CHAPTER 6 RELATIONS
F. CHAPTER 6 RELATIONS
Summary of P–V–T equations for real gases
P R = P/P c ,T R = T/T C ,v R ' = v/v c ', v c ' = RT c /P c , Z(T,P) = v(T,P)/v 0 (T,P)
Inflection conditions: ( ∂P/∂v) 2
b ≈ (2/3)N πσ Avog 3 . Pitzer Factor w = –1.0 – log (P sat )
= –1 – 0.4343 ln (P sat ) .Z=Z 10 (0) R TR=0.7 R TR=0.7 (T R ,P R )+wZ (1) (T R ,P R )
If R is a universal gas constant, then a, v, b are based upon mole basis. If R is simply a gas constant, then a, v, b are based upon mass basis Summary of Equations of State
Name
State Equations
Constants
Remarks
2 1. 3 Virial Eq. Pv= RT + B
1 ´P + C 1 ´P + D 1 ´P + ..., where B 1 ’, C 1 ’, D 1 ’ are called 2nd, 3rd, P/ u T=Z
B 1 ´, C 1 ´, … are functions of T or Pv = RT + B/ and 4th virial coefficients that
v + C/v 2 + ... , and B, C, ... are functions of T.
represent corrections to ideal gas behavior
B*(T*) = B(T) / b o , b 0 in Virial Eq. 3
2. Approximate
Pv/RT = Z = 1 + B(T)/P
m /kmole, T* = T/( ε/k) For
Z = 1 + (P R /T R ) (0.083 – (0.422/T r ))
b o , ε/k. See R.E. Sonntag and G. Van Wylen
where T R = T/T c, P R = P/P c
3. Clausius I
P = RT/(v–b)b = body volume
Cannot satisfy inflection conditions.
a = (27/64)v c ' 2 P c = (27/64) R 2 Does not agree with (VW)
4. Van der Waals P = RT/(v–b) – a/v 2 , b is a correction for
volume occupied by molecular and repulsive
c /P c ≈ 2.667πεσ N Avog , b= v c '/8 (P c v c /RT c ) exp = 0.2 to 0.3
forces, a/v 2 is a correction for attractive forces
=(1/8) RT c /P c , Z c =3/8, A = for most gases. Another
Z – (B* +1) Z + A* Z – A*B =0
(27/64) P R /T 2 R , B* = (1/8) P R /T R
form:of the VW relation is v 3 + v 2 (– b P – R T)/P + v ( a /P) (– ab /P) = 0
5. Berthelot
P = RT/(v–b) – (a/T) (1/v 2 )a = (27/64) (v c ' 2 T 2
3 c P c ) = = (27/64)R
T c /P c ,b=v c '/8 = RT c /8P c
Name State Equations
Constants
Remarks
6. Dieterici P = (RT/(v–b)) exp {–a/(RTv)}
2 c /P c b 2 Developed to provide better
a = (4/e 2 )v c ' 2 P c =(4/e 2 )R 2 T 2
=v c '/e = RT c /(e P c ), Z c = 0.271,
agreement with
e = exp(1)=2.3026
experiments.
7. Redlich–Kwong P = RT/(v–b) –a/(T 1/2 v (v + b))
a= 0.4275 v c ' 2 T 0.5 c P
2.5 c = 0.4275R
2 Good accuracy over wide
(RK)
range and at high pressure. or
T c /P c ,
b = 0.08664 v c ' = 0.08664 RT c /P c ,
Z * –Z 2 + (A*– B* 2 – B*) Z – A*B 2.5 * =0 Z
c =1/3, A = 0.4275 P R /T R , B* =
0.08664 P R /T R
8. Clausius II
P = RT(v–b)–a/(T(v+c) 2 )a = 27/64 v ' 2 T P =(27/64) R c 2 c c
T 3 c /P c b=v c '(Z c –1/4) = (RT c /P c ) (Z c –1/4) c=v c '(3/8 –Z c )= (RT c /P c ) (3/8 –Z c )
9. Peng Robinson P = (RT/(v–b) ) – (a α(w,TR)/((v+b(1+√2)) (v
a = 0.45724 v c ' 2 Pc = 0.45724R 2
+ b (1– √2)))
T c /P c α(w,T R ) = (1 + f(w) (1 – T (1/2) R )) 2
f(w) = 0.37464 + 1.54226 w – 0.26992 w 2
b = 0.07780 v c ' =0.0778 (RT c /P c ), Z c = 0.26
10. SRK equation P = RT/(v–b) – a α(w,T R )/(v(v+b))
2 A = 0.4275v 2
Pc = 0.4275R
T 2 c /P c , b = 0.08664 v c ' =0.08664 (RT c /P c ), Z c = 1/3
α(w,T
R ) = (1 + f(w) (1 – T R )) 2
f(w) =(0.480 + 1.574 w – 0.176 w 2 )
