CHEM 481 Lecture Notes _Engle 2nd - Chapter 3_

CHEM 481 Lecture Notes _Engle 2nd - Chapter 3_ - Chapter 3-...

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Last printed 2/9/2010 2:19 PM 3-1 Chapter 3- The First Law (cont) C ALCULATING M EASURABLE C HANGE Calculating Change along a Path: Consider calculating the change in elevation during a hike dh “height differential”: infinitesimal change in h that occurs as system moves infinitesimal amount along path h measurable change obtained by summing up (or integrating) all the dh’s It doesn’t matter what path you take, Δ h is always the same. We refer to functions that have this feature as having “ exact differentials ” or being “ state functions In thermodynamics, we express the measurable change in a variable in the same way as a sum of infinitesimal changes (i.e. an integral) f i Ud U Δ= o dU is an exact differential o All state functions have exact differentials State Functions: Properties whose values depend only upon the current state of the system, not on how it was prepared -or, equivalently- The change in property in going from i f is independent of path (Examples of state functions are: U, H, P.) Path Functions : Properties that relate to the preparation of a system. That is, they depend upon path. (Examples are q & w) i f N f i i i Change in elev for each step along path hh d h δ =
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Last printed 2/9/2010 2:19 PM 3-2 N f i Infintesimal amount of work done at each step along path Total Work wd w Done ⎛⎞ = ⎜⎟ ⎝⎠ o Path functions have “ inexact differentials Note: We don’t usually refer to a system as possessing “heat” or “work”. These are quantities of energy supplied along the path. Therefore, U = q + w is WRONG! D IFFERENTIALS Much of the development of thermodynamic theory centers around differential quantities such as dU, dH, dS, etc. Mathematics of Differentials : Consider a function, f(x,y) [Could be U(V,T)] We are interested in how f changes when x or y are changed. For change in x only (, ) y Slope f fxy fxy dx x =+ ±²³ This basically the equation for a line y df f f f df dx x (x-change only) =− = For change in y only (,) x Slope f d y y x df f f f df dy y (y-change only) = If BOTH x and y change, then the “f-differential” is y x ff df dx dy xy ∂∂ Euler’s criterion for exactness : df is exact if the mixed-partial derivatives are equal, i.e. df dy f f’ y f df dx f f’ x f
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Last printed 2/9/2010 2:19 PM 3-3 y x y x ff yd x xd y ⎛⎞ ∂∂ = ⎜⎟ ⎝⎠ Problem 7: Show that f(x,y)=5xy+3x 2 y 3 has an exact differential I NTERNAL E NERGY D IFFERENTIAL Consider a closed system of constant composition. If we write U = U(V,T), then we can express changes in U w/ V & T as: TV 2st Term 1nd Term UU dU dV dT VT =+ ±²³²´ 2nd Term : When system is heated (i.e. temperature change), the internal energy increases. The amount of increase depends upon the conditions under which the heating takes place (i.e. Constant Volume or Constant Pressure) V V U C Constant-Volume Heat Capacity T = 1st Term : The 2 nd term incorporates the “internal pressure”: T T U V π= “Internal Pressue” π T is a measure of how U changes when the V changes.
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CHEM 481 Lecture Notes _Engle 2nd - Chapter 3_ - Chapter 3-...

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