Iron, a large component of steel, is obtained by reducing iron(III) oxide (present in hematite ore) with hydrogen in a blast furnace. Steam is a byproduct of the reaction. Calculate ΔG° at 230°C for the reduction of one mole of Fe2O3.

Question

Iron, a large component of steel, is obtained by reducing iron(III) oxide (present in hematite ore) with hydrogen in a blast furnace. Steam is a byproduct of the reaction. Calculate ΔG° at 230°C for the reduction of one mole of [latex]Fe_{2}O_{3}[/latex].

ANALYSIS
n [latex]Fe_{2}O_{3}[/latex] (one mole); temperature (230°C)	Information given:
Table 8.3 ([latex]ΔH°_{f}[/latex] values) Table 16.1 (S° values)	Information implied
ΔG°	Asked for

Table 8.3 Standard Enthalpies of Formation at 25°C (kJ/mol) of Compounds at 1 atm, Aqueous Ions at 1 M

Compounds
-365.6	[latex]NH_{4}NO_{3}(s)[/latex]	-241.8	[latex]H_{2}O(g)[/latex]	-795.8	[latex]CaCl_{2}(s)[/latex]	-100.4	AgBr(s)
+90.2	NO(g)	-285.8	[latex]H_{2}O(l)[/latex]	-1206.9	[latex]CaCO_{3}(s)[/latex]	-127.1	AgCl(s)
+33.2	[latex]NO_{2}(g)[/latex]	-187.8	[latex]H_{2}O_{2}(l)[/latex]	-635.1	CaO(s)	-61.8	Agl(s)
+9.2	[latex]N_{2}O_{4}(g)[/latex]	-20.6	[latex]H_{2}S(g)[/latex]	-986.1	[latex]Ca(OH)_{2}(s)[/latex]	-124.4	[latex]AgNO_{3}(s)[/latex]
-411.2	NaCl(s)	-814.0	[latex]H_{2}SO_{4}(l)[/latex]	-1434.1	[latex]CaSO_{4}(s)[/latex]	-31.0	[latex]Ag_{2}O(s)[/latex]
-573.6	NaF(s)	-90.8	HgO(s)	-391.5	[latex]CdCl_{2}(s)[/latex]	-1675.7	[latex]Al_{2}O_{3}(s)[/latex]
-425.6	NaOH(s)	-393.8	KBr(s)	-258.2	CdO(s)	-858.6	[latex]BaCl_{2}(s)[/latex]
-239.7	NiO(s)	-436.7	KCl(s)	-1139.7	[latex]Cr_{2}O_{3}(s)[/latex]	-1216.3	[latex]BaCO_{3}(s)[/latex]
-278.7	[latex]PbBr_{2}(s)[/latex]	-397.7	[latex]KClO_{3}(s)[/latex]	-157.3	CuO(s)	-553.5	BaO(s)
-359.4	[latex]PbCl_{2}(s)[/latex]	-432.8	[latex]KClO_{4}(s)[/latex]	-168.6	[latex]Cu_{2}O(s)[/latex]	-1473.2	[latex]BaSO_{4}(s)[/latex]
-219.0	PbO(s)	-494.6	[latex]KNO_{3}(s)[/latex]	-53.1	CuS(s)	-135.4	[latex]CCl_{4}(l)[/latex]
-277.4	[latex]PbO_{2}(s)[/latex]	-641.3	[latex]MgCl_{2}(s)[/latex]	-79.5	[latex]Cu_{2}S(s)[/latex]	-134.5	[latex]CHCl_{3}(l)[/latex]
-287.0	[latex]PCl_{3}(g)[/latex]	-1095.8	[latex]MgCO_{3}(s)[/latex]	-771.4	[latex]CuSO_{4}(s)[/latex]	-74.8	[latex]CH_{4}(g)[/latex]
-374.9	[latex]PCl_{5}(g)[/latex]	-601.7	MgO(s)	-823.0	[latex]Fe(OH)_{3}(s)[/latex]	+226.7	[latex]C_{2}H_{2}(g)[/latex]
-910.9	[latex]SiO_{2}(s)[/latex]	-924.5	[latex]Mg(OH)_{2}(s)[/latex]	-824.2	[latex]Fe_{2}O_{3}(s)[/latex]	+52.3	[latex]C_{2}H_{4}(g)[/latex]
-580.7	[latex]SnO_{2}(s)[/latex]	-1284.9	[latex]MgSO_{4}(s)[/latex]	-1118.4	[latex]Fe_{3}O_{4}(s)[/latex]	-84.7	[latex]C_{2}H_{6}(g)[/latex]
