Chemical Engineering Thermodynamics
Lecturer: Zhenxi Jiang (Ph.D. U.K.)
School of Chemical Engineering
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Chapter 12
Solution Thermodynamics: Application
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12.4 Heat Effects of Mixing Processes
The heat of mixing, defined in accord with Eq.
(12.29), is:
(12.39)
H  H   xi Hi
i
It gives the enthalpy change when pure species
are mixed at constant T and P to form one mole
(or a unit mass) of solution. Data are most
commonly available for binary systems, for
which Eq. (12.39) solved for H becomes:
H =x1H1+x2H2 + ΔH
(12.40)
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12.4 Heat Effects of Mixing Processes
This equation provides for the calculation of the
enthalpies of binary mixtures from enthalpy data
for pure species 1 and 2 and from the heats of
mixing. Treatment is here restricted to binary
systems.
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12.4 Heat Effects of Mixing Processes
Data for heats of mixing are usually available for
a very limited number of temperatures. If the
heat capacities of the pure species and of the
mixture are known, heats of mixing are
calculated for other temperatures by a method
analogous to the calculation of standard heats of
reaction at elevated temperatures from the
value at 25°C.
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12.4 Heat Effects of Mixing Processes
Heats of mixing are similar in many respects to
heats of reaction. When a chemical reaction
occurs, the energy of the products is different
from the energy of the reactants at the same T
and P because of the chemical rearrangement of
the constituent atoms.
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12.4 Heat Effects of Mixing Processes
When a mixture is formed, a similar energy
change occurs because interactions between the
force fields of like and unlike molecules are
different. These energy changes are generally
much smaller than those associated with
chemical bonds; thus heats of mixing are
generally much smaller than heats of reaction.
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12.4 Heat Effects of Mixing Processes
Heat of Solution
When solids or gases are dissolved in liquids,
the heat effect is called a heat of solution, and is
based on the dissolution of 1 mol of solute. If
species 1 is the solute, then x1 is the moles of
solute per mole of solution. Becauseis the heat
effect per mole of solution, H/x1 is the heat effect
per mole of solute. Thus,
H
～
H 
where
x1
is the heat of solution on the basis of a mole of solute
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12.4 Heat Effects of Mixing Processes
Solution processes are conveniently represented
by physical-change equations analogous to
chemical-reaction equations. When 1 mol of
LiCl(s) is mixed with 12 mol of H2O, the process
is represented by:
LiCl (s) + 12H20 (l) → LiCl (12H20)
The designation LiCl (12H20) represents a
solution of 1 mol of LiCl dissolved in 12 mol of
H2O.
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12.4 Heat Effects of Mixing Processes
The heat of solution for this process at 25°C
and 1 bar is
= - 33,614 J. This means that
the enthalpy of 1 mol of LiCl in 12 mol of H2O is
33,614 J less than the combined enthalpies of 1
mol of pure LiCl (s) and 12 mol of pure H2O(J).
Equations for physical changes such as this are
readily combined with equations for chemical
reactions. This is illustrated in the following
example, which incorporates the dissolution
process just described.
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12.4 Heat Effects of Mixing Processes
Example 12.4
Calculate the heat of formation of LiCI in 12 mol
of H20 at 25°C.
Solution 12.4
The process implied by the problem statement
results in the formation from its constituent
elements of 1 mol of LiCl in solution in 12 mol of
H2O. The equation representing this process is
obtained as follows:
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12.4 Heat Effects of Mixing Processes
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12.4 Heat Effects of Mixing Processes
The first reaction describes a chemical change
resulting in the formation of LiCl(s) from its
elements, and the enthalpy change
accompanying this reaction is the standard heat
of formation of LiCl(s) at 25°C.
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12.4 Heat Effects of Mixing Processes
The second reaction represents the physical
change resulting in the dissolution of 1 mol of
LiCl(s) in 12 mol of H20(l), and the enthalpy
change is a heat of solution. The overall
enthalpy change, -442,224 J, is the heat of
formation of LiCl in 12 mol of H20. This figure
does not include the heat of formation of the
H20.
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12.4 Heat Effects of Mixing Processes
Often heats of solution are not reported directly
but must be determined from heats of formation
by the reverse of the calculation just illustrated.
