9.3 Consi Conside derr a syste system m with with comp compone onent ntss Cu, Cu, Ni and and Zn tha thatt contai contains ns four four phas phases: es: α, α, β, ε and L. (a List the !aria"#e re$uired to specify the state of each phase% count them. (" List the conditions for e$ui#i"rium for this system% count them. (c &rom these counts, compute the num"er of de'rees of freedom. (d Compare this resu#t with the num"er of de'rees of freedom computed from the i""s phase ru#e. )o#ution: (a
("
No. of components(c) = 3 No. of phases(P)= 4 Degree of freedom: F = c+2-P F = 3+2-1= 4 So for each phase the !ar"a#$es re%&"red to spec"f' the state "s 4. amp$e* for the state of & the !ar"a#$es are: , P P N" n. ,he cond"t"ons for e%&"$"#r"&m for the s'stem are:
,/=,0=,=, P/=P0=,=, &/= &0=&=& N"/= N"0= N"= N" n/=n0=n=n ,he tota$ no. of cond"t"ons are (c+2)(P-1) = 1
*+.* *+.*
(c
Degree of freedom "s = (,ota$ no. of !ar"a#$es "n the s'stem) 5 (No. of e%&at"ons re$at"ng to the !ar"a#$es) F= 16 -1 = 1
(d
From 7"##s phase r&$e: F+P= +2 F= 3+2-4= 1
ep#o ep#ott the the - cur!es cur!es shown shown in &i'ure &i'ure *+./ *+./ for the fo##owi fo##owin' n' choice choicess of refer referen ence ce states for components * and /. a. 0L%L1 ". 0L%α1
c. 0α%α1 cur!es are computed with the fo##owin' information: so#ution mode#s are simp#e re'u#ar so#utions. 2 m* *4++5% 2m/ 64+5% 7) m* 9 8mo# % 7) m/ ;8mo# % a+α 6<++ 8mo#% a +L *+,4++ 8mo#. )o#ution:
7"!en: components: 1 8 2 are "n phases / and . For reg&$ar so$&t"on mode$: D7m"/9/*/=a o/1/2/+;,(1/$n1/+2/$n2/) D7m"/9/*/=<41/2/+;,(1/$n1/+2/$n2/) D7m"/9/*/=<41/2/+<.3141(1/$n1/+2/$n2/) / / / / / D7m"/ 9/*/=<4(1-2/) 2 +<.3141>(1- 2 ) $n(1-2 ) + 2 $n 2 ?
D7m"9*=ao12+;,(1$n1+2$n2) D7m"9*=112+;,(1$n1+2$n2) D7m"9*=112+<.3141(1$n1+2$n2) D7m" 9*=1(1-2) 2 +<.3141>(1-2 )$n(1-2)+2$n2) a. For reference state 9*: 7"##s free energ' of m""ng of "%&"d Phase ta@en 9* as reference. D7m"9*= D7m"9* D7m" 9*=1(1-2) 2 +<.3141>(1-2 )$n(1-2)+2$n2)
7"##s free energ' of m""ng of / phase ta@en reference as 9*
D7m"/9*=1/(71/-71o) + 2/(72/-72o) D7m"/9*=1/(71/-71o/+ 71o/ -71o) + 2/(72/ -72o/+ 72o/ -72o) D7m"/9*= 1/(71/-71o/)+ 1/(71o/ -71o)+ 2/(72/ 72o/)+ 2/(72o/ -72o) D7m"/9*=D7m"/9/*/ + 1/(71o/ -71o)+ 2/(72o/ -72o) D7m"/9*=D7m"/9/*/- 1/DSm1(Tm1-T) -2/DSm2(Tm2-T) D7m"/9*=D7m"/9/*/- 1/9(1500-1000) -2/7(850-1000)
D7m"/9*=D7m"/9/*/- 1/9(500) +2/7(150) / / / / / / / / D7m" 9*=<4(1- 2 ) 2 +<.3141>(1-2 ) $n(1-2 ) + 2 $n 2 ? - (1-2 ) 9(500) +2/7(150)
b.
