3. (25 marks) A container of volume V contains N gas molecules (with average den- sity p = N/V), which behave independen

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3 25 Marks A Container Of Volume V Contains N Gas Molecules With Average Den Sity P N V Which Behave Independen 1 (112.91 KiB) Viewed 80 times

3. (25 marks) A container of volume V contains N gas molecules (with average den- sity p = N/V), which behave independently from each other (it's an ideal gas). Supposing a gas molecule is equally likely to be found anywhere in the container: (a) What is the probability that a given molecule is found in some small volume element v? (b) What is the probability of there being exactly n molecules in the small volume element v? From this, find the average number of molecules ñ found in v. (c) Show that if v « V, we can write the probability from b) as a Poisson distribu- tion (remember to clearly state you assumptions). Find the variance An of the number of molecules in v. As a numerical application, calculate ñ and An for an ideal gas (say an oxygen molecule 02 - though it really doesn't matter in our case) at standard pressure and room temperature (20°C) in a room of volume Im