Heat of Mixing in the System Nitrogen-Methane Thermodynamic theory can be put to work in making plant and equipment design more accurate. “Transfer application” of this theory to other fields and other problems may be possible HELMUT KNAPP American Messer Corp., New York, N. Y.
To
DESIGN a plant that separates nitrogen-methane mixtures it is important to know the enthalpy, h, of the mixture as a function of T , p , and N, where N = concentration in mole fractions. Because N1 = 1 - Nz in binary mixtures, only one concentration is required as a variable. I n many cases it might be satisfactory to apply ideal gas laws and to calculate the enthalpy of the mixture by adding the enthalpies of the pure components. For more accurate design it is of interest to know how much the enthalpy of mix-
t9
gibg2 Ni
-
Ni
N2
Figure 1. Molal free energy of a binary mixture at constant temperature and pressure is shown as a function of composition
ing contributes to the real enthalpy of the mixture. As much work has been done to determine the equilibrium data (7, 2, 5) and the equation of state for the mixture ( 4 ) , sufficient material is available to attempt calculation of the heat of mixing by using the laws of thermodynamics.
Method of Calculation
librium between two phases of a binary mixture, molal free energy, g, is plotted as a function of N in Figure 1 at constant pressure p and temperature T. The molal free energy or thermodynamic potential can be expressed by
+
g( T , $ 3 N ) = N1.d T , P) ( 1 - NlkZ ( T , $ ) A , d T , P ,
+
iv-1
(1)
Using Equation of State to Determine Partial Molal Heat of Mixing. T h e equation of state for nitrogen-methane mixtures ( 4 ) could be used to plot the activity coefficients, y, as a function of the temperature at constant concentration and calculate the partial molal heat of mixing, by the formula
An,,
Stotler and Benedict ( 4 ) ,however, found the activity coefficients essentially independent of temperature and pressure and presented the activity coefficients as a function of the composition only. This indicates that the heat of mixing which is determined by the AR;s must be very small. But as the y’s were found by averaging over a great range of temperatures and pressures, it cannot be concluded that heat of mixing is zero. Using Equilibrium Data to DeterAnother apmine Heat of Mixing. proach is therefore used. Tot illustrate the conditions that determine the equi-
,
I
I
I
NI
Ni‘
Figure 2. liquid and equilibrium are shown position
-
I
N: Nl
I N2
Molal free energies for vapor phase and isobaric curve for a binary mixture as a function of the com-
VOL. 51, NO. 6
0
JUNE 1959
783
where gl = molal free energy of the pure component 1 at T and p , and 1 and 2 refer to components 1 (nitrogen) and 2 (methane). In Figure 2,a and b, the conditions for liquid and vapor phases are demonstrated, and isobaric equilibrium curves for pressure 61 are plotted in a T-N diagram. Considering the conditions at temperature TI, it is evident that the two-phase region exists between concentrations N"I < N 1 < X r l . For iV1 < N"1 only the liquid phase and for N1 > N'1 only the vapor phase is present. If the free energy curves were known as shown in Figure 2,a, points N ' I and N"1 could immediately be determined by drawing a line tangent to vapor and liquid curve simultaneously. The equilibrium is defined by the condition g = minimum and it is obvious that the free energy, g, of a mixture consisting of liquid with concentration NI = NIr1 and vapor with concentration h'l = -VI is always lower than the free energy of the liquid or the vapor (The alone within N"1 < N1 < N ' 1 . free energy of the mixture is represented by the tangent.) For concentrations N1 < N"1 the liquid and for iYl > N'I the vapor phase is stable. In Figure 2,a, the parts of the curves corresponding to stable conditions are drawn as a heavy line. The light lines correspond to unstable conditions such as superheated liquid and undercooled vapor mixtures. The conditions for equilibrium in the two-phase region can be derived for constant temperature and pressure.
Figure 3. Difference of enthalpies and entropies of liquid and vapor at equilibrium temperature and pressure were calculated for nitrogen and methane
(3)
p IN
ASth
To derive Ag in respect to N1, Ag must be expressed as a function of concentration. =
T
- ( N 1 c l-t -Vzcz)ldT
b
bN1
a
2 Ah1'
INDUSTRIAL AND ENGINEERING CHEMISTRY
P,
RT In a, 01' - m", 02' - o2lJ, dlV1 B.t.u./Lb. B.t.u./Lb. B.t.u:/Lb. B.t.u./Lb. Mole Mole Mole Mole
T , O El.
