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Unformatted text preview: 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 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 ? Lets consider an example that is of considerable interest to us: an ideal gas with N atoms enclosed in a container of volume V T . Lets also divide the box into two parts, of volumes V and V , such that V + V = 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 ....
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This note was uploaded on 04/11/2010 for the course CHEM 1101 taught by Professor Leroy during the Spring '10 term at University of Toronto Toronto.
 Spring '10
 Leroy

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