Examine the conditions under which a liquid Fe– Mn alloy can be in equilibrium with an FeO– MnO liquid solution in an atmosphere containing oxygen at 1800°C. For
Mn_{(l)}+\frac{1}{2}O_{2(g)}=MnO_{(l)} (i)
\Delta G^{\circ }_{(i)} =-344,800+55.90T J
and for
Fe_{(l)}+\frac{1}{2}O_{2(g)}=FeO_{(l)} (ii)
\Delta G^{\circ }_{(ii)} =-232,700+45.13T J
The pertinent equilibrium is
FeO_{(l)}+Mn_{(l)}=MnO_{(l)}=Fe_{(l)} (iii)
for which
\Delta G^\circ _{(iii),2073 K}=\Delta G^\circ _{(i),2073 K}-\Delta G^\circ _{(ii),2073 K}= -228,900 + 138,800 = -90,100 J
=-8.3144\times 2073\ln K_{(iii),2073 K} J
Therefore,
K_{(iii),2073 K}=186=\frac{(a_{MnO})[a_{Fe}]}{(a_{FeO})[a_{Mn}]}where:
(a_{MnO}) = the activity of MnO in the liquid oxide phase with respect to pure MnO
v(a_{FeO}) = the activity of FeO in the liquid oxide phase with respect to ironsaturated liquid iron oxide
[a_{Mn}] = the activity of Mn in the liquid metal phase with respect to pure liquid Mn
[a_{Fe}] = the activity of Fe in the liquid metal phase with respect to pure liquid Fe.
Since both the liquid metal solution and the liquid oxide solution exhibit Raoultian behavior, the condition for phase equilibrium between the two is
\frac{(X_{MnO})[X_{Fe}]}{(X_{FeO})[X_{Mn}]}=186 (iv)
or
\frac{[X_{Fe}]}{[X_{Mn}]}=186 \frac{(X_{FeO})}{(X_{MnO})}A series of tie-lines joining the compositions of equilibrated metal and oxide solutions is shown in Figure 13.2. Consider the metallic alloy of composition X_{Fe}=0.5 . Equation (iv) gives
1=186\frac{(X_{FeO})}{1-(X_{FeO})}or
(X_{FeO}) =0.00535and thus, when a metallic alloy of composition X_{Fe}=0.5 is equilibrated with an oxide solution, the composition of the latter is X_{FeO}=0.00535 . Consider now the influence of the partial pressure of oxygen in the gaseous atmosphere:
\Delta G^\circ _{(i)}=-RT\ln K_{(i)}=-RT\ln \frac{(a_{MnO})}{[a_{Mn}]p^{1/2}_{O_2}}Thus, since
\Delta G^\circ _{(i),2073 K}=-228,900 J
K^\circ _{(i),2073K}=5.856\times 10^{5}=\frac{(a_{MnO})}{[a_{Mn}]p^{1/2}_{O_2}}or, since the metallic and oxide solutions are ideal,
\frac{(X_{MnO})}{[X_{Mn}]} =5.856\times 10^{5}p^{1/2}_{O_2} (v)
Similarly,
\Delta G^\circ _{(ii)}=-RT\ln \frac{(a_{FeO})}{[a_{Fe}]p^{1/2}_{O_2}}and since
\Delta G^\circ _{(ii),2073 K}=-138,800 J
then
K _{(ii),2073 K}=3143=\frac{(a_{FeO})}{[a_{Fe}]p^{1/2}_{O_2}}or
\frac{(a_{FeO})}{[a_{Fe}]} =3143p^{1/2}_{O_2}The ideal Raoultian behavior of the two solutions allows Equation (vi) to be written as
\frac{(X_{FeO})}{[X_{Fe}]} =3143 p^{1/2}_{O_2}Consider the equilibrium between the metallic alloy of X_{Fe}=0.5 and the oxide solution of X_{FeO}=0.00535 . From Equation (v),
p_{O_2}=\left\lgroup\frac{0.99465}{0.5}\times \frac{1}{5.856\times 10^{5}} \right\rgroup ^2=1.15\times 10^{-11} atm
and from Equation (vi)
p_{O_2}=\left\lgroup\frac{0.00535}{0.5}\times \frac{1}{3143} \right\rgroup ^2=1.15\times 10^{-11} atm
Thus, the tie-line connecting the compositions of the equilibrated metallic and oxide alloy is also the oxygen isobar in Figure 13.2. Thus, at any fixed oxygen pressure, the individual ratios
\frac{(a_{FeO})}{[a_{Fe}]}\left\lgroup=\frac{(X_{FeO})}{[X_{Fe}]} \right\rgroup and \frac{(a_{MnO})}{[a_{Mn}]}\left\lgroup=\frac{(X_{MnO})}{[X_{Mn}]} \right\rgroup
are fixed by Equations (v) and (vi), combination of which gives