P.S.I.A.
a
180
20 30 40 50 60 70 80 90 100
32.0 28.3 23.7 19.5 15.5 11.9 9.5 7.8 6.45
1325 1265 1200 1125 1025 937 853 777 706
30 50 70 80 100 125 150 175 200
20.5 19.8 17.4 15.8 13.2 10.2 7.86 6.1 4.45
1200 1188 1135 1100 1028 925 820 720 595
50 80 100 125 150 200 250 300 350 400
14.0 12.8 11.9 10.6 9.5 8.36 7.3 5.62 4.40 3.68 3.1
1155 1118 1085 1035 986 930 871 756 649 571 495
70 100 150 200 250 300 350 400 450 500
10.2 9.6 8.7 7.8 6.7 5.56 4.62 3.80 2.95 2.23
1110 1080 1035 981 909 820 731 638 516 383
200
220
175
+
+
Ah
from (n, RT In 01 calculated. Differences of molal free energies from Figure 4. ?)/?)Ar1 Ah" determined from Equation 5
Ah - TAs
where As, is the entropy increase upon mixing due to changes in the configuration. If the molecules are distributed at random, As, = - RINl In Nl (1 - Nl) In (1 - N , ) ] . I t is known from (4)that the heat of mixing is small and it therefore can be assumed that only very small deviations from ideal behavior occur. Asth is the entropy change upon mixing due to changes in the thermal characteristics.
Here again it can be assumed that deviations from ideal behavior-Le., from - (A'ICI f Nzcg) = the condition xci, 0-are very small. Based on these assumptions,
Data Used for Calculating -,
where Ah = molal enthalpy of mixing to be determined and As = molal enASth. thalpy of mixing = &,
784
['mix
where c = molal specific heat of the mixture and the pure components.
'The relation between the first two terms is used for this calculation. From Equation l it follows that
Ag
so"
=
PSIA
240
- 580 - 400 - 320 - 230 - 185 - 140 - 100 - 60 - 20 - 650
- 460 - 360
- 320 - 240
- 180 - 120 - 75 - 30
- 670 - 520 - 420 - 360 - 300 - 255 -210
- 145 - 80
-
50
- 20
- 740 - 590 - 440 - 330 - 265 - 200 - 165 - 130
- 95 - 60
N1"
480 570 655 740 785 830 860 890 910
265 295 225 155 55 - 33
- 107 - 173 - 224
0.079 0.144 0.221 0.312 0.427 0.560 0.682 0.793 0.901
260 460 570 600 660 710 760 785 810
290 268 205 180 128 - 35 - 60 - 140 - 245
0.047 0.115 0.190 0.234 0.338 0.486 0.638 0.780 0.902
180 340 420 485 550 580 610 640 670 680 690
305 258 245 190 136 95
0.036 0.099 0.144 0.203 0.272 0.348 0.432 0.600 0.763 0.893 0,989
40 170 330 410 455 500 520 540 560 580
51
- 29 - 101 - 159 -215 330 320 265 241 189 120 46 - 32 - 139 - 257
0* 008 0.050 0.125 0.205 0.301 0.406 0.513 0.621 0.725 0.822
H E A T OF M I X I N G 2000
1800
1600 1400 1200 1000 W
600
2
600
z
400
$ 200 m -400
140
160
180 200 220 TEMPERATURE IN ‘R
240
L o
260
. o
Figure 4. Extrapolations are necessary to determine difference of molal free energies of liquid and vapor as a function of temperature and pressure A.
- gi” for Nz.
81’
B.
92‘
- gz” for CHI
c”-200 -400 -600 -600 -1000 180
Inserting Equations 3 and 4 into the equilibrium conditions of Equation 2: gl’ - gz‘
+ RT In N1 + 1-
lNl,
or
= g1”
The same formula could have been obtained by working with chemical potentials.
