Chemical Equilibrium, ΔG, and Standard State — Lecture Notes

Context and setup

• The instructor references compiling past notes and scanning pages, with a goal of sharing scanned copies via email. This frames the lecture as a consolidation of ideas already introduced in class.
• The main thermodynamics focus is equilibrium, free energy changes (ΔG), and how temperature, pressure, and concentrations influence spontaneity and the direction of a reaction.

Equilibrium at the melting point of water and basic definitions

• At the melting point of water, exactly $T = 0^ ext{o}C$ and $P = 1~ ext{atm}$, ice (solid) and liquid water are in dynamic equilibrium: ice tends to melt, while liquid water tends to freeze, at the same rate.
• If you could maintain 0 °C for a substantial time, you would observe a persistent equilibrium where neither complete melting nor complete freezing occurs.
• Thermodynamic statement: at equilibrium, the change in Gibbs free energy for the process is zero: oxed{\Delta G = 0}
• If the system is not at equilibrium, the sign of ΔG determines spontaneity for the forward process (ice → water):
  • If \Delta G < 0, the forward process is spontaneous.
  • If \Delta G > 0, the forward process is non-spontaneous (the reverse process is spontaneous).
• For the ice ⇌ water system at 1 atm, raising the temperature makes the forward process more spontaneous (ΔG becomes negative); lowering the temperature makes the forward process less favorable (ΔG becomes more positive).
• The forward process discussed here is ice melting into liquid water; the reverse process is liquid water freezing into ice.

What constitutes equilibrium in a chemical reaction

• Consider a generic reaction: $A \rightleftharpoons B$
  • Forward reaction: $A \rightarrow B$ (conversion of reactant A to product B).
  • Reverse reaction: $B \rightarrow A$.
• Equilibrium occurs when the speeds (rates) of the forward and reverse reactions are equal, i.e., $v<em>{f} = v</em>{r}$. At this point, there is no net change in the amounts of A and B over time, even though both reactions may be occurring.
• The forward reaction may start out very fast (high spontaneity, large negative ΔG) and gradually slow as the system approaches equilibrium; the reverse reaction may accelerate as products accumulate, leading to equal rates at equilibrium.
• The state at equilibrium is characterized by a constant ratio of products to reactants, not necessarily by the absence of reaction, but by the balance of forward and reverse processes.

Equilibrium constant and the visual of the reaction progress

• When a reaction reaches equilibrium, the ongoing conversion of A to B and B to A occurs at equal rates; the net production of B stops increasing.
• The equilibrium constant K is defined (for the reaction A ⇌ B) as the ratio of product concentration to reactant concentration at equilibrium:
  • $\boxed{K = \frac{[B]}{[A]}}$
  • The square brackets denote molar concentrations (i.e., mol/L).
• For a general reaction where concentrations are used, the same idea applies: the ratio of products to reactants at equilibrium defines K.
• The term equilibrium constant is a state function for a given reaction at a specified temperature; it does not explicitly depend on the path to equilibrium.

Reaction quotient (Q) and its relation to spontaneity

• Reaction quotient Q is the same ratio as K but measured at any time before equilibrium:
  • $\boxed{Q = \frac{[B]}{[A]}}$
• Key distinction: Q generally differs from K while the reaction is not at equilibrium; only at equilibrium is Q = K.
• Consequences of Q relative to K:
  • If Q < K, the forward reaction is spontaneous (ΔG < 0).
  • If Q > K, the forward reaction is non-spontaneous (ΔG > 0); the reverse reaction is spontaneous.
  • If $Q = K$, the system is at equilibrium (ΔG = 0).
• Rationale: when there is too much reactant A (relative to B), the system tends to form more B to reach equilibrium; when there is too much product B, the system tends to convert some B back to A.
• Practical note: measuring Q at any time provides a snapshot of spontaneity without waiting for equilibrium; measuring K requires waiting for equilibrium or calculating under standard-state assumptions.