-296.8	[latex]SO_{2}(g)[/latex]	-385.2	MnO(s)	-36.4	HBr(g)	-103.8	[latex]C_{2}H_{8}(g)[/latex]
-395.7	[latex]SO_{3}(g)[/latex]	-520.0	[latex]MnO_{2}(s)[/latex]	-92.3	HCl(g)	-238.7	[latex]CH_{3}OH(l)[/latex]
-208.0	[latex]ZnI_{2}(s)[/latex]	-46.1	[latex]NH_{3}(g)[/latex]	-271.1	HF(g)	-277.7	[latex]C_{2}H_{5}OH(l)[/latex]
-348.3	ZnO(s)	+50.6	[latex]N_{2}H_{4}(l)[/latex]	+26.5	HI(g)	-110.5	CO(g)
-206.0	ZnS(s)	-314.4	[latex]NH_{4}Cl(s)[/latex]	-174.1	[latex]HNO_{3}(l)[/latex]	-393.5	[latex]CO_{2}(g)[/latex]
Anions				Cations
-1292.1	[latex]HPO_{4}^{2-}(aq)[/latex]	-121.6	[latex]Br^{-}(aq)[/latex]	+171.1	[latex]Hg^{2+}(aq)[/latex]	+105.6	[latex]Ag^{+}(aq)[/latex]
-887.3	[latex]HSO_{4}^{-}(aq)[/latex]	-677.1	[latex]CO_{3}^{2-}(aq)[/latex]	-252.4	[latex]K^{+}(aq)[/latex]	-531.0	[latex]Al^{3+}(aq)[/latex]
-55.2	[latex]I^{-}(aq)[/latex]	-167.2	[latex]Cl^{-}(aq)[/latex]	-466.8	[latex]Mg^{2+}(aq)[/latex]	-537.6	[latex]Ba^{2+}(aq)[/latex]
-541.4	[latex]MnO_{4}^{-}(aq)[/latex]	-104.0	[latex]ClO_{3}^{-}(aq)[/latex]	-220.8	[latex]Mn^{2+}(aq)[/latex]	-542.8	[latex]Ca^{2+}(aq)[/latex]
-104.6	[latex]NO_{2}^{-}(aq)[/latex]	-129.3	[latex]ClO_{4}^{-}(aq)[/latex]	-240.1	[latex]Na^{+}(aq)[/latex]	-75.9	[latex]Cd^{2+}(aq)[/latex]
-205.0	[latex]NO_{3}^{-}(aq)[/latex]	-881.2	[latex]CrO_{4}^{2-}(aq)[/latex]	-132.5	[latex]NH_{4}^{+}(aq)[/latex]	+71.7	[latex]Cu^{+}(aq)[/latex]
-230.0	[latex]OH^{-}(aq)[/latex]	-1490.3	[latex]Cr_{2}O_{7}^{2-}(aq)[/latex]	-54.0	[latex]Ni^{2+}(aq)[/latex]	+64.8	[latex]Cu^{2+}(aq)[/latex]
-1277.4	[latex]PO_{4}^{3-}(aq)[/latex]	-332.6	[latex]F^{-}(aq)[/latex]	-1.7	[latex]Pb^{2+}(aq)[/latex]	-89.1	[latex]Fe^{2+}(aq)[/latex]
+33.1	[latex]S^{2-}(aq)[/latex]	-692.0	[latex]HCO_{3}^{-}(aq)[/latex]	-8.8	[latex]Sn^{2+}(aq)[/latex]	-48.5	[latex]Fe^{3+}(aq)[/latex]
-909.3	[latex]SO_{4}^{2-}(aq)[/latex]	-1296.3	[latex]H_{2}PO_{4}^{-}(aq)[/latex]	-153.9	[latex]Zn^{2+}(aq)[/latex]	0.0	[latex]H^{+}(aq)[/latex]

Table 16.1 Standard Entropies at 25°C (J/mol·K) of Elements and Compounds at 1 atm, Aqueous Ions at 1 M

Elements

Ag(s) 42.6 [latex]Cl_{2}(g)[/latex] 223.0 [latex]I_{2}(s)[/latex] 116.1 [latex]O_{2}(g)[/latex] 205.0
Al(s) 28.3 Cr(s) 23.8 K(s) 64.2 Pb(s) 64.8
Ba(s) 62.8 Cu(s) 33.2 Mg(s) 32.7 [latex]P_{4}(s)[/latex] 164.4
[latex]Br_{2}(l)[/latex] 152.2 [latex]F_{2}(g)[/latex] 202.7 Mn(s) 32.0 S(s) 31.8
C(s) 5.7 Fe(s) 27.3 [latex]N_{2}(g)[/latex] 191.5 Si(s) 18.8
Ca(s) 41.4 [latex]H_{2}(g)[/latex] 130.6 Na(s) 51.2 Sn(s) 51.6
Cd(s) 51.8 Hg(l) 76.0 Ni(s) 29.9 Zn(s) 41.6

