HOMOGENEOUS CLOSED SYSTEM

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1 CHAE II

2 A closed system is one that does not exchange matter with its surroundings, although it may exchange energy. W n in = 0 HOMOGENEOUS CLOSED SYSEM System n out = 0 Q dn i = 0 (2.1) i = 1, 2, 3,... No internal energy transported accross boundary. All energy exchange between a closed system and its surroundings appears as heat and work. he total energy change of the surroundings equals the net energy transferred to or from it as heat and work.

3 First and second laws of hermodynamcs: du ds d (2.2) For reversible process: du = ds d (2.3) With ds = dq rev : heat absorbed by the system d = dw rev : work done by the system If the interaction occurs irreversibly: du < ds d (2.4)

4 he internal energy change can be calculated by integrating eq. (2.2): U S 2 U2 U1 ds d S (2.5) For process occuring at constant S and : du S, 0 (2.6) At constant S and, U tends toward a minimum in an actual or irreversible process in a closed system, and remains constant in a reversible process. Eq. (2.6) provides a criterion for equilibrium in a closed system.

5 Definition: H U + (2.7) Differentiating eq. (2.7) yields: dh = du + d + d Combining the above equation with eq. (2.3) leads to: dh = ( ds d) + d + d dh = ds + d (2.8) For a closed system at constant S and : dh,s 0 (2.9)

6 the Helmholtz free energy (A) is a thermodynamic potential that measures the useful work obtainable from a closed system at a constant temperature and volume. A = the maximum amount of work extractable from a thermodynamic process in which temperature and volume are held constant.. Under these conditions, it is minimized at equilibrium.

7 Definition: A = U S Differentiating eq. (2.10) yields: (2.10) da = du d(s) = dq + dw ds S d = ds d ds S d da = S d d (2.11) For a closed system at constant dan : da, 0 (2.12)

8 Definition: G A + (2.13) Gibbs free energy (G) is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure (isothermal, isobaric). he Gibbs free energy is the maximum amount of nonexpansion work that can be extracted from a closed system; this maximum can be attained only in a completely reversible process.

9 Differentiating eq. (2.13) yields: dg = da + d() = S d d + d + d dg = S d + d (2.14) For a closed system at constant and : dg, 0 (2.15)

10 If F = F(x,y), the total differential of F is: dy y F dx x F df x y with y x F M x y F N (2.16) Ndy Mdx F

11 Further differentiation yields y x F y M 2 x y x F x N 2 y y x x N y M (2.17) Hence from equation: Ndy Mdx F we obtain: y x x N y M (2.17) (2.16)

12 esume: du = ds d (2.3) dh = ds + d (2.8) da = S d d (2.11) dg = S d + d (2.14) According to eq. (2.17): S S S S S S (2.18) (2.21) (2.20) (2.19)

13 ENHALY As a function of and, we can express: H H dh, otal differential of eq. (2.22): H d H d (H/) is obtained from the definition of C : H C (2.22) (2.23) (2.24)

14 dh = ds + d (2.8) Differentiation with respect of at constant yields: S H Combining eq. (2.25) with Maxwell equation (2.21): H (H/) is derived from fundamental equation: (2.25) (2.26) Introducing eqs. (2.24) and (2.26) into eq. (2.23) results in : d d C dh (2.27)

15 ENOY Entropy as a function of and : S S, otal differential of eq. (2.28): (2.28) ds S d S d (2.29) (S/) is obtained from Maxwell eq. (2.21): S (2.21)

16 dh = ds + d (2.8) Differentiation with respect of at constant yields: S H Combining eqs. (2.30) with (2.24): (S/) is derived from fundamental equation: (2.30) (2.31) Introducing eqs. (2.21) and (2.31) into eq. (2.29) results in : (2.32) C S d d C ds

17 INENAL ENEGY As a function of and, we can express: U U, otal differential of eq. (2.33): du U d U d (U/) is obtained from the definition of C : U C (2.33) (2.34) (2.35)

