Chemistry

Problem

The thermal decomposition of limestone is endergonic under standard conditions:

\mathrm{CaCO_3}(\text{s}) \to \mathrm{CaO}(\text{s}) + \mathrm{CO_2}(\text{g}) \quad\Delta G^\circ = +130\,\text{kJ}\,\text{mol}^{-1}

A chemist wishes to drive this decomposition at room temperature by coupling it to one of the three exergonic reactions listed in the table.

Candidate reaction	Equation	$\Delta G^\circ$ (kJ·mol⁻¹)
1	$CO_2(\text{g}) + 4\,H_2(\text{g}) \to CH_4(\text{g}) + 2\,H_2O(\text{l})$	$-130$
2	$2\,C(\text{s}) + O_2(\text{g}) \to 2\,CO(\text{g})$	$-510$
3	$C(\text{s}) + O_2(\text{g}) \to CO_2(\text{g})$	$-394$

All reactions take place in the same closed vessel, and the chemist may use any whole-number stoichiometric coefficients. a. Part 1: (a) Select the candidate reaction that can be coupled directly to the decomposition of

\mathrm{CaCO_3}

so that a common intermediate is eliminated. Justify your choice by referring explicitly to the species that cancels. b. Part 2: (b) Add the two appropriate reactions together using integer stoichiometric coefficients to obtain a single overall reaction with no intermediates. Show your work. c. Part 3: (c) Calculate the standard free-energy change,

\Delta G^\circ

, for the overall reaction obtained in part (b). State whether the coupled process is spontaneous under standard conditions and explain your reasoning. d. Part 4: (d) Briefly explain why each of the two reactions you did not select in part (a) cannot, by itself, be coupled to

\mathrm{CaCO_3}

decomposition to yield a spontaneous overall process without introducing additional steps.

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