Compounds

AgBr(s) 107.1 [latex]CaCl_{2}(s)[/latex] 104.6 [latex]H_{2}O(g)[/latex] 188.7 [latex]NH_{4}NO_{3}(s)[/latex] 151.1
AgCl(s) 96.2 [latex]CaCO_{3}(s)[/latex] 92.9 [latex]H_{2}O(l)[/latex] 69.9 NO(g) 210.7
AgI(s) 115.5 CaO(s) 39.8 [latex]H_{2}O_{2}(l)[/latex] 109.6 [latex]NO_{2}(g)[/latex] 240.0
[latex]AgNO_{3}(s)[/latex] 140.9 [latex]Ca(OH)_{2}(s)[/latex] 83.4 [latex]H_{2}S(g)[/latex] 205.7 [latex]N_{2}O_{4}(g)[/latex] 304.2
[latex]Ag_{2}O(s)[/latex] 121.3 [latex]CaSO_{4}(s)[/latex] 106.7 [latex]H_{2}SO_{4}(l)[/latex] 156.9 NaCl(s) 72.1
[latex]Al_{2}O_{3}(s)[/latex] 50.9 [latex]CdCl_{2}(s)[/latex] 115.3 HgO(s) 70.3 NaF(s) 51.5
[latex]BaCl_{2}(s)[/latex] 123.7 CdO(s) 54.8 KBr(s) 95.9 NaOH(s) 64.5
[latex]BaCO_{3}(s)[/latex] 112.1 [latex]Cr_{2}O_{3}(s)[/latex] 81.2 KCl(s) 82.6 NiO(s) 38.0
BaO(s) 70.4 CuO(s) 42.6 [latex]KClO_{3}(s)[/latex] 143.1 [latex]PbBr_{2}(s)[/latex] 136.0
[latex]BaSO_{4}(s)[/latex] 132.2 [latex]Cu_{2}O(s)[/latex] 93.1 [latex]KClO_{4}(s)[/latex] 151.0 [latex]PbCl_{2}(s)[/latex] 136.0
[latex]CCl_{4}(l)[/latex] 216.4 CuS(s) 66.5 [latex]KNO_{3}(s)[/latex] 133.0 PbO(s) 66.5
[latex]CHCl_{3}(l)[/latex] 201.7 [latex]Cu_{2}S(s)[/latex] 120.9 [latex]MgCl_{2}(s)[/latex] 89.6 [latex]PbO_{2}(s)[/latex] 68.6
[latex]CH_{4}(g)[/latex] 186.2 [latex]CuSO_{4}(s)[/latex] 107.6 [latex]MgCO_{3}(s)[/latex] 65.7 [latex]PCl_{3}(g)[/latex] 311.7
[latex]C_{2}H_{2}(g)[/latex] 200.8 [latex]Fe(OH)_{3}(s)[/latex] 106.7 MgO(s) 26.9 [latex]PCl_{5}(g)[/latex] 364.5
[latex]C_{2}H_{4}(g)[/latex] 219.5 [latex]Fe_{2}O_{3}(s)[/latex] 87.4 [latex]Mg(OH)_{2}(s)[/latex] 63.2 [latex]SiO_{2}(s)[/latex] 41.8
[latex]C_{2}H_{6}(g)[/latex] 229.5 [latex]Fe_{3}O_{4}(s)[/latex] 146.4 [latex]MgSO_{4}(s)[/latex] 91.6 [latex]SnO_{2}(s)[/latex] 52.3
[latex]C_{3}H_{8}(g)[/latex] 269.9 HBr(g) 198.6 [latex]MnO(s)[/latex] 59.7 [latex]SO_{2}(g)[/latex] 248.1
[latex]CH_{3}OH(l)[/latex] 126.8 HCl(g) 186.8 [latex]MnO_{2}(s)[/latex] 53.0 [latex]SO_{3}(g)[/latex] 256.7
[latex]C_{2}H_{5}OH(l)[/latex] 160.7 HF(g) 173.7 [latex]NH_{3}(g)[/latex] 192.3 [latex]ZnI_{2}(s)[/latex] 161.1
CO(g) 197.6 HI(g) 206.5 [latex]N_{2}H_{4}(l)[/latex] 121.2 ZnO(s) 43.6
[latex]CO_{2}(g)[/latex] 213.6 [latex]HNO_{3}(l)[/latex] 155.6 [latex]NH_{4}Cl(s)[/latex] 94.6 ZnS(s) 57.7

