Exercises: Thermodynamic and Kinetic Control; Coupled Reactions

At 298 K and 1 atm, the conversion of diamond to graphite has $\Delta G^\circ < 0$, yet a diamond ring does not turn into graphite on your finger. Is the conversion thermodynamically favorable? Is it kinetically accessible at room temperature? Name the quantity that explains why no change is observed.

Label each statement as describing thermodynamic favorability (T), kinetic accessibility (K), or neither (N):

(i) The reaction has a very large activation energy.

(ii) The reaction has $\Delta G^\circ = -250\ \text{kJ/mol}$.

(iii) A catalyst makes the reaction reach equilibrium within seconds.

(iv) The reaction has $K \gg 1$.

In cells, the phosphorylation of glucose is coupled to ATP hydrolysis:

$\text{glucose} + \text{P}_i \rightarrow \text{glucose-6-phosphate} + \text{H}_2\text{O}$, $\Delta G^\circ = +13.8\ \text{kJ/mol}$

$\text{ATP} + \text{H}_2\text{O} \rightarrow \text{ADP} + \text{P}_i$, $\Delta G^\circ = -30.5\ \text{kJ/mol}$

Calculate $\Delta G^\circ$ for the coupled process and state whether it is thermodynamically favorable.

A reactant R can form product A or product B. The pathway to A has $E_a = 50\ \text{kJ/mol}$ and $\Delta G^\circ = -20\ \text{kJ/mol}$. The pathway to B has $E_a = 90\ \text{kJ/mol}$ and $\Delta G^\circ = -45\ \text{kJ/mol}$.

(a) Which product forms faster? (b) Which product is more stable and would dominate if the system reached equilibrium? (c) Identify the kinetic and the thermodynamic product. (d) Explain why a low-temperature experiment and a high-temperature experiment might give different product mixtures.

Copper metal can be obtained from copper(I) sulfide by coupling two reactions at 298 K:

$\text{Cu}_2\text{S}(s) \rightarrow 2\,\text{Cu}(s) + \text{S}(s)$, $\Delta G^\circ = +86.2\ \text{kJ}$

$\text{S}(s) + \text{O}_2(g) \rightarrow \text{SO}_2(g)$, $\Delta G^\circ = -300.1\ \text{kJ}$

(a) Write the net reaction. (b) Calculate the net $\Delta G^\circ$. (c) Explain how the species shared by both reactions makes the first process possible, and what a very negative $\Delta G^\circ$ implies about $K$.

A mixture of $\text{H}_2(g)$ and $\text{O}_2(g)$ at room temperature can sit for years without visible change, even though $2\,\text{H}_2(g) + \text{O}_2(g) \rightarrow 2\,\text{H}_2\text{O}(l)$ has $\Delta G^\circ \approx -474\ \text{kJ}$. A spark, or a bit of platinum powder, makes it react at once.

(a) Explain the lack of reaction in terms of thermodynamics and kinetics. (b) State how the platinum catalyst changes $E_a$, $\Delta G^\circ$, and $K$. (c) Explain what the spark does.

A biochemical conversion $\text{X} + \text{P}_i \rightarrow \text{Y} + \text{H}_2\text{O}$ has $\Delta G^\circ = +25.0\ \text{kJ/mol}$ at 298 K. A cell couples it to ATP hydrolysis, $\Delta G^\circ = -30.5\ \text{kJ/mol}$, through a shared intermediate: $\text{X} + \text{ATP} \rightarrow \text{X-P} + \text{ADP}$, followed by $\text{X-P} + \text{H}_2\text{O} \rightarrow \text{Y} + \text{P}_i$.

(a) Calculate $\Delta G^\circ$ and $K$ for the coupled process, and $K$ for the uncoupled reaction. Use $R = 8.314\ \text{J/(mol·K)}$. (b) By roughly what factor does coupling increase $K$? (c) Aaliyah says, "ATP just adds energy to the reaction." Evaluate this idea and explain why a shared intermediate is needed. (d) Explain what would happen to the energy of ATP hydrolysis if it occurred with no coupling.

For $\text{C(graphite)} \rightarrow \text{C(diamond)}$ at 298 K, $\Delta H^\circ = +1.9\ \text{kJ/mol}$ and $\Delta S^\circ = -3.4\ \text{J/(mol·K)}$.

(a) Calculate $\Delta G^\circ$ at 298 K. (b) Using the signs of $\Delta H^\circ$ and $\Delta S^\circ$, explain whether there is any temperature at which this conversion becomes favorable at standard conditions. (c) Synthetic diamonds are made at very high pressure; explain why pressure helps, given that diamond is denser than graphite. (d) Explain why natural diamonds do not convert to graphite even though the reverse process is favorable.