ACTIVITY VS. COMPOSITION RELATIONSHIPS .1 Introduction

In the equations given earlier, activity is an abstract thermodynamic parameter, and is of no practical significance. What is important is to know the compositions of various phases—metal, slag, gas, etc. which would co-exist at chemical equilibrium. This requires knowledge of activity as a function of composition in various phases of importance in ironmaking and steelmaking at high temperatures.

Roughly speaking, activity is a measure of ‘free concentration’. In ideal gases, molecules are free, and hence activity of a species i (i.e. a_i) = p_i, as already noted in the earlier section.

Pyro-metallurgical processing is quite fast and gets completed in a reasonably short period of time (of the order of hours/minutes). Solid reactants and reagents in these processes are compounds (either pure or mixture). Solid products are also either almost pure metals or compounds, since with solid-state diffusion being very slow, there is almost no opportunity for solid solution formation. Hence, all participating solids are assumed to be pure and by convention, are at their respective standard states, and hence, their activities are to be taken as 1.

Therefore, the main concern is with activity-versus-composition relations in liquid slags and liquid alloys. From a thermodynamic viewpoint, molten slags are solutions of oxides. Liquid alloys are metallic solutions.

a_Cu

aMn

a_Si

0 0.2 0.4 0.6 0.8 1.0

0 0.2 0.4 0.6 0.8 1.0

aB

Xa

Fe B

4.3.2 Ideal, Non-ideal and Regular Solutions

An ideal solution obeys Raoult’s law, which may be stated as

a_i = X_i (4.25)

In practical applications, the weight percent constitutes the composition scale. The X_i is related to W_i by the following equation:

X_i =

=

= + + + +

Â

1

/ /

( / ) ( / ) ( / )

( / )

i i i i

k i i k k

i i

i

W M W M

W M W M W M

W M

(4.26)

where, W_i and M_i are the weight percent and molecular mass respectively of component i.

Most real solutions do not obey Raoult’s law. They exhibit either positive or negative deviations from it. For a binary solution, this is illustrated in Figure 4.2 (Ward 1963) for some Fe–B liquid binary alloys at 1600°C (i.e. 1873 K). In this figure, B is Cu, Mn or Si. The Fe–Mn solution obeys Raoult’s law, the Fe–Cu exhibits strong positive deviation and the Fe–Si strong negative deviation from Raoult’s law. In binary liquid oxides, FeO–MnO behaves ideally, whereas most binary silicates (i.e. CaO–SiO₂, FeO–SiO₂, MgO–SiO₂, MnO–SiO₂, etc.) exhibit strong negative deviations from Raoult’s law.

Figure 4.2 Raoultian activities of copper, manganese and silicon in solution in molten iron at 1600°C.

Departures from Raoult’s law are quantified using a thermodynamic parameter, known as the activity coefficient, which is defined as

= ⁱ

i i

a

g X (4.27)

where, g_i is the activity coefficient of component i in a solution.

Over the past several decades, various attempts have been made to propose analytical correlations between activity and composition for non-ideal solutions. None of them, however, can be taken as valid for all types of solutions, since they invariably contain some simplifying assumptions. The simplest of all of these solution models is the regular solution model. For a binary solution with components A and B, it predicts

RT ln g_A = a X_B²; RT ln g_B = a X_A² (4.28) where, a is a constant at constant temperature. It may be noted further that a varies inversely with T, and that X_A + X_B = 1.

4.3.3 Activities in Molten Slag Solutions

Molten slags are, by definition, concentrated solutions since the minimum concentration of any component oxide is more than a few weight percent. Hence, their activity–composition relations are handled on the basis of the equations proposed earlier. Mole fraction is the composition scale, activity and activity coefficient defined with Raoult’s law as the reference state [Eq. (4.27)], and pure components as standard states (Eq. (4.1)).