Typical are data for the heats of formation of 1
mol of LiCI:
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12.4 Heat Effects of Mixing Processes
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12.4 Heat Effects of Mixing Processes
Heats of solution are readily calculated from
these data. The reaction representing the
dissolution of 1 mol of LiCl(s) in 5 mol of H20(l)
is obtained as follows:
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12.4 Heat Effects of Mixing Processes
This calculation can be carried out for each
quantity of H2O for which data are given. The
results are then conveniently represented
graphically by a plot of
, the heat of solution
per mole of solute, vs. n, the moles of solvent
per mole of solute. The composition variable,
n  n / n , is related to x1:
2
1
x2 (n1  n2 ) 1  x1
n

x1 (n1  n2 )
x1
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12.4 Heat Effects of Mixing Processes
whence
1
x1 
1 n
The following equations therefore relate, the
heat of mixing based on 1 mol of solution, and,
, the heat of solution based on 1 mol of
solute:
H
H 
 H (1  n)
x1
H
or H 
1 n
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12.4 Heat Effects of Mixing Processes
Figure 12.14 shows plots of
vs. n for LiCl(s)
and HCl(g) dissolved in water at 25°C. Data in
this form are readily applied to the solution of
practical problems.
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12.4 Heat Effects of Mixing Processes
Because water of hydration in solids is an
integral part of a chemical compound, the heat
of formation of a hydrated salt includes the heat
of formation of the water of hydration. The
dissolution of 1 mol of LiCl2H2O(s) in 8 mol of
H2O produces a solution containing 1 mol LiCl in
10 mol of H2O, represented by LiCl(10H2O). The
equations which sum to give this process are:
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12.4 Heat Effects of Mixing Processes
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12.4 Heat Effects of Mixing Processes
Heats of Solution
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12.4 Heat Effects of Mixing Processes
Example 12.5
A single-effect evaporator operating at
atmospheric pressure concentrates a 15% (by
weight) LiCl solution to 40%. The feed enters
the evaporator at the rate of 2 kg s-1 at 25°C.
The normal boiling point of a 40% LiCl solution
is about 132°C, and its specific heat is
estimated as 2.72 kJ kg-1 °C-1. What is the
heat-transfer rate in the evaporator?
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12.4 Heat Effects of Mixing Processes
Solution 12.5
The 2 kg of 15% LiCl solution entering the
evaporator each second consists of 0.30 kg LiCl
and 1.70 kg H20. A material balance shows that
1.25 kg of H20 is evaporated and that 0.75 kg of
40% LiCl solution is produced. The process is
represented by Fig. 12.15.
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12.4 Heat Effects of Mixing Processes
The energy balance for this flow process is
ΔHt = Q, where ΔHt is the total enthalpy of
the product streams minus the total enthalpy of
the feed stream. Thus the problem reduces to
finding ΔHt from available data.
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12.4 Heat Effects of Mixing Processes
Because enthalpy is a state function, the
calculational path for is immaterial and is
selected for convenience and without reference
to the actual path followed in the evaporator.
The data available are heats of solution of LiCl in
H20 at 25°C (Fig. 12.14), and the calculational
path, shown in Fig. 12.16, allows their direct use.
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12.4 Heat Effects of Mixing Processes
The enthalpy changes for the individual steps
shown in Fig. 12.16 must add to give the total
enthalpy change:
The individual enthalpy changes are determined
as follows.
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12.4 Heat Effects of Mixing Processes
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12.4 Heat Effects of Mixing Processes
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12.4 Heat Effects of Mixing Processes
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12.4 Heat Effects of Mixing Processes
Heats of Solution
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12.4 Heat Effects of Mixing Processes
Heats of Solution
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12.4 Heat Effects of Mixing Processes
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12.4 Heat Effects of Mixing Processes
Enthalpy/Concentration Diagrams
The enthalpy/concentration (Hx) diagram is a
useful way to represent enthalpy data for binary
solutions. It plots enthalpy as a function of
composition (mole fraction or mass fraction of
one species) with temperature as parameter.
The pressure is a constant and is usually l(atm).
Figure 12.17 shows a partial diagram for the
H2SO4/H2O system, where enthalpy values here
are for a unit mass of solution. Equation (12.40)
is therefore directly applicable:
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12.4 Heat Effects of Mixing Processes
Enthalpy/Concentration Diagrams
H  x1H 1  x2 H 2 H
(12.40)
Values of H for the solution depend not only on
heats of mixing, but also on enthalpies H1 and
H2 of the pure species. Once these are known
for a given T and P, H is fixed for all solutions at
the same T and P, because has a unique and
measurable value for each composition.
