The_First_Law_Part_II

The_First_Law_Part_II - Exact and Inexact Differentials The...

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1 Exact and Inexact Differentials The test for exactness The total differential dz of a quantity z can be determined by the differentials dx and dy in two other quantities x and y . In general, dz=M(x,y)dx+N(x,y)dy, M and N are functions of x and y To show the test for exactness, consider a function z that has an exact differential. If z has an exact value in the x-y plane, then it is a function of x and y . If z=f(x,y) , then dy y z dx x z dz x y + = On comparison of the last two equations we can conclude that dx x z y x M y = ) , ( and dy y z y x N x = ) , ( Since y x x y y z x x z y = then y x x N y M = Euler’s criterion for exactness Show that the expression 23 33 dF x y dx y x dy =+ is exact. Since (, ) 3 Mxy xy = and Nx yy x = then 22 9 x M y  =   and 9 y N x y x = . Therefore, Euler’s criterion is satisfied and dF is exact.
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2 Consider, P, T and V m We will find that Thermodynamics enables to relate many thermodynamic properties of substances to partial derivatives of the above physical quantities. P m T V ) ( , T m P V ) ( , T m V P ) ( , m V T P ) ( , P m V T ) ( , m V P T ) ( By utilizing the relation y y z x x z ) / 1 ) ( = we see that three of the above six are the reciprocals of the other three. And T m V P ) ( P m T V ) ( m V P T ) ( = -1 Hence, there are only two independent partial derivatives P m T V ) ( and T m P V ) ( . The other four can be calculated from these two and need not be measured. Thermal expansitivity , ( α ) P m m n P T V V T V V P T ) ( 1 ) ( 1 ) , ( , Isothermal compressibility , ( κ ) T m m n T P V V P V V P T ) ( 1 ) ( 1 ) , ( , α and κ tell you how fast the volume of a substance increases by temperature and decreases with pressure It can be shown that = m V T P ) ( Reversible and Irreversible Processes A reversible process is one where the system is always infinitesimally close to equilibrium, and an infinitesimal change in condition can reverse the process to restore both system and surroundings to their initial states. ( A reversible process is obviously an idealization )
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3 PAdx Dw = , A is the cross sectional area l = b – x, x is piston’s position V = Al = Ab – Ax and dV=d(Ab-Ax)=-Adx Dw rev = -PdV for closed system, rev. process Work is a function of the path The sign of work depends on the sign of dV For expansion , dV > 0 piston moves to – x ( dx < 0) Dw < 0 - work done on the system by the surroundings (work done on the surroundings is positive) For compression , dV < 0, moves to + x ( dx > 0) and Dw > 0 - work done on the system is positive (work done on the surroundings is negative) = 2 1 PdV w rev closed syst., rev. process
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This note was uploaded on 04/07/2012 for the course CHEM 340 taught by Professor Staff during the Spring '08 term at Ill. Chicago.

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The_First_Law_Part_II - Exact and Inexact Differentials The...

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