\frac{(X_{MnO})[X_{Fe}]}{(X_{FeO})[X_{Mn}]}=\frac{5.856\times 10^{5}}{3143} =186in accordance with Equation (iv). Consider the oxidation of a finite quantity of a liquid metallic alloy of composition X_{Fe}=0.5 by an infinite oxygen-containing gaseous atmosphere in which the partial pressure of oxygen is slowly increased. From Figure 13.2, the metal phase is stable when the partial pressure of oxygen is less than 1.15\times 10^{-11} atm. At p_{O_2}=1.15\times 10^{-11} atm, the metallic alloy is in equilibrium with an oxide solution of X_{FeO}=0.00535 . Increasing the partial pressure of oxygen to 1.79\times 10^{-11} atm moves the state of the system to the state b on the (v) isobar in Figure 13.2. In this state, a metallic alloy of X_{Fe}=0.6 (at a ) is in equilibrium with an oxide solution of X_{FeO}=0.0053 (at c ), and the relative quantities of the two phases are given by application of the lever rule to the tieline (v); that is, the fraction of the system occurring as the metallic alloy in state a is bc /ac , and the fraction occurring as the oxide solution in state c is ab /ac . Increasing the oxygen pressure to 7.0\times 10^{-11} atm moves the system to the state e on the (iii) isobar, where a metallic alloy of X_{Fe}=0.8 (at d ) is in equilibrium with an oxide solution of X_{FeO}=0.021 (at f ). The ratio of metallic alloy to oxide solution occurring is ef/de . Continued increase in the partial pressure of oxygen moves the state of the system upward along the broken line in Figure 13.2, during which the ratio of oxide-to-metal phase increases and the mole fraction of Fe in the metal phase and the mole fraction of FeO in the oxide phase increase. When the composition of the oxide reaches X_{FeO}=0.5 (at g ), the infinitesimal amount of equilibrium metal phase has the composition X_{Mn}=0.00535 and the oxygen pressure is 2.55\times 10^{-8} atm. The oxidation of Fe– Mn alloys at 2073 K occurs between the limits of oxygen pressure 2.92\times 10^{-12} atm for the equilibrium between pure Mn and pure MnO, and 1.02\times 10^{-7} atm for the equilibrium between pure Fe and pure FeO. The establishment of equilibrium (iii) requires that the ΔG – T lines for the oxidation of Fe and Mn intersect at 2073 K— that is, that \Delta G_{(iii),2073 K}=0. For any oxidation 2M+O_2=2MO, clockwise rotation of the ΔG – T line (e.g., the line ab in Figure 13.1) about its point of intersection with the T = 0 axis occurs when the ratio a_{MO}/a_{M} is decreased to a value less than unity, and, conversely, anticlockwise rotation of the line occurs when the ratio a_{MO}/a_{M} is increased to a value greater than unity. Also, since the equilibrium constant K is a function only of temperature, then, at any oxygen pressure p_{O_2} in the system M-MO_2-O_2 at the temperature T , the equilibrium ratio a_{MO}/a_{M} must be
\frac{a_{MO}}{a_M}=\frac{p^{1/2}_{O_2}}{p^{1/2}_{O_2(eq T pure M/pure MO)}}where {p_{O_2(eq T pure M/pure MO)}} is that unique oxygen pressure at the temperature T required for equilibrium between pure M and pure MO. Thus, for any oxygen pressure within the allowed limits, equilibrium (iii) occurs when Equations (v) and (vi) are satisfied, and, under these conditions, the Δ G – T lines for the oxidation of Fe and Mn intersect at 1800°C. Thus, as a consequence of the ability to vary a_M and a_MO , equilibrium (iii) can be established at any T and any p_{O_2} (within the aforementioned limits). This is in contrast to the situation illustrated in Figures 12.4 and 12.5, in which, if both metals and both oxides are present in their pure states, then an equilibrium such as (iii) can only be achieved at the single unique state (unique T and unique p_{O_2} ) at which the Ellingham lines for the two oxidation reactions intersect with one another. The restrictions on general multicomponent multiphase equilibria are discussed in Section 13.4.