For reference state {L;α}: D7m"/9*/= D7m"/9/*/ - 1/D71o/ D7m"/9*/= D7m"/9/*/ - 1/DSm1(,m1-,) D7m"/9*/= D7m"/9/*/ - 1/9(1-1) D7m"/9*/= D7m"/9/*/ - 1/9()
/ / / / / / / D7m"/ 9*/=<4(1-2 ) 2 +<.3141>(1- 2 ) $n(1-2 ) +2 $n2 ?- (1- 2 ) 9()
D7m"9*/= D7m"9* +2D72o/ D7m"9*/= D7m"9*+2DSm2(,m2-,) D7m"9*/= D7m"9*+27(<-1) D7m"9*/= D7m"9*-27(1)
D7m" 9*/=1(1-2) 2 +<.3141>(1-2 )$n(1-2)+2$n2) -27(1)
c. For reference state 9/*/: D7m"/9/*/= D7m"/9/*/ / / / / / D7m"/9/*/= <4(1-2/) 2 +<.3141>(1- 2 ) $n(1-2 ) + 2 $n2 ?
D7m"9/*/= D7m"9*+1D71o/ +2D72o/ D7m"9/*/= D7m"9*+1DSm1(,m1-,) + 2DSm2(,m2-,) D7m"9/*/= D7m"9*+ 1/9(1-1)+ 27(<-1) D7m"9/*/= D7m"9*+ 1/9() -27(1)
/ )$n(1-2)+2$n2) -27(1)+ (1-2 ) D7m" 9/*/=1(1-2) 2 +<.3141>(1-2 9()
*+./
2he =-> system forms re'u#ar so#utions in "oth the β and #i$uid phase forms. ?arameters for the heat of mi@in' for these phases are a +α -6/++ 8mo# and a +L -*+,4++ 8mo#. a. Compute and p#ot cur!es for 7 mi@ for the β and #i$uid phases for these mode#s at 6++5. ?ure = me#ts at *+4+5 with a heat of fusion of 6/++ 8mo#. ?ure > me#ts at AA+5 with a heat of fusion of A6++ 8mo#. ". Chan'e the reference states appropriate#y to p#ot and compare 7 mi@β0β%L1 and 7mi@L0β%L1. Ne'#ect differences in heat capacities for the phases for the pure components.
)o#ution:
a.
For reg&$ar so$&t"on mode$ Ae ha!e the B-C s'stem "n 0 phase:
Dm!"#{#;#} $ a0#%&#%'# T(%&#*n%&#%'#*n%'#) Dm!"#{#;#} $ -8200%&#%'# T(%&#*n%&#%'#*n%'#)
S"m"$ar$' for the s'stem B-C "n phase Ae ha!e:
Dm!"L{L;L} $ a0L%&L%'L T(%&L*n%&L%'L*n%'L) Dm!"L{L;L} $ -10+500%&L%'L T(%&L*n%&L%'L*n%'L)
b.
Dm!"#{#;L} $ Dm!"#{#;#} -%'#D0#
L
$ Dm!"#{#;#} -%'#(&0L- &0#) $ Dm!"#{#;#} -%'#DSm0(Tm'-T) $ Dm!"#{#;#} -%'#D,m0(Tm'-T) Tm' $Dm!"#{#;#} -%'# .800 (..0-T)..0 # # # # # # # # Dm!" {#;L} $ -8200% & % ' T(%& *n%& % ' *n% ' ) - %' .800 (..0 T)..0
Dm!"L{#;L} $
Dm!"L{L;L} %&LD0#
L
$ Dm!"L{L;L} %&LDSm0(Tm&-T) $ Dm!"L{L;L} %&LD,m0(Tm&-T) Tm& L L L L Dm!"L{#;L} $ -10+500%& % T(%&L *n%&L %' *n% ' ' ) L %& 8200(1050-T) 1050
