Thermodynamics and Free Energy Notes

Free Energy & Concentration

• Entropy (disorder) increases with volume and is a function of concentration.
• Free energy, therefore, depends on concentration.

Free Energy Change ($\Delta G$)

• $\Delta G$ is based on the standard free-energy change ($\Delta G^\circ$) and initial concentrations ($[A_i]^a$) of products and reactants.
• Equation: $\Delta G = \Delta G^\circ + RT \ln \frac{[C<em>i]^c [D</em>i]^d}{[A<em>i]^a [B</em>i]^b}$
  • $R$: gas constant ($8.315$ J·mol$^{-1}$·K$^{-1}$).
  • $T$: Temperature in Kelvin.
• Composed of two parts:
  1. Constant Term: Standard Free Energy Change ($\Delta G^\circ$), characteristic for each reaction.
  2. Variable Term: Based on initial reactant/product concentrations, stoichiometry, and temperature.

Standard States ($\Delta G^\circ$)

• A reference state for comparing thermodynamic parameters of different reactions.
• Defined conditions:
  • $1$ M reactants and products.
  • Specific temperature (usually $298 \text{ K}$).
  • Specific pressure (usually $1 \text{ atm}$).
• Symboled with a degree sign (e.g., $\Delta G^\circ$).
• Not representative of normal cellular conditions.

Biochemists' Standard-State ($\Delta G^\circ\prime$)

• Modified standard conditions to reflect biochemical environments.
• Assumptions:
  • $[H^+] = 1$ (equivalent to pH $7.0$).
  • $[H_2O] = 1$ (due to very high concentration, approx. $55.5 \text{ M}$).
• Symboled as $\Delta G^\circ\prime$.
• $\Delta G^\circ$ and $\Delta G^\circ\prime$ are often used interchangeably in biochemical contexts.

Equilibrium

• A state where forward and reverse reaction rates are equal.
• Concentrations of reactants and products remain constant over time.
• All chemical reactions proceed until equilibrium is reached.
• At equilibrium, $\Delta G = 0$.

Equilibrium Constant ($K_{eq}$)

• Ratio of product to reactant concentrations at equilibrium.
• For $aA + bB \rightleftharpoons cC + dD$: $K<em>{eq} = \frac{[C</em>{eq}]^c [D<em>{eq}]^d}{[A</em>{eq}]^a [B_{eq}]^b}$.
• K_{eq} >> 1: Favors product formation ($\Delta G^\circ$ is large and negative).
• K_{eq} << 1: Favors reactant formation ($\Delta G^\circ$ is large and positive).
• $K_{eq}$ is temperature-dependent.

Relationship Between Free Energy and Equilibrium

• At equilibrium ($\Delta G = 0$), the relationship is: $\Delta G^\circ = -RT \ln K_{eq}$.
• This can also be expressed as: $K_{eq} = e^{-\Delta G^\circ/RT}$.
• This relationship is temperature-dependent; assume $T = 298 \text{ K}$ unless otherwise indicated.

Additivity of Thermodynamic State Functions

• Thermodynamic state functions (e.g., $\Delta G$ and $\Delta G^\circ$) are additive.
• For consecutive reactions ($1$ and $2$): $\Delta G<em>{Total} = \Delta G</em>1 + \Delta G_2$.
• Allows unfavorable reactions to occur by coupling them with highly favorable ones.

Driving Unfavorable Reactions Forward

1. Increased concentrations of reactants: Compartmentation can maintain high local concentrations.
2. Coupling of reactions: A thermodynamically favorable reaction can drive an unfavorable one.

Usage of Free Energy Equations

1. $\Delta G = \Delta H - T\Delta S$: Use when entropy ($\Delta S$) or enthalpy ($\Delta H$) information is required or provided.
2. $\Delta G = \Delta G^\circ + RT \ln \frac{[P]}{[R]}$: Use for non-equilibrium conditions.
3. $\Delta G^\circ = -RT \ln K_{eq}$: Use ONLY for equilibrium conditions.

• Multiple equations may be required to solve complex problems.