If y = Bx n (where B = constant)

Transcription

1 Pages of ext Book Lec 14 Mon 24se18 First: One variable: Calculus Review: If y Bx n (where B constant) dy n? Bnx 1 dx How do you KNOW (not memorize) that it is x n-1? What are the units of dx? 1

2 Partial Derivatives (needed if a function deends on more than one variable.) (Examle: Dividing the Ideal Gas Law by gives an equation for ) ( n,, ) nr If n and are held constant, then B d nr B d Cool way: n, nr the LOPE of in the direction he curly d is the same a the ordinary d. he curliness just tells you that other variables besides can affect, but are being held constant. he subscrits further emhasize this by telling which variables are constant

3 imilarly, if n and are held constant nr 1 1 B' If n and are held constant, then B 1/ 1 d 1 nr d 1 nr d nr B' 2 d d d n, nr 2 the LOPE of in the direction he curly d is the same a the ordinary d. he curliness just tells you that other variables besides can affect, but are being held constant. he subscrits further emhasize this by telling which variables are constant

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5 loes are not indeendent!!! wo aths from 1 2 tee u, stee down, or less stee down, less stee u his is a way of saying the order of differentiation does not matter. 5

6 loes are not indeendent: Maxwell s Relations d d + d P nr nr d d d 2 because nr Maxwell : P nr nr 2 2 the order of differentiation does not matter. i.e., the sloe changes with exactly as the sloe changes with

7 he Maxwell Relations aly to all such equations: the sloe of he sloe of the sloe of sloe of he because d d d d d + is also 7 Alied to the Fundamental Equation: therefore imlies : Recall this means that? and? and

8 imilar relationshis come from definitions of H, G, and A IF we add d(p) to both sides of the Fundamental Equation, we magically transform it into a similar equation for dh: d d d dq + dw d + d( ) dh rev dh d d + d + d dh d + d rev hen it follows that : and P H H and P P he last 3 lines show how to answer roblem 4 on homework but how do we get to something useful? 8

9 Generalize Let Z be anything that deends on X and Y, i.e., Z Z(X,Y) hen it is ALWAYRE that : dz If Z X, Y, and Z are any state functions whatsoever. For examle: Z elevation above sea level X distance heading EA Y distance heading NORH Z X Z X Y Y dx X Z + Y + X Z Y dy changes are small, but not infinitely small, it is X Y true that :

10 Comare: dz With: adding Given that d(p) to q Conclude that: rev d subtract d() from d subtract d() from dh Z X + w Y dx d dh Z + Y da dg P and P and H H and and P P A A P and P and G G and and P P rev gives : gives : gives : X d d d dy d + P P P P d dp d + dp H P A G P d d d d + H + P + his dazzling array of information surrisingly leads to something of great ractical value CAN YO EE I? A + G P dp d dp 10 d What is measureable?

11 Most of the items in the big dazzling array involve entroy, B, WE HAE NO ENROPY MEER A A P and P and From: G G and and P P P dg d + d From: da d d You can measure how ENROPY changes with at constant if you: Measure as you raise the at constant You can measure how ENROPY changes with at constant if you: Encase the material in a rigid constant container equied with a PRERE GAGE. Measure P as you raise the 11

12 First and econd Laws combined to give (,) d d d (the Fundamental Equation ). But we talk a lot about (,). his is more useful because we know how to change and and how to kee and constant. d d + d already know C v What is for an ideal gas? 0, of course What is for any material? 12

13 What is d d d Divide by d and hold constant But, there is no entroy meter. However, from: for any material? da d d How do you exerimentally determine Encase the material in a rigid constant container equied with a PRERE GAGE. Measure P as you raise the Good that it is all in terms of MEAREABLE variables.? 13

14 Does this give zero for an ideal gas? nr / For ANYHING!! nr / ; nr /

15 Getting how E changes with and how H changes with for anything se a Pressure Meter at different temeratures se a olume Meter at different 15 temeratures