11. Generalized Eq.
R =T R /( v ′ R –b )–a α (w,T R )/(T R ( v R ′ +c ) a = a/(P c v ′ c T c ), b = b/ v ′ c ,c =
Name State Equations
Constants
Remarks
of state *) ( v ′ R +d ) ,
c/ v ′ c , and d * = d/ v ′ c . See table
below
12. Compressibility Pv = ZRT or P R v R '=ZT R Z=v/v ideal for T R > 2.5, Z > factor
v R ' = v/v c ', v c ' =RT c /P c
1, for T R < 2.5 Z < 1 and
has a minimum value. At Also fv = ( φ(T , P ) Z(T
P R = P/P c ,T R = T/T c
R , P R )) RT (Chapter
T = 1, P = 1, Z can vary
widely. For P R > 10 always 07)
use real gas relations.
13. Benedict Webb P = RT/v + (B
Good accuracy over wide Rubin
2 RT–A 2 –C 2 /T )/v 2 + (B 3 R T–
8 constants. See Table 20A
A 3 )/v 3 + A 3 C 6 /v 6 + (D
3 /(v T 2 ))(1+E 2 /v 2 )
P–V–T condition.
exp(–E /v 2 2 )
14. Martin–Hou
P = RT/(v–b) + (A + B + C e –KT )/(v–b) 2 +
12 constants evaluated from P–v–T Mainly developed for
(A +B T+C e –KT 2 2 3 2 3 )/(v–b) 3 +A 4 /(v–b) 3 4 + (A 5 data of fluids
refrigerants, 1 % accuracy
+B 5 T+C 5 e –KT )/(v–b) 5
for v > 0.67 v c and T < 1.5
15. Lee Kessler
P R =(T R /v R’ )(1+A/ v ′ R +B/ v ′ 2 R +C/ v ′ 5 R +(D/ v ′ R )(
See Table A–21
β+γ/ v ′ 2 R )exp(– γ/ v ′ 2 R )), Z = P R v ′ R /T R
16. Beattie P v = R T( v +B
Accurate for v > 1.25 v c Bridgeman
2 0 (1– ( b / v )) (1– c/( v T ))–
See Table A–20B
(A 0 / v )(1–(a/ v )) equation
Generalized cubic equation of state P R = (T R /(v R '– b * )) – a * α(w,T R n )/(T R (v R '+c * )(v * R +d * )),
where a * = a/(P v ' 2 T c n c c ), b * = b/v c ', c * = c/v c ', d * = d/v c '
Equation 2 c = c/v
d =c/v c'
α α α α (w,T R ) b =b/v c a =a/(P c T c v c ' )
Clausius–I
(27/64) Clausius–II
1 1 Z c –1/4 27/64 Horvath–Lin γβ (note 3)
(3/8–Z c )
(3/8–Z c )
0 1 1 β (note 3) α (note 3) RK
(1+ √2)0.07780 (1–√2)0.0778 0 0.07780 0.45724 Note 1: T R b'/v c ' 2 – a'/v c ' 2
Note 2: c'/v 2
c ' –T R Note 3: γ = Z c –4.72 /360, (1+(
γ+1) 4 ) –1 , 2 α = ((1+γf) (1–2f– 2 γf 2 ))/((1–f) (2+ γf) ), β = (1–2f– 2 γf )/((2+
γ)+(2+4γ)f +(γ+2γ 2 )f 2 )
Liquids and Solids dv = v β P dT – v β T dP, β P = (1/v)( ∂v/∂T) P , β T = –(1/v) ( ∂v/∂P) T , κ T = 1/( β T P) = (–v/P) ( ∂P/∂T) T