Question

Some bacteria use light energy to convert carbon dioxide and water to glucose and oxygen: \(6 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(a q)+6 \mathrm{O}_{2}(g) \quad \Delta G^{\circ}=2870 \mathrm{~kJ}\) at \(25^{\circ} \mathrm{C}\) Other bacteria, those that do not have light available to them, couple the reaction $$ \mathrm{H}_{2} \mathrm{~S}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \longrightarrow \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{S}(s) $$ to the glucose synthesis above. Coupling the two reactions, the overall reaction is \(24 \mathrm{H}_{2} \mathrm{~S}(g)+6 \mathrm{CO}_{2}(g)+6 \mathrm{O}_{2}(g) \longrightarrow \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(a q)+18 \mathrm{H}_{2} \mathrm{O}(l)+24 \mathrm{~S}(s)\) Show that the reaction is spontaneous at \(25^{\circ} \mathrm{C}\).

Short answer

Answer: Yes, the coupled reaction is spontaneous at 25°C because the calculated Gibbs free energy change (ΔG_total) is negative (-2021.2 kJ).

Step-by-step solution

1. Combine the two reactions

To combine the two reactions, we will first multiply the second reaction by some factor so that the number of water molecules in both reactions is equal. Then we'll add the first reaction to the adjusted second reaction to obtain the overall reaction. Looking at the given reactions: Reaction 1: \(6 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(a q)+6 \mathrm{O}_{2}(g)\) Reaction 2: \(\mathrm{H}_{2} \mathrm{S}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \longrightarrow \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{S}(s)\) We can adjust the stoichiometry of Reaction 2: \(24 \times\) (Reaction 2): \(24 \mathrm{H}_{2} \mathrm{S}(g)+12 \mathrm{O}_{2}(g) \longrightarrow 24 \mathrm{H}_{2} \mathrm{O}(l)+24 \mathrm{S}(s)\) Now, we can add Reaction 1 to the adjusted Reaction 2: Reaction 1 \(+\) \(24 \times\) (Reaction 2): \(24 \mathrm{H}_{2} \mathrm{S}(g)+6 \mathrm{CO}_{2}(g)+6 \mathrm{O}_{2}(g) \longrightarrow \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(a q)+18 \mathrm{H}_{2} \mathrm{O}(l)+24 \mathrm{S}(s)\) This is the overall reaction.

2. Calculate ΔG for the combined reaction

We are given the ΔG value for Reaction 1 at 25ºC: ΔG¹ = 2870 kJ To find the ΔG for \(24 \times\) (Reaction 2), we can use tabulated standard Gibbs free energy values of formation for each compound: ΔG = Σ[nΔG(products)] - Σ[nΔG(reactants)] ΔG² = 24*(ΔG(1 mol of H₂O) + ΔG(1 mol of S)) - 24*(ΔG(1 mol of H₂S) + 0.5*ΔG(1 mol of O₂)) Looking up the values in a table, we have: - ΔG(1 mol of H₂O) = -237.2 kJ/mol - ΔG(1 mol of S) = 0 kJ/mol (since it's a standard state) - ΔG(1 mol of H₂S) = -33.6 kJ/mol - ΔG(1 mol of O₂) = 0 kJ/mol (since it's a standard state) Using these values: ΔG² = 24*(-237.2 + 0 - (-33.6) - 0.5*0) = -4891.2 kJ Now, we can add the ΔG values of reactions to calculate the ΔG for the overall reaction: ΔG_total = ΔG¹ + ΔG² = 2870 - 4891.2 = -2021.2 kJ

3. Determine if the reaction is spontaneous

Since ΔG_total < 0, the reaction is spontaneous at 25ºC.

Key concepts

Spontaneous Reactions

In chemistry, a spontaneous reaction is one that occurs without the need for external force or energy. This concept primarily depends on the Gibbs Free Energy (ΔG) of the reaction. If the value of ΔG is negative, the reaction will proceed spontaneously.

Spontaneity is an important concept because it dictates the feasibility of a reaction under given conditions. When a reaction is spontaneous, it means that the reactants can convert into products without additional energy input. In the case of the bacteria involved in the exercise, they leverage spontaneous reactions to drive processes essential for life.

Understanding spontaneous reactions allows us to predict whether a chemical process can occur naturally. This understanding is foundational for many biological and chemical applications across various fields.

Gibbs Free Energy

Gibbs Free Energy (G) is a thermodynamic function used to predict the direction and extent of chemical reactions. It combines the system's enthalpy and entropy to determine free energy change, represented as ΔG.

The equation for Gibbs Free Energy is:

• ΔG = ΔH - TΔS

Where:
- ΔH is the change in enthalpy (heat content),
- T is the temperature in Kelvin,
- ΔS is the change in entropy (disorder) in the system.

When ΔG is negative, the process is spontaneous; when positive, it is non-spontaneous; and when zero, the system is at equilibrium. The exercise shows how calculating the ΔG can demonstrate the feasibility of reactions bacteria use to synthesize glucose.

This concept is crucial in fields like bioenergetics, where understanding energy changes helps explain how organisms like bacteria sustain themselves.

Chemotrophy

Chemotrophy refers to organisms obtaining their energy by chemical means, distinct from reliance on sunlight. Chemotrophs harness energy from chemical reactions, often involving inorganic molecules, to form organic compounds used in metabolism and cellular processes.

Some bacteria exemplify chemotrophy by oxidizing inorganic substances such as hydrogen sulfide (H₂S) from which they derive energy to convert CO₂ into organic material through chemical reactions. This approach is especially vital in environments where sunlight, the primary energy source for photosynthesis, is unavailable.

These organisms demonstrate the diversity and adaptability of life forms, significantly contributing to nutrient cycling and environmental balance. Understanding chemotrophy provides insights into the adaptability and survival strategies of microorganisms in diverse ecosystems.

Reaction Coupling

Reaction coupling is a vital biochemical concept where two reactions are linked so that an energetically unfavorable reaction can proceed by being paired with an energetically favorable one. In essence, the energy released from the favorable reaction helps drive the unfavorable one.

The coupling of reactions helps organisms perform essential metabolic processes that wouldn't occur spontaneously due to positive ΔG. In the exercise, certain bacteria couple the reaction of glucose synthesis with another reaction involving sulfur compounds to ensure the overall process is energetically feasible.

By effectively managing their energy resources using reaction coupling, bacteria can thrive in various environments, demonstrating nature's efficiency in utilizing available resources. This concept is fundamental in understanding cellular metabolism and energy management strategies in living organisms.