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The reduction of Iron(III) oxide ($Fe_2 O_3$) to pure iron during the first step of steelmaking,
$2 Fe_2 O_3(s) \rightarrow 4 Fe(s)+3 O_2(g)$
Is driven by the high-temperature combustion of coke, a purified form of coal:
$C(s)+O_2(g) \rightarrow CO_2(g)$
Suppose at the temperature of a blast furnace the Gibbs free energies of formation $\Delta G_f$ of $CO_2$ and $Fe_2 O_3$ are $-415. kJ/mol$ and $-840. kJ/mol$, respectively. Calculate the minimum mass of coke needed to produce $440. t$ of pure iron. (One metric ton, symbol t, equals 1000 kg.)
Round your answer to 2 significant digits.
30. $\times 10^5 Ikg$
Answer (1)
Calculate the moles of iron produced: n ( F e ) = 0.055845 k g / m o l 440 × 1 0 3 k g = 7878956.03 m o l .
Determine the moles of F e 2 O 3 required: n ( F e 2 O 3 ) = 2 1 n ( F e ) = 3939478.015 m o l .
Calculate the moles of carbon required: n ( C ) = 4.04819 × n ( F e 2 O 3 ) = 15942077.56 m o l .
Calculate the mass of carbon required: ma ss ( C ) = n ( C ) × 0.012011 k g / m o l = 191470.60 k g ≈ 1.9 × 1 0 5 k g . The final answer is 1.9 × 1 0 5 k g .
Explanation
Problem Setup and Data We are given the reaction for the reduction of Iron(III) oxide ( F e 2 O 3 ) to pure iron: 2 F e 2 O 3 ( s ) i g h t ha r p oo n u p 4 F e ( s ) + 3 O 2 ( g ) and the combustion of coke: C ( s ) + O 2 ( g ) i g h t ha r p oo n u pC O 2 ( g ) We are also given the Gibbs free energies of formation for C O 2 and F e 2 O 3 : Δ G f ( C O 2 ) = − 415 k J / m o l and Δ G f ( F e 2 O 3 ) = − 840 k J / m o l . Our goal is to calculate the minimum mass of coke needed to produce 440 t of pure iron.
Moles of Iron Produced First, calculate the number of moles of iron produced. We know that we want to produce 440 t of iron, which is equal to 440 × 1 0 3 k g . The molar mass of iron is approximately 55.845 g / m o l or 0.055845 k g / m o l . Therefore, the number of moles of iron produced is: n ( F e ) = 0.055845 k g / m o l 440 × 1 0 3 k g = 7878956.03 m o l
Moles of Iron(III) Oxide Required Next, determine the number of moles of F e 2 O 3 required to produce this amount of iron. From the stoichiometry of the first reaction, 2 moles of F e 2 O 3 produce 4 moles of F e . Therefore, the number of moles of F e 2 O 3 needed is: n ( F e 2 O 3 ) = 2 1 n ( F e ) = 2 1 × 7878956.03 m o l = 3939478.015 m o l
Gibbs Free Energy Change for Iron Oxide Reduction Now, calculate the Gibbs free energy change for the first reaction. Since the Gibbs free energy of formation of elements in their standard state is zero, we have: Δ G 1 = 4 × Δ G f ( F e ) + 3 × Δ G f ( O 2 ) − 2 × Δ G f ( F e 2 O 3 ) = − 2 × ( − 840 k J / m o l ) = 1680 k J / m o l
Gibbs Free Energy Change for Coke Combustion Calculate the Gibbs free energy change for the second reaction: Δ G 2 = Δ G f ( C O 2 ) − Δ G f ( C ) − Δ G f ( O 2 ) = − 415 k J / m o l
Moles of Carbon Required To determine the minimum mass of coke needed, we need to find the number of moles of carbon required to make the overall reaction spontaneous. The overall reaction can be written as: 2 F e 2 O 3 ( s ) + n C ( s ) + n O 2 ( g ) → 4 F e ( s ) + 3 O 2 ( g ) + n C O 2 ( g ) The overall Gibbs free energy change is Δ G = Δ G 1 + n Δ G 2 . For the reaction to occur spontaneously, Δ G < 0 . Therefore, Δ G 1 + n Δ G 2 < 0 , which means: -\frac{\Delta G_1}{\Delta G_2} = -\frac{1680}{-415} = 4.04819"> n > − Δ G 2 Δ G 1 = − − 415 1680 = 4.04819 So, for each mole of F e 2 O 3 reduced, we need at least 4.04819 moles of carbon. Therefore, the total number of moles of carbon required is: n ( C ) = 4.04819 × n ( F e 2 O 3 ) = 4.04819 × 3939478.015 m o l = 15942077.56 m o l
Mass of Carbon Required Finally, calculate the mass of carbon required. The molar mass of carbon is approximately 12.011 g / m o l or 0.012011 k g / m o l . Therefore, the mass of carbon required is: ma ss ( C ) = n ( C ) × M ( C ) = 15942077.56 m o l × 0.012011 k g / m o l = 191470.60 k g Rounding to two significant figures, we get 1.9 × 1 0 5 k g or 190000 k g .
Final Answer Therefore, the minimum mass of coke needed to produce 440 t of pure iron is approximately 1.9 × 1 0 5 k g .
Examples
The principles used in this problem are crucial in the steelmaking industry. Understanding the thermodynamics of reactions, particularly the Gibbs free energy, allows engineers to optimize the process of reducing iron ore to pure iron. By carefully controlling the amount of coke used, they can minimize costs and environmental impact while maximizing the efficiency of steel production. This also applies to other metallurgical processes where the reduction of metal oxides is necessary.