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12.4 Heat Effects of Mixing Processes
Absolute enthalpies are unknown, and arbitrary
zero points are chosen for the enthalpies of the
pure species. Thus, the basis of an
enthalpy/concentration diagram is H1 = 0 for
some specified state of species 1 and H2 =0 for
some specified state of species 2. The same
temperature need not be selected for these
states for both species.
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12.4 Heat Effects of Mixing Processes
In the case of the H2S04( 1)/H2O(2) diagram shown in
Fig. 12.17, H2 =0 for pure liquid H20 at the triple point
[32(°F)], and H1= 0 for pure liquid H2S04 at 25°C
[77(°F)]. In this case the 32(°F) isotherm terminates
at H = 0 at the pure- H2O edge of the diagram, and the
77(°F) isotherm terminates at H = 0 at the pureH2S04 edge of the diagram. The advantage of taking H
= 0 for pure liquid water at its triple point is that this is
the base of the steam tables. Enthalpy values from the
steam tables can then be used in conjunction with
values taken from the enthalpy/concentration diagram.
Were some other base used for the diagram, one
would have to apply a correction to the steam-table
values to put them on the same basis as the diagram.
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12.4 Heat Effects of Mixing Processes
Enthalpy/Concentration Diagrams
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12.4 Heat Effects of Mixing Processes
For an ideal solution, isotherms on an
enthalpy/concentration diagram are straight
lines connecting the enthalpy of pure species 2
at =0 with the enthalpy of pure species 1 at =
1, as illustrated for a single isotherm in Fig.
12.18 by the dashed line. The solid curve
represents an isotherm for a real solution. Also
shown is a tangent line from which partial
enthalpies maybe determined in accord with
Eqs, (11.15) and (11.16). Equations (11.82) and
(12.40) combine to give ; is therefore the
vertical distance between the curve and the
dashed line of Fig. 12.18.
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12.4 Heat Effects of Mixing Processes
Here, the actual isotherm lies below the idealsolution isotherm, and H is everywhere negative.
This means that heat is evolved whenever the
pure species at the given temperature are mixed
to form a solution at the same temperature.
Such a system is exothermic. The H2SO4/H2O
system is an example. An endothermic system is
one for which the heats of solution are positive;
in this case heat is absorbed to keep the
temperature constant. An example is the
methanol/benzene system.
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12.4 Heat Effects of Mixing Processes
One useful feature of an enthalpy/concentration
diagram is that all solutions formed by adiabatic
mixing of two other solutions are represented by
points lying on a straight line connecting the
points that represent the initial solutions. This is
shown as follows.
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12.4 Heat Effects of Mixing Processes
Let the superscripts a and b denote two initial
binary solutions, consisting of na and nb moles
respectively. Let superscript c denote the final
solution obtained by simple mixing of solutions a
and b in an adiabatic process, either batch
"mixing at constant P or steady-flow mixing with
no shaft work or change in potential or kinetic
energy. In either case, Ht = Q = 0, and the total
energy balance is:
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12.4 Heat Effects of Mixing Processes
Enthalpy/Concentration Diagrams
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12.4 Heat Effects of Mixing Processes
Our purpose now is to show that the three
points c, a, and b represented by (Hc, x1c),
(Ha, x1a), and (Hb, x1b) lie along a straight line on
an Hx diagram. The equation for a straight line
in these coordinates is:
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12.4 Heat Effects of Mixing Processes
H = m x1 + k
(B)
If this line passes through points a and b,
Ha = m x1a + k
and
Hb = m x1b + k
Each of these equations may be subtracted from
the general equation, Eq. (B):
H – Ha = m (x1 - x1a)
H - Hb = m (x1 - x1b)
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12.4 Heat Effects of Mixing Processes
Enthalpy/Concentration Diagrams
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12.4 Heat Effects of Mixing Processes
Enthalpy/Concentration Diagrams
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12.4 Heat Effects of Mixing Processes
Enthalpy/Concentration Diagrams
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12.4 Heat Effects of Mixing Processes
Enthalpy/Concentration Diagrams
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12.4 Heat Effects of Mixing Processes
Enthalpy/Concentration Diagrams
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12.4 Heat Effects of Mixing Processes
Enthalpy/Concentration Diagrams
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12.4 Heat Effects of Mixing Processes
Enthalpy/Concentration Diagrams
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12.4 Heat Effects of Mixing Processes
Enthalpy/Concentration Diagrams
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12.4 Heat Effects of Mixing Processes
Homework?
No
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12.4 Heat Effects of Mixing Processes
Thanks!
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