18 du = ds d (2.3) Differentiation with respect of at constant yields: Combining eqs. (2.36) and Maxwell equation (2.20): (U/) is derived from fundamental equation: (2.36) (2.37) Introducing eqs. (2.35) and (2.37) into eq. (2.34) results in : (2.38) S U U d d C du

19 Entropy as a function of and : ENOY, S S d S d S ds otal differential of eq. (2.39): (2.39) (2.40) (S/) is obtained from Maxwell eq. (2.20): (2.20) S

20 du = ds d (2.3) Differentiation with respect of at constant yields: Combining eqs. (2.41) and (2.35): (S/) is derived from fundamental equation: (2.41) (2.42) Introducing eqs. (2.20) and (2.42) into eq. (2.40) results in : (2.43) S U C S d d C ds

21 Fundamental equation: dg = S d + d (2.14) expressed as functional relation G = G(, ) (2.44) hus the special, or canonical, variables for the Gibbs energy are temperature and pressure. Since these variables can be directly measured and controlled, the Gibbs energy is a thermodynamic property of great potential utility

22 An alternative form of Eq. (2.14), a fundamental property relation, follows from the mathematical identity: G 1 G d dg 2 d (2.45) Substitution for dg by Eq. (2.14) and for G by Eq. (2.13) gives, after algebraic reduction G d H d 2 d (2.46) he advantage of this equation is that all terms are dimensionless; moreover, in contrast to eq. (2.14), the enthalpy rather than the entropy appears on the right side.

23 From eq. (2.46): G (2.47) H G (2.48) When G/ is known as a function of and, / and H/ follow by simple differentiation. he remaining properties are given by defining equations. In particular, S H G and U H

24 No experimental method for the direct measurement of numerical values of G or GI is known, and the equations which follow directly from the Gibbs energy are ofblittle practical use. by definition the residual Gibbs energy is: G = G G ig (2.49) where G and Gig are the actual and the ideal-gas values of the Gibbs energy at the same temperature and pressure.

25 Other residual properties are defined in an analogous way. he residual volume, for example, is: ig (2.50) Since = Z/, the residual volume and the compressibility factor are related: Z 1 (2.51)

26 he definition for the generic residual property is: M = M M ig (2.52) where M is the molar value of any extensive thermodynamic property, e.g.,, U, H, S, or G. Note that M and M ig, the actual and ideal-gas properties, are at the same temperature and pressure. Equation (2.46), written for the special case of an ideal gas, becomes ig ig ig G H d d 2 d (2.53)

27 esidual roperty: G d H d 2 d (2.54) his fundamental property relation for residual properties applies to fluids of constant composition. Useful restricted forms are: G (2.55) H G (2.56)

28 esidual Gibbs energy: G = H S (2.57) he residual entropy is therefore S H G (2.58) For constant, eq. (2.54) becomes: d G d (contant ) (2.59)

29 Integration from zero pressure to arbitrary pressure yields: G 0 d (constant ) (2.60) where at the lower limit G / is equal to zero because the zero-pressure state is an ideal-gas state. In view of eq. (2.51): G 0 Z 1 d (constant ) (2.61) Differentiation of eq. (2.61) with respect to temperature gives G Z (2.62)

30 Combining eqs. (2.62) and (2.54) gives H Z 0 d (constant ) (2.63) he residual entropy is found by combination of eqs. (2.58), (2.61), and (2.63): S Z 0 d Z 0 1 d p (constant ) (2.64)

31 ESIDUAL OEY FOM IIAL EOS For two-term virial equation Z 1 B From eq. (2.61): G 0 Z 1 d ( konstan) (2.61) We can get: G B (2.65)

32 Differentiation of eq. (2.56) with respect to at constant gives Introducing eq. (2.62) into eq. (2.56): H H 1 db B 2 d B db d Introducing eqs. (2.65) and (2.67) into eq. (2.58) S G db d B 2 db d (2.66) (2.67) (2.68)

