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Math  /  Data & Statistics

QuestionCalculate ΔGrxn for the following reaction at 298 K. Use the ΔGf∘ values in this table of thermodynamic properties.\text{Calculate } \Delta G_{\mathrm{rxn}} \text{ for the following reaction at 298 K. Use the } \Delta G_{\mathrm{f}}^{\circ} \text{ values in this table of thermodynamic properties.}
Cr2O3(s)+3CO(g)⟶2Cr(s)+3CO2(g)\mathrm{Cr}_{2} \mathrm{O}_{3}(\mathrm{s}) + 3 \mathrm{CO}(\mathrm{g}) \longrightarrow 2 \mathrm{Cr}(\mathrm{s}) + 3 \mathrm{CO}_{2}(\mathrm{g})
ΔGrxn∘=□\Delta G_{\mathrm{rxn}}^{\circ} = \square
□ kJ\square \text{ kJ}
\text{Is this reaction spontaneous or nonspontaneous at 298 K?}
\text{nonspontaneous}
\text{spontaneous}
\text{The standard Gibbs free energy of formation values are:}
ΔGf∘ for Cr2O3(s) is −1058.1 kJ/mol\Delta G_{\mathrm{f}}^{\circ} \text{ for } \mathrm{Cr}_{2} \mathrm{O}_{3}(\mathrm{s}) \text{ is } -1058.1 \text{ kJ/mol}
ΔGf∘ for CO(g) is −137.2 kJ/mol\Delta G_{\mathrm{f}}^{\circ} \text{ for } \mathrm{CO}(\mathrm{g}) \text{ is } -137.2 \text{ kJ/mol}
ΔGf∘ for Cr(s) is 0 kJ/mol\Delta G_{\mathrm{f}}^{\circ} \text{ for } \mathrm{Cr}(\mathrm{s}) \text{ is } 0 \text{ kJ/mol}
ΔGf∘ for CO2(g) is −394.4 kJ/mol\Delta G_{\mathrm{f}}^{\circ} \text{ for } \mathrm{CO}_{2}(\mathrm{g}) \text{ is } -394.4 \text{ kJ/mol}

Studdy Solution

STEP 1

1. The reaction is at standard conditions (298 K).
2. The standard Gibbs free energy of formation (ΔGf∘\Delta G_{\mathrm{f}}^{\circ}) values are provided for each substance in the reaction.
3. The formula for calculating the standard Gibbs free energy change of the reaction (ΔGrxn∘\Delta G_{\mathrm{rxn}}^{\circ}) is:
ΔGrxn∘=∑ΔGf∘(products)−∑ΔGf∘(reactants)\Delta G_{\mathrm{rxn}}^{\circ} = \sum \Delta G_{\mathrm{f}}^{\circ}(\text{products}) - \sum \Delta G_{\mathrm{f}}^{\circ}(\text{reactants})

STEP 2

1. Identify the ΔGf∘\Delta G_{\mathrm{f}}^{\circ} values for each reactant and product.
2. Calculate the total ΔGf∘\Delta G_{\mathrm{f}}^{\circ} for products and reactants.
3. Compute ΔGrxn∘\Delta G_{\mathrm{rxn}}^{\circ}.
4. Determine if the reaction is spontaneous or nonspontaneous.

STEP 3

Identify the ΔGf∘\Delta G_{\mathrm{f}}^{\circ} values for each substance:
- ΔGf∘ for Cr2O3(s)=−1058.1 kJ/mol\Delta G_{\mathrm{f}}^{\circ} \text{ for } \mathrm{Cr}_{2} \mathrm{O}_{3}(\mathrm{s}) = -1058.1 \text{ kJ/mol} - ΔGf∘ for CO(g)=−137.2 kJ/mol\Delta G_{\mathrm{f}}^{\circ} \text{ for } \mathrm{CO}(\mathrm{g}) = -137.2 \text{ kJ/mol} - ΔGf∘ for Cr(s)=0 kJ/mol\Delta G_{\mathrm{f}}^{\circ} \text{ for } \mathrm{Cr}(\mathrm{s}) = 0 \text{ kJ/mol} - ΔGf∘ for CO2(g)=−394.4 kJ/mol\Delta G_{\mathrm{f}}^{\circ} \text{ for } \mathrm{CO}_{2}(\mathrm{g}) = -394.4 \text{ kJ/mol}

STEP 4

Calculate the total ΔGf∘\Delta G_{\mathrm{f}}^{\circ} for products:
ΔGf∘(products)=[2×0+3×(−394.4)] kJ/mol\Delta G_{\mathrm{f}}^{\circ}(\text{products}) = [2 \times 0 + 3 \times (-394.4)] \text{ kJ/mol}
=0+(−1183.2)=−1183.2 kJ/mol= 0 + (-1183.2) = -1183.2 \text{ kJ/mol}

STEP 5

Calculate the total ΔGf∘\Delta G_{\mathrm{f}}^{\circ} for reactants:
ΔGf∘(reactants)=[−1058.1+3×(−137.2)] kJ/mol\Delta G_{\mathrm{f}}^{\circ}(\text{reactants}) = [-1058.1 + 3 \times (-137.2)] \text{ kJ/mol}
=−1058.1+(−411.6)=−1469.7 kJ/mol= -1058.1 + (-411.6) = -1469.7 \text{ kJ/mol}

STEP 6

Compute ΔGrxn∘\Delta G_{\mathrm{rxn}}^{\circ}:
ΔGrxn∘=−1183.2−(−1469.7) kJ/mol\Delta G_{\mathrm{rxn}}^{\circ} = -1183.2 - (-1469.7) \text{ kJ/mol}
=−1183.2+1469.7=286.5 kJ/mol= -1183.2 + 1469.7 = 286.5 \text{ kJ/mol}

STEP 7

Determine if the reaction is spontaneous or nonspontaneous:
Since ΔGrxn∘=286.5 kJ/mol\Delta G_{\mathrm{rxn}}^{\circ} = 286.5 \text{ kJ/mol} is positive, the reaction is nonspontaneous at 298 K.
The standard Gibbs free energy change for the reaction is:
ΔGrxn∘=286.5 kJ/mol\Delta G_{\mathrm{rxn}}^{\circ} = \boxed{286.5} \text{ kJ/mol}
The reaction is nonspontaneous at 298 K.

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