Probability Distribution and Constants

[Rachael_Victoria]

Hi,
I just started a physical chemistry class and we are working on probability theory. The questions I am having a hard time with are as follow:
we are given that E is proportional to exp[-E/RT]. It is stated that this is a simple system having only four energy states numbered 1 through 4.
The values for each energy state are E1=0 J/mol, E2= 1000 J/mol, E3= 2000 J/mol, and E4= 3000 J/mol.
The first question asks us to obtain the normalized probability distribution for the system in state i, and evaluate the normalization constant C at T=298K.
So I did it like this P(Ei) = C exp [-Ei/RT], and C= the sum of [(-E1/RT)-(E2/RT)-(E3/RT)-(E4/RT)]^-1
this once you crunch the numbers equals 1/2.4119835807817
So P(Ei)= (exp [-E/RT])/2.4119835807817
So my first question is did I do this correctly? The normalized probability distribution for the system in state i would simply be the formula above correct?
The second question asks us to calculate the average energy per system that a large number of such systems would have at 298K.
My question is should I do an integral from 0 to infinity?
so <P(E)>= (integral [E exp(-E/RT)])/(integral [-E/RT]) from 0 to infinity?
This might all be completely wrong. I don't have the solution manual and therefore cannot check my answers. If it is wrong if someone could explain how it is wrong I would really appreciate it.
Thanks,
Rachael

 

[Tide]

The probabilities should add up to 1, i.e. it is a certainty that the system will be in one of those states. That means multiplying by the reciprocal of the number you found rather than multiplying.

 

[Rachael_Victoria]

The probabilities should add up to 1, i.e. it is a certainty that the system will be in one of those states. That means multiplying by the reciprocal of the number you found rather than multiplying.

I see what you are saying, they way our book presents it is that C is equal to 1 over the equation. So we got the entire C=1/[P(Xi)] thing. Here is the exact question as it is worded in my book: "...we shall find that the probability of a system occupying an energy state with energy E is proportional to exp[-E/RT]. Consider a simple system with having only four possible energy states..."

A) Obtain the normalized probability distribution for the system in state i.

I got P(Ei) = C exp [-Ei/RT]

and for C I got 1/( C= the sum of [(-E1/RT)-(E2/RT)-(E3/RT)-(E4/RT)])

B) Calculate the average energy per sysem that a large number of such systems would have at 298K.

My question here is am I calculating C using the integral of exp [-E/RT] from 0 to infinity and then using that C to find P(Xi)?
THanks
Rachael

 

[Tide]

No, you won't be integrating anything because the energies are discrete - not continuous. Basically what you will do is to multiply the energy of each state by its probability and sum over all four states. This will give you the average energy for the given temperature provided you normalized properly per our eariler discussion.