where G = nigl f nzgz 4- (nl 4- n,) Ag. = partial molal free enthalpy or chemical potential Making the same assumptions as before,
+ RTIn N I +
G1 =
a
- (nl ani
+ nz)Ah = gt + R T I n N I + a Ah ( 7 ) Ah + (1 - Ni) GI‘ Q,l
g l r - gl”
d,” = 0,”
=
N ’, = Ah‘‘ + RT In 4 Ni
Ah’+ ( 1 - N 1 ” ) -
a
bNi
-
Ah” -
a Ah’ - Ni’) bNi ” = Ah“ gz’ - g2” + RT In 4, Nz a AhIf - N1’ a Ah’ Ah’ + N1” aNi dN1 (1
Numerical Calculations The right side of Equation 5 must now be calculated for the total range of concentrations, in order to determine the molal enthalpy of mixing by integration from N I = 0 to N1 = 1. I t is necessary to know g‘ - g”, the difference of molal free energies of vapor and liquid phase of the pure components as a function of temperature and pressure. This is known at the condensation point where liquid and vapor are in equilibrium and where g’ - g” = 0. I t can be calculated for conditions where at least one phase is unstable, if certain assumptions are made. g‘ - g“ = h‘
- h“ -
T(s’
-
-
h’ - h” and s‘ st‘ are known for equilibrium conditions. T o find g’ - g K for a certain pressure above or below the vaporization temperature, To,the following calculation could be made : g ’ ( T 1 ) - g ” ( T i ) = h’( T i ) - h”(T1) TiIs’(T1) - s ” ( T i ) ]
with the equilibrium conditions and
Subtracting Equation 8a from 8 gives Equation 5. It was preferred, however, to demonstrate the equilibrium conditions by the graphical method.
N1
- gz” 4a Ah” RT in 1 +1 - N1“ aN1 [ N ,~ (gl’ - gl”) - (ge’ - ~ 2 ’ ’ ) + Ni’ 1 - N N R T 1 n - P 1 - N1‘ Ni” Ah’
200
(8)
(sa)
220
240 260 260 TEMPERATURE IN O R
300
320
Here all quantities are known except the specific heat of a superheated liquid. The same situation results if g’ - g” is considered below the condensation temperature where c D r , the specific heat of the subcooled vapor, is not available. I t is possible to perform the integration when extrapolating the cp into the unstable state. For higher pressures, however, cp changes considerably near the condensation point and unreasonable results would be obtained by using this method for more excessive g” was therefore interpolations. g’ calculated by assuming that h’(T,) h”(To),the molal enthalpy of liquefaction or vaporization, and s’(To) s”(T,), the molal entropy of liquefaction or vaporization, do not vary at constant pressure for temperatures above or below the condensation point. In this case, g’ - g” is simply given by c”,
-
g ’ ( T , P o ) - g ” ( T , P o ) = h’(To,Po) h”( To, Po) - T[s’(To, Po) - .’’(To, Po11
In Figure 3, h’ - h” and s’ - s” for nitrogen and methane are plotted as a function of p . T h e curves are extrapolated for pressures approaching the critical pressure, pori*. Using thwe values, g’ - g” is plotted in Figure 4,A and B, as a function of temperature for constant pressures. g’ g” becomes zero at the condensation temperatures. For g’ - g”, extrapolations are necessary into the state of superheated liquid. These extrapolations are more difficult than those for methane into the states of subcooled liquid and subcooled vapor. For the nitrogen a correction on the linear extrapolation has therefore been made, as it was assumed that beyond a certain g’ - g” the s’ - s” becomes smaller VOL. 51, NO. 6
JUNE 1959
785
,
100
0
W -I
0 -100
2
5
F.
m
-200
-300 0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 M O L E F R A C T I O N N1
0.8
0.9
1.0
Figure 5. d / b N l , Ah” is shown as a function of the mole fraction of the liquid, N”1
4 Figure 6.
Ah”, TAs”, and Ag”at T
as a function of
-350
I
I
0.2
0
than s’(T,) - s‘’(T,) (broken lines in Figure 4 4 ) . I t is further necessary to know the quantity
This is the relative volatility: a , and the smoothed relative volatility values of (2) were used for these calculations. I n the table the data are listed from which Figure 5 was plotted. Figure 5 shows
a
- (Ah” - Ah’) bN1 as a function of liquid composition, d\rl’‘. Actually Ah is a function of T and p T h e result demonstrated in Figure 6 covers a wide range of temperatures (TI > T > Tz)and pressures ($1 > 1, > $2). As the points do not scatter too much, it can be concluded that Ah does not vary within the considered range. The points represent the difference of
t\vo derivations taken at two different concentrations. T h e points can be evaluated, assuming that Ah’