Cations Anions

[latex]Ag^{+}(aq)[/latex] 72.7 [latex]Hg^{2+}(aq)[/latex] -32.2 [latex]Br^{-}(aq)[/latex] 82.4 [latex]HPO_{4}^{2-}(aq)[/latex] -33.5
[latex]Al^{3+}(aq)[/latex] -321.7 [latex]K^{+}(aq)[/latex] 102.5 [latex]CO_{3}^{2-}(aq)[/latex] -56.9 [latex]HSO_{4}^{-}(aq)[/latex] 131.8
[latex]Ba^{2+}(aq)[/latex] 9.6 [latex]Mg^{2+}(aq)[/latex] -138.1 [latex]Cl^{-}(aq)[/latex] 56.9 [latex]I^{-}(aq)[/latex] 111.3
[latex]Ca^{2+}(aq)[/latex] -53.1 [latex]Mn^{2+}(aq)[/latex] -73.6 [latex]ClO_{3}^{-}(aq)[/latex] 162.3 [latex]MnO_{4}^{-}(aq)[/latex] 191.2
[latex]Cd^{2+}(aq)[/latex] -73.2 [latex]Na^{+}(aq)[/latex] 59.0 [latex]ClO_{4}^{-}(aq)[/latex] 182.0 [latex]NO_{2}^{-}(aq)[/latex] 123.0
[latex]Cu^{+}(aq)[/latex] 40.6 [latex]NH_{4}^{+}(aq)[/latex] 113.4 [latex]CrO_{4}^{2-}(aq)[/latex] 50.2 [latex]NO_{3} ^{-}(aq)[/latex] 146.4
[latex]Cu^{2+}(aq)[/latex] -99.6 [latex]Ni^{2+}(aq)[/latex] -128.9 [latex]Cr_{2}O_{7}^{2-}(aq)[/latex] 261.9 [latex]OH^{-}(aq)[/latex] -10.8
[latex]Fe^{2+}(aq)[/latex] -137.7 [latex]Pb^{2+}(aq)[/latex] 10.5 [latex]F^{-}(aq)[/latex] -13.8 [latex]PO_{4}^{3-}(aq)[/latex] -222
[latex]Fe^{3+}(aq)[/latex] -315.9 [latex]Sn^{2+}(aq)[/latex] -17.4 [latex]HCO_{3}^{-}(aq)[/latex] 91.2 [latex]S^{2-}(aq)[/latex] -14.6
[latex]H^{+}(aq)[/latex] 0.0 [latex]Zn^{2+}(aq)[/latex] -112.1 [latex]H_{2}PO_{4}^{-}(aq)[/latex] 90.4 [latex]SO_{4}^{2-}(aq)[/latex] 20.1

STRATEGY

1. Write a balanced equation for the reaction.
2. Find Δ[latex]H°_{f}[/latex] values in Table 8.3 (or Appendix 1) and substitute into Equation 8.3 to obtain ΔH°.

[latex]q_{calorimeter} = C_{cal}[/latex] × Δt (8.3)
3. Find S° values in Table 16.1 (or Appendix 1) and substitute into Equation 16.1 to obtain ΔS°. (Remember to convert J/K to kJ/K.)
4. Change °C to K and substitute the values for ΔH° and ΔS° into the Gibbs-Helmholtz equation (Equation 16.2) to obtain ΔG°.

ΔG° = ΔH° - TΔS° (16.2)

Accepted Answer

[latex]Fe_{2}O_{3}(s) + 3H_{2}(g) → 2Fe(s) + 3H_{2}O(g)[/latex]	1. Equation
ΔH° = 3Δ[latex]H°_{f} H2O(g)[/latex] - Δ[latex]H°_{f} Fe_{2}O_{3}(s) = -725.4 kJ + 824.2 kJ = +98.8 kJ[/latex]	2. ΔH°
ΔS° = 3S° [latex]H_{2}O(g) + 2S° Fe(s) - 3S° H_{2}(g) - S° Fe_{2}O_{3}[/latex](s) = 566.1 J/K + 54.6 J/K - 391.8 J/K - 87.4 J/K = +141.5 J/K = +0.1415 kJ/K	3. ΔS°
ΔG° = ΔH° - TΔS° = 98.8 kJ - (273 + 230)K (0.1415 kJ/K) = +27.6 kJ	4. ΔG°

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