Activity vs. composition data in binary slag systems are, therefore, graphically represented as shown in Figure 4.2 for binary iron alloys. However, industrial slags are generally multi-component solutions, and cannot be represented in the same manner as in Figure 4.2. The ternary CaO–SiO₂–Al₂O₃ constitutes the basis for blast furnace slags, and also some slags encountered in steelmaking processes. Figure 4.3 presents the values of activity of SiO₂ in this system at 1550°C (1823 K). These are in the form of iso-activity lines for SiO₂. Similary, there would be diagrams presenting iso-activity lines for CaO and Al₂O₃. For other temperatures, separate diagrams are required.

In Figure 4.3, the liquid field (i.e. molten slag field) is bounded by liquidus lines. In this diagram, Al₂O₃ has been written as AlO_1.5 because the molecular masses of CaO, SiO₂ and AlO_1.5 are close, being equal to 56, 60 and 51 respectively. Therefore, the mole fraction scale becomes approximately the same as the weight fraction scale. Further discussions on thermodynamics of slag solutions shall be taken up later.

4.3.4 Activity–Composition Relationships in Dilute Solutions Activities with one weight percent standard state

Liquid steel and to a reasonable extent liquid pig iron primarily fall in the category of dilute solutions, where concentrations of solutes (carbon, oxygen, silicon, manganese, sulphur, phosphorus, etc.) are mostly below 1 wt. % or so, except for high-alloy steels.

SiO₂ SiO₂

aSiO2

XSiO

X^CaO 2

AlO1-5

AlO_1-5 CaO

AlO1-5

CaO.4AlO1-5

CaO.2AlO_1-5 CaO.12AlO1-5

2CaO.SiO₂

2CaO.2AlO SiO_{1-5 2} 0.003

3CaO.SiO₂ CaO

0.70 0.90

0.80 0.70

(SiC) 0.60 0.50 0.40 0.30 0.20 0.10 0.025 0.05

0.01

10^–3 10^–4

2SiO .6AlO_{2 1-5}

Figure 4.3 Iso-activity lines of SiO₂ in CaO–SiO₂–Al₂O₃ ternary at 1823 K; the liquid at various locations on the liquidus is saturated with compounds as shown.

Solutes in dilute binary solutions obey Henry’s law, which is stated as follows:

a_i = g_i⁰X_i (4.29)

where, g_i⁰ is a constant, known as the Henry’s law constant. Deviation from Henry’s law occurs when the solute concentration increases.

Therefore, activities of dissolved elements in liquid steel are expressed with reference to Henry’s law, and not Raoult’s law. Since it is the intention to find values directly in weight percent, the composition scale is weight percent. With these modifications, i is in solution with Fe as solvent in the binary Fe–i solution. If:

1. Henry’s law is obeyed by the solute inthe binary Fe–i solution, then

h_i = W_i (4.30)

2. Henry’s law is not obeyed by the solute i, then

h_i = f_iW_i (4.31)

where, h_i is activity, W_i is weight percent and f_i is activity coefficient in the so-called one weight percent standard state. This is because, at 1 wt. %, h_i = l, if Henry’s law is obeyed.

Again, it can be shown that f_i is related to Raoultian activity coefficient g as f_i = ₀ⁱ

i

g

g (4.32)

Let us consider the case of oxidation of Si to SiO₂ (i.e. Eq. (4.19)). There, the standard state of Si was taken as pure silicon (solid below 1710°C). The equilibrium relation at temperature T would be:

where (a_SiO2) denotes activity of SiO₂ in oxide solution, and [a_Si] activity of Si in metallic phase (say, in liquid iron).

Now, if the activity of Si in liquid iron has to be expressed in 1 weight % standard state, then Eq. (4.19) would be modified as

[Si]1 wt. % std. state + O₂(g) = SiO₂ (4.34) Therefore, for equilibrium calculations involving solution in liquid iron in 1 wt. % standard state, the values of partial molar free energies of mixing (G_i^m) of solutes in liquid iron are required as well. Some of these are tabulated in Table 4.2 (Engh 1992).