In view of the introduction of interaction parameters, the example of the Si– O equilibrium in liquid Fe, discussed in Section 13.3, can now be reexamined. In this example, it was determined that, for
Si_{(1 wt\% in Fe)}+O_{2(g)}=SiO_{2(s)}
\Delta G^\circ =-833,400+229.5T J
For
½O_{2(g)}=\left[O\right]_{(1 wt\% in Fe)}
\Delta G^\circ =-111,300-6.41T J (i)
and thus, for
Si_{(1 wt\% in Fe))}+2O_{(1 wt\% in Fe))}=SiO_{2(s)}
\Delta G^\circ =-610,800+242.32T J (ii)
From Equation (i) at 1600°C,
\frac{h_{O(1 wt\%)}}{p^{1/2}_{o_2}}=2.746\times 10^{3} (iii)
and, from Equation (ii) at 1600°C,
\frac{a_{SiO_2}}{h_{Si(1 wt\%)}h^{2}_{O(1 wt\%)}}=2.380\times 10^{4} (iv)
Thus, with p_{O_2}=5.57\times 10^{-12} atm and a_{SiO_2}=1 ,Equation (iii) gives
h_{O(1 wt\%)}=6.48\times 10^3 (v)
and Equation (iv) gives
h^{2}_{O(1 wt\%)}h_{Si(1 wt\%)}=4.0\times 10^{-5} (vi)
Division of Equation (vi) by h^{2}_{O(1 wt\%)} from Equation (v) gives
h_{Si(1 wt\%)} =1 (vii)
In the previous treatment the assumption that Si obeys Henry’ s law leads to the conclusion that
h_{Si(1 wt\%)} =wt\%Si=1
At 1600° C, from Table 13.1,
e^{Si}_{O}=-0.14 e^{O}_{O}=-0.2
e^{O}_{Si}=-0.25 e^{Si}_{Si}=-0.32
Thus, from Equation (v),
\log fo+\log \left[wt\% O\right]=\log (6.48\times 10^{-3})
or
-0.2\times \left[wt\% O\right]-0.14\times \left[wt\% Si\right]+\log \left[wt\% O\right]=-2.188 (viii)
and from Equation (vii),
\log f_{Si}+\log \left[wt\% Si\right]=\log (1)
or
0.32\left[wt\% Si\right]-0.25 \left[wt\% O\right]+\log \left[wt\% Si\right] =0 (ix)
Computer solution of Equations (viii) and (ix) gives
\left[wt\% Si\right]=0.631 and \left[wt\% O\right]=0.00798
In the example in Section 13.3, in which the effect of dissolved oxygen was ignored and it was assumed that f_{Si}=1 , the equilibrium weight percentage of Si in iron when a_{SiO_2}=1 and p_{O_2}=5.57\times 10^{-12} atm was 1.0. It is of interest to determine which of the two initial assumptions, that (1) e^{Si}_{Si}=0 and (2) e^{\circ }_{Si}=e^{Si}_{O}=0 , contributes more to the error in the initial calculation. Use e^{Si}_{Si}=0.32 and assume that e^{O}_{Si} and e^{Si}_{O} are zero. From Equation (ix):
0.32\left[wt\% Si\right]+\log [wt\% Si]=0
which gives [wt% Si] = 0.629, and from Equation (viii),
-0.20\times [wt\% O]-0.24\times 0.629+\log [wt\% O]=-2.188
which gives [wt% O] = 0.00797. The error introduced by ignoring the interaction between Si and O in solution in Fe is thus seen to be negligible in comparison with that introduced by assuming that Si obeys Henry’ s law over some initial range of composition.
| table 13.1 Interaction Coefficients for Dilute Solutions of elements Dissolved in Liquid Iron at 1600° C | ||||||||||||
| element( i) | element(j) | |||||||||||
| Al | C | Co | Cr | H | Mn | N | Ni | O | P | S | Si | |
| Al | 4.8 | 11 | ــــ | ــــ | (34) | ــــ | (0.5) | ــــ | -160 | ــــ | 4.9 | 6 |
| C | (4.8) | 22 | 1.2 | -2.4 | (72) | ــــ | (11.1) | 1.2 | (-9.7) | ــــ | 9 | 10 |
| Co | ــــ | (6) | ــــ | ــــ | (11) | ــــ | (4.7) | ــــ | (2.6) | ــــ | ــــ | ــــ |
| Cr | ــــ | (-10) | ــــ | ــــ | (-11) | ــــ | (-16.6) | ــــ | (-13) | ــــ | (-3.55) | ــــ |
| H | 1.3 | 6 | 0.18 | -0.22 | 0 | -0.14 | ــــ | 0 | ــــ | 1.1 | 0.8 | 2.7 |
| Mn | ــــ | ــــ | ــــ | ــــ | (-7.7) | ــــ | (-7.8) | ــــ | (0) | ــــ | (-4.3) | (0) |
| N | 0.3 | 13 | 1.1 | -4.5 | ــــ | -2 | 0 | 1 | 5 | 5.1 | 1.3 | 4.7 |
| Ni | ــــ | (5.9) | ــــ | ــــ | (0) | ــــ | (4.2) | 0 | (2.1) | ــــ | (0) | (1.0) |
| O | -94 | -13 | 0.7 | -4.1 | ــــ | 0 | (5.7) | 0.6 | -20 | 7 | -9.1 | -14 |
| P | ــــ | ــــ | ــــ | ــــ | (34) | ــــ | (11.3) | ــــ | (13.5) | ــــ | (4.3) | (9.5) |
| S | 5.8 | (24) | ــــ | -2.2 | (26) | -2.5 | (3.0) | 0 | (-18) | 4.5 | -2.8 | 6.6 |
| Si | (6.3) | 24 | ــــ | ــــ | (76) | 0 | (9.3) | 0.5 | (-25) | 8.6 | (5.7) | 32 |
| some interaction coefficients e_i ^j \times 10^2 for dilute solutions of elements dissolved in liquid iron at 1600° C. values in parentheses are calculated from e_i ^j = (MW_{i}/MW_{j}) e_{j}^{i}.(from J. f. elliott, m. Gleiser, and v. ramakrishna, Thermochemistry for Steelmaking , vol. 2, addison-wesley, reading, ma, 1963.) | ||||||||||||