33 ESIDUAL OEY FOM CUBIC EOS Equations (2.61), (2.62), and (2.64) are incompatible with pressure-explicit equations of state, and must be transformed to make the variable of integration. Z Z d dz d (constant ) 2 d dz Z d 2 (constant ) (2.69) d dz Z d (constant ) (2.70)

34 Introducing eq. (2.70) into eq. (2.61): G 0 Z 1 dz Z d (constant ) (2.71) he lower limit of the integration is = 0 is the condition of ideal gas: = 0 = and Z = 1 Hence, eq. (2.71) becomes: G Z dz 1 Z Z 1 Z 1 d (constant )

35 G Z Z dz Z 1 d (constant ) G Z 1 lnz Z 1 d (constant ) G Z 1 lnz Z 1 d (constant ) (2.72)

36 Generalized form of cubic eos: b a b b Z 1 b a b b Substituting the above equation into eq. (2.72) yields: G Z 1 lnz 1 b a b b 1 d (constant ) (2.73)

37 Consider the terms in the bracket of right hand side: 1 b 1 b a 1 b b a 1 b b b 1 1 Integration of the equation: 1 b a b b b d ln b a ln b b b ln

38 ln b a ln b b b ln b a ln b b b Introducing the last equation into eq. (2.73): G Z 1 lnz ln b a ln b b b (constant ) (2.74)

39 he corresponding equation for H follows from Eq. (2.54), which in view of Eq. (2.51) may be written: H 2 d d G Z 1 d Division by d and restriction to constant yields: H 2 (2.75) Z 1 G (constant ) (2.76) Differentiation of eq. (2.69) provides the first derivative on the right, and differentiation of eq. (2.72) provides the second. Substitution leads to: H Z Z 1 d (2.75)

40 Z is derived from a cubic eos Z b a b b Z a 2 b b 1 Z 2 a b b

41 b 1 b 1 b a Z 2 he integration part of eq. (2.75): d b 1 b 1 b a d Z b b ln b a

42 Introducing the last eq. into eq. (2.75): H Z 1 a b ln b b (2.76)

43 S is calculated using the equation: S H G Z 1 Z 1 lnz a ln b b a ln b ln b b b b S lnz ln b a b ln b b (2.77)

44 EXAMLE Calculate H and S for n-butane at 500K and 50 bar using the K eos. SOLUION b a b b For the K eos: = 1 = 0 r c a = 0,42748 b = 0, For n-butane: c = 425,1 K c = 37,96 bar

45 a 2 2 c a c = , ,14 425,1 37,96 2 b b c c 0,08664 is calculated numerically: 83,14425,1 37,96 80,667 b a b b Initial value of : 83, ,4 cm 3 /mole

46 i b a i1 i 1 b b i1 i i-1 i error E E E E E E E E E E-05

47 ada iterasi ke-10 diperoleh hasil = 569,7 cm 3 /mol Z 50569,7 83, ,6850 H Z 1 a b ln b b H r 0.5 Z 1 0,5 0.5 c a b r r 0,5 0.5 r ln b

48 H Z 1 b 1,5a 0.5 r ln b H 0, , ,66783, = 1, ln 569,7 569,7 80,667 H = (8,314 J mol -1 K -1 ) (500 K) ( 1,0833) = 4.503,3 J mol -1

49 S lnz ln b a b ln b b S lnz ln b 0,5a b 0.5 r ln b = 0,78735 S = (8,314 J mole -1 K -1 ) ( 1,0833) = J mole -1 K -1

Thermodynamic Variables and Relations

MME 231: Lecture 10 Thermodynamic Variables and Relations A. K. M. B. Rashid Professor, Department of MME BUET, Dhaka Today s Topics Thermodynamic relations derived from the Laws of Thermodynamics Definitions