear in a volume V/3. Sketch the probability distribution of n and of n/N. Indicate how the curves change as N increases.

ok. $\bar{n}=\sum_i n_i P_i$

do i use $P_i$ form the grand canonical distribution or from teh canonical distribution here? i was going to go with the former as the subsystem of volume V/3 is an open system i.e. it can exchange energy AND particles with the rest of the system - but i wanted to check??

for the variance, do i just use

$\bar{\Delta n^2}= \bar{n^2} - \bar{n}^2$?

what would the graphs look like?

cheers

 

[NateD]

The mean and standard deviation of the number of molecules n to appear in a volume V/3 can be calculated using the grand canonical distribution: Mean: \bar{n} = N\frac{V}{3V_{eq}} Standard Deviation: \Delta n = \sqrt{N\frac{V}{3V_{eq}}} The probability distribution of n and of n/N can be represented by a Poisson distribution. As N increases, the probability distribution of n/N becomes narrower and more peaked. The probability distribution of n becomes wider and less peaked.