Standard state, standard-state properties, and nonstandard state

• Standard state is a reference state used for thermodynamic measurements. It defines a baseline for comparing reactions under different conditions.
  • Common standard-state conventions: temperature T°, pressure P°, and concentrations [ ]° such that:
  • Temperature: typically $T^ ext{o} = 298~ ext{K}$ (25°C). This is the standard temperature.
  • Pressure: typically $P^ ext{o} = 1~\text{atm}$ for gases.
  • Concentration: typically $[C]^ ext{o} = 1~\text{M}$ for solutions.
• Example for standard state: a gas-phase reaction such as water vapor turning into H2 and O2 can be analyzed under standard state conditions of 298 K and 1 atm, with product concentrations referenced to 1 M where applicable.
• Nonstandard state is any condition different from the standard state (different T, P, or concentrations).
  • Example given: T = 500 K and P = 2 atm would be nonstandard for a reaction typically reported under standard-state references.
• Thermodynamic quantities with standard-state notation are marked with a degree symbol over the symbol (e.g., ΔH°, ΔS°, ΔG°). This denotes quantities measured under standard-state conditions.
• A key question is how to convert from standard-state Gibbs energy change ΔG° to the Gibbs energy change under nonstandard conditions ΔG at a given T and practical state variables.

Relationship between standard-state and nonstandard Gibbs energy: the core equation

• The fundamental equation that connects standard-state ΔG° to the nonstandard ΔG at temperature T and reaction quotient Q is:
  • $\boxed{\Delta G = \Delta G^{\circ} + RT \ln Q}$
• Here:
  • $\Delta G$ is the Gibbs energy change at the actual conditions (T, P, concentrations).
  • $\Delta G^{\circ}$ is the Gibbs energy change under standard-state conditions.
  • $R$ is the gas constant; its value depends on the units used:
  • In SI (J, mol, K): $\boxed{R = 8.314\,\mathrm{J\,mol^{-1}\,K^{-1}}}$
  • In L·atm units: $\boxed{R = 0.08206\,\mathrm{L\,atm\,mol^{-1}\,K^{-1}}}$
  • $T$ is the absolute temperature in kelvin (K).
  • $Q$ is the reaction quotient at the current conditions (not necessarily equilibrium).
• Conceptual purpose of the equation:
  • It shows how ΔG changes from the standard state when you move away from standard conditions, by accounting for temperature and the current composition via ln Q.
  • At standard-state conditions (Q = 1), the equation reduces to ΔG = ΔG°. This is why ΔG° is defined as the free energy change under those reference conditions.
• Practical implications:
  • Knowing ΔG° and the current Q and T allows you to predict spontaneity at nonstandard conditions.
  • The equation is central to understanding how systems respond to changes in temperature and composition, and it is especially important for chapters covering nonstandard thermodynamics and chemical equilibria.
• Important reminder from the lecture notes: this equation is emphasized as very important for understanding topics likely covered in chapters 20 and 21.

Notes on terminology and why the reverse reaction matters

• Reversible reactions are common in chemistry; many organic and inorganic reactions can proceed in both directions depending on conditions.
• The instructor emphasizes that even when a forward reaction dominates initially (large negative ΔG), the reverse reaction can become significant as products accumulate, eventually balancing rates at equilibrium.
• In the context of biology and medicine, reversibility (e.g., getting sick then recovering) is used as a metaphor to illustrate that forward processes often have reverse pathways, supporting the idea of equilibrium and dynamic balance.

Quick recap of the key ideas to remember for exams

• Equilibrium condition: $\Delta G = 0$ and forward and reverse rates are equal ($v<em>f = v</em>r$).
• For the ice ⇌ water system at 1 atm, at 0°C the system is at equilibrium; above 0°C, the forward process (ice → water) becomes spontaneous (\Delta G < 0).
• Forward spontaneity can be controlled by temperature: increasing T tends to drive the forward process (for ice ⇌ water) toward spontaneity at 1 atm; decreasing T can make the forward process non-spontaneous (\Delta G > 0).
• Equilibrium constant: $K = \frac{[B]}{[A]}$ for a reaction A ⇌ B; K is defined at a given temperature for the standard-state conditions.
• Reaction quotient: $Q = \frac{[B]}{[A]}$ at any moment; Q ≠ K out of equilibrium; Q = K at equilibrium.
• Relationship between ΔG°, ΔG, and Q: $\Delta G = \Delta G^{\circ} + RT \ln Q$; this links standard-state thermodynamics to current conditions.
• Standard state definitions: $T^{\circ} = 298\,\text{K}$, $P^{\circ} = 1\,\text{atm}$, and concentrations $[C]^{\circ} = 1\,\text{M}$; standard-state quantities are denoted with a degree symbol (e.g., $\Delta G^{\circ}$).
• The standard-state equation is particularly useful for applying thermodynamics to chapter 21 concepts and for transitioning from standard-state values to actual conditions.