{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

07 Lecture 10 to 11 January 27,29

# 07 Lecture 10 to 11 January 27,29 - CHEM 350 Lectures 11...

This preview shows pages 1–3. Sign up to view the full content.

CHEM 350 Lectures 11 and 12, January 27 and 29, 2010 Ensemble Averages. variable v : having M possible values v 1 , v 2 , · · · , v M : corresponding probabilities p 1 , p 2 , · · · , p M N systems in ensemble with variable v having value v r in N r = N p r of them – with so many possibilities in principle, we should be willing to settle for a little less detail. For example, we might be interested in first instance only in the average value of the variable (other names: mean value , ENSEMBLE AVERAGE ) defined by h v i ≡ 1 N ( N 1 v 1 + N 2 v 2 + · · · + N M v M ) = M X r =1 p r v r . average of a function of v : h f ( v ) i = M X r =1 p r f ( v r ) . This represents a linear operation, since h cf ( v ) i = c h f ( v ) i h af ( v ) + bg ( v ) i = a h f ( v ) i + b h g ( v ) i . CAUTION: this does not necessarily mean that evaluation of products is simple! h f ( u ) g ( v ) i ≡ M X r =1 M X s =1 p rs f ( u r ) g ( v s ) . Now, if we assume that the variables are statistically independent, then p rs = p r p s , and h f ( u ) g ( v ) i = M X r =1 M X s =1 p r p s f ( u r ) g ( v s ) = h f ( u ) i h g ( v ) i 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
All this is fine if we know all values of p r . BUT, how do we evaluate such averages without knowing all values of p r ? Let’s consider an example that is of considerable interest to us: an ideal gas with N atoms enclosed in a container of volume V T . Let’s also divide the box into two parts, of volumes V and V 0 , such that V + V 0 = V T . Treat the atoms as statistically independent, and construct an ensemble of such containers of the ideal gas. If p is the probability that an atom is found in volume V , then q = 1 - p is the probability that an atom is found in volume V 0 . At equilibrium, we can write for p and q the expressions p = V V T , q = V 0 V T Question : What is the probability p(N,n) that n out of the N atoms will be found in volume V (with N-n hence in V’)?
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 8

07 Lecture 10 to 11 January 27,29 - CHEM 350 Lectures 11...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online