Question 13.14.2: Determine the conditions under which an Fe– Cr– O melt is in...

At 1600°C, Fe– Cr melts exhibit Raoultian ideality, and the molar heat of melting of Cr, at its equilibrium melting temperature of 2173 K, is 21,000 J. Thus, for Cr_{(s)}=Cr_{(l)} ,

\Delta G^\circ _{m}=\Delta H^\circ _{m}-T\frac{\Delta H^\circ _{m}}{T_{m}}=21,000-9.66T   J

and for Cr_{(l)}=\left[Cr\right]_{(1  wt\%  in  Fe)} ,

\Delta G=RT\ln \frac{55.85}{100\times 52.01}=-37.70T   J

Therefore, for Cr_{(s)}- \left[Cr\right]_{(1  wt\%  in  Fe)} ,

\Delta G^\circ _{(iii)}=21,000-47.36T   J              (iii)

The standard Gibbs free energy change for the reaction

2\left[Cr\right]_{(1wt\%)} +3\left[O\right]_{(1wt\%)}=Cr_2O_{3(s)}                  (iv)

is thus

= -829,090 + 372.13T  J

or, at 1873 K,

\log \frac{h^2_{Cr(1  wt\%)}.h^3_{O(1  wt\%)}}{a_{Cr_2O_3}} =-3.68                       (v)

Saturation of the melt with solid Cr_2O_3  occurs at {a_{Cr_2O_3}}=1   , and, if the interactions between Cr and O in solution are ignored, and it is assumed that oxygen obeys Henry’ s law, Equation (v) can be written as

\log \left[wt\%  Cr\right] =-1.5\log \left[wt\%  O\right] -1.84                      (vi)

which is the variation of [wt% Cr] with [wt% O] in liquid iron required for equilibrium with solid Cr_2O_3  at 1600°C. Equation (vi) is drawn as line (vi) in Figure 13.35.

For

\Delta G^\circ _{(vii)}=-1,409,420+318.07T  J               (vii)

and thus, for the reaction

= -1,007,140 + 436.27T  J               (viii)

or, at 1873 K,

\log \frac{a_{Fe}.h^2_{Cr(1wt\%)}.h^4_{O{(1wt\%)}} }{a_{FeO.Cr_2O_3}} =-5.30              (ix)

Saturation of the melt with FeO.Cr_2O_3   occurs at a_{FeO.Cr_2O_3} =1  and, with the same assumptions as before, and a_{Fe}=X_{Fe}=1-X_{Cr},  the variation of [wt% Cr] with [wt% O] required for equilibrium with solid FeO.Cr_2O_3   at 1600°C is

\log (1-X_{Cr})+2\log \left[wt\%  Cr\right]+4\log \left[wt\%  O\right]=-5.30                 (x)

In solutions sufficiently dilute that X_{Fe}\sim 1 ,  Equation (x) can be simplified as

\log \left[wt\%  Cr\right]=-2\log \left[wt\%  O\right] -2.65                (xi)

Equation (xi) is drawn as line (x) in Figure 13.35. Lines (vi) and (x) intersect at the point A , log [wt% Cr] = 0.59, log [wt% O] = – 1.62 (wt% O = 0.024, wt% Cr = 3.89), which is the composition of the melt which is simultaneously saturated with solid Cr_2O_3   and  FeO.Cr_2O_3 .  From the phase rule, equilibrium in a three-component system (Fe– Cr– O) among four phases (liquid Fe– Cr– O, solid Cr_2O_3 , solid FeO.Cr_2O_3 ,  and a gas phase) has one degree of freedom, which, in the present case, has been used by specifying the temperature to be 1873 K. Thus, the activities of Fe, Cr, and O are uniquely fixed, and hence [wt% Cr] and [wt% O] are uniquely fixed. The equilibrium oxygen pressure in the gas phase is obtained from Equation (ii) as

which, with [wt% O] = 0.024, gives p_{O_2(eq)}=8.96\times 10^{-11}  atm .  The positions of the lines in Figure 13.35 are such that, in melts of [wt% Cr] > 3.89, Cr_2O_3   is the stable phase in equilibrium with saturated melts along the line AB and, in melts in which [wt% Cr] < 3.89, FeO.Cr_2O_3   is the stable phase in equilibrium with saturated meltsalong the line AC . Alternatively Cr_2O_3   is the stable phase in equilibrium with saturated melts of [wt% O] < 0.024, and FeO.Cr_2O_3   is the stable phase in equilibrium with saturated melts of [wt% O] > 0.024. Consider a melt in which log [wt% Cr] = 1.5. From Figure 13.35, or Equation (vi), the oxygen content at this chromium level required for equilibrium with Cr_2O_3   (at the point B in Figure 13.35) is 5.93\times 10^{-3}  wt%, or log [wt% O] = – 2.25. From Equation (v), the activity of Cr_2O_3   in this melt with respect to solid Cr_2O_3   is unity, and hence the melt is saturated with respect to solid Cr_2O_3 .  However, from Equation (ix), in the same melt (i.e., X_{Fe}=0.668,  [wt% Cr] = 31.6, [wt% O] = 0.00593), the activity of FeO.Cr_2O_3   with respect to solid FeO.Cr_2O_3   is only 0.2. Thus, the melt is saturated with respect to Cr_2O_3   and is undersaturated with respect to FeO.Cr_2O_3 .   Moving along the line BA from B toward A,   a_{Cr_2O_3}=1  , and a_{FeO.Cr_2O_3}=1  increases from 0.2 at B to unity at A in the doubly saturated melt. Consider a melt in which log [wt% Cr] = – 0.5. From Figure 13.35, the oxygen content required for saturation with FeO.Cr_2O_3   is 0.084 wt% (log [wt% O] = – 1.075 at the point C in Figure 13.35). From Equation (ix), the activity of FeO.Cr_2O_3   in this melt is unity. However, from Equation (v), the activity of Cr_2O_3   in the melt, with respect to solid Cr_2O_3 ,  is only 0.285. Thus, this melt is saturated with FeO.Cr_2O_3   and is undersaturated with Cr_2O_3   On moving along the line CA from C toward A ,a_{FeO.Cr_2O_3}  is unity and a_{Cr_2O_3}  increases from 0.285 at C to unity at A . If the various solute– solute interactions had been considered, Equation (v), with a_{Cr_2O_3}=1, would be written as

or

or

With

e^{0}_{Cr}=O         e^{O}_{O}=-0.2     e^{Cr}_{O}=-0.041     and  e^{O}_{Cr}=-0.13

this gives

-0.43\left[wt\%  O\right]+0.0615\left[wt\%  Cr\right] +\log \left[wt\%Cr\right]+1.5\log \left[wt\%  O\right] =-1.84               (xii)

which is drawn as line (xii) in Figure 13.36.

Similarly, with a_{FeO.Cr_2O_3} =1   , Equation (ix) would be written as

or

or

which is drawn as line (xiii) in Figure 13.36. Lines (xii) and (xiii) intersect at log [wt% Cr] = 0.615, log [wt% O] = – 1.455 ([wt% Cr] = 4.12, [wt% O] = 0.035). When the interactions among the solute were ignored, the point of intersection, A , was obtained as [wt% Cr] = 3.89, [wt% O] = 0.024.