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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.
Answer (1)
Calculate the moles of iron produced: n ( F e ) = 55.845 440 × 1 0 6 ≈ 7878950.67 mol.
Determine the moles of iron(III) oxide required: n ( F e 2 O 3 ) = 2 1 n ( F e ) ≈ 3939475.33 mol.
Calculate the required moles of carbon: n ( C ) = 4.048 × n ( F e 2 O 3 ) ≈ 15947755.57 mol.
Compute the mass of carbon needed and round to two significant digits: ma ss ( C ) = n ( C ) × 12.011 ≈ 190 tons. The final answer is 190
Explanation
Problem Analysis We are given the reaction for the reduction of iron(III) oxide to pure iron and the combustion of coke to carbon dioxide. We are also given the Gibbs free energies of formation for C O 2 and F e 2 O 3 . The goal is to find the minimum mass of coke needed to produce 440 tons of pure iron.
Moles of Iron First, calculate the number of moles of iron to be produced: n ( F e ) = m o l a r _ ma ss ( F e ) ma ss ( F e ) = 55.845 g / m o l 440 × 1 0 6 g = 7878950.667 m o l
Moles of Iron(III) Oxide Next, calculate the number of moles of F e 2 O 3 required based on the stoichiometry of the reaction 2 F e 2 O 3 ( s ) → 4 F e ( s ) + 3 O 2 ( g ) :
n ( F e 2 O 3 ) = 2 1 n ( F e ) = 2 1 ( 7878950.667 m o l ) = 3939475.3335 m o l
Gibbs Free Energy Change for Iron Oxide Reduction Calculate the Gibbs free energy change for the reaction 2 F e 2 O 3 ( s ) → 4 F e ( s ) + 3 O 2 ( g ) . Since the Gibbs free energy of formation of elements in their standard state is zero, Δ G f ( F e ) = 0 and Δ G f ( O 2 ) = 0 . Therefore, 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 × 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 reaction C ( s ) + O 2 ( g ) → C O 2 ( g ) . Since the Gibbs free energy of formation of elements in their standard state is zero, Δ G f ( C ) = 0 and Δ G f ( O 2 ) = 0 . Therefore, Δ G 2 = Δ G f ( C O 2 ) = − 415 k J / m o l
Overall Reaction and Stoichiometry To determine the minimum amount of coke needed, we need to consider the overall reaction. We need to reverse the iron oxide reduction reaction and divide by 2 to get 2 F e ( s ) + 2 3 O 2 ( g ) → F e 2 O 3 ( s ) . The Gibbs free energy change for this reaction is − 2 1 Δ G 1 = − 840 k J / m o l . Now, we need to couple this reaction with the coke combustion reaction such that the overall reaction is spontaneous. Let's consider the reaction: F e 2 O 3 ( s ) + x C ( s ) → 2 F e ( s ) + x C O 2 ( g ) . The Gibbs free energy change for this reaction is Δ G = Δ G 1 + x Δ G 2 . For the reaction to be spontaneous, Δ G < 0 . Thus, 1680 + x ( − 415 ) < 0 , which means \frac{1680}{415} \approx 4.048"> x > 415 1680 ≈ 4.048 . This means that for every mole of F e 2 O 3 , we need 4.048 moles of C.
Moles of Carbon Required Calculate the number of moles of carbon required: n ( C ) = x × n ( F e 2 O 3 ) = 4.048 × 3939475.3335 m o l = 15947755.567 m o l
Mass of Carbon Required Calculate the mass of carbon required: ma ss ( C ) = n ( C ) × m o l a r _ ma ss ( C ) = 15947755.567 m o l × 12.011 g / m o l = 191548492.115 g
Mass of Carbon in Tons Convert the mass of carbon to tons: ma ss ( C ) = 1 0 6 g / t o n 191548492.115 g = 191.548 t o n s
Final Answer Round the result to 2 significant digits: 191.548 ≈ 190 tons.
Examples
In steelmaking, determining the precise amount of coke needed to reduce iron ore is crucial for efficient production. This calculation ensures that the process is both economically viable and environmentally sustainable by minimizing waste and energy consumption. By understanding the stoichiometry and thermodynamics of the reactions involved, engineers can optimize the process to produce high-quality steel with minimal resource expenditure. This principle extends to other industrial processes where controlling reaction conditions and material inputs is essential for achieving desired outcomes.
