Thermodynamics: Enthalpy, Entropy, Free Energy, and Spontaneity

Thermodynamic Constructs Introduced: The lecture revisits entropy ($<br>ho$) and enthalpy ($ext{H}$), initially discussed on Wednesday.

Entropy and Spontaneity: Change in entropy of a system ($<br>ho_{ ext{system}}$) alone does not necessarily indicate spontaneity, though it can influence it.

Enthalpy ($ ext{H}$) - Exothermic vs. Endothermic Processes:

• Exothermic Process: Defined by a negative change in enthalpy ($ext{H} < 0$).

  • Heat is a product of the process.

  • If a reaction in a test tube is exothermic, holding the test tube will feel hot because heat is released into the surroundings (your hand).

• Endothermic Process: Defined by a positive change in enthalpy ($ext{H} > 0$).

  • Heat appears as a reactant; it is absorbed by the process.

  • If a reaction in a test tube is endothermic, holding the test tube will feel cold.

  • This cold sensation is because heat is absorbed from the surroundings (your hand) by the reaction. Heat flows from warmer (your hand) to colder (the reacting system).

Gibbs Free Energy ($ ext{G}$) - The Indicator of Spontaneity:

• The relationship between enthalpy, entropy, and free energy is given by the equation: $\Delta G = \Delta H - T\Delta S$ where $T$ is the absolute temperature in Kelvin.

• Both entropy and enthalpy directly influence the free energy change ($\Delta G$).

• Spontaneity Criteria:

  • If $\Delta G < 0$: The process is spontaneous (also called exergonic) in the forward direction (reactants $\rightarrow$ products).

  • If $\Delta G > 0$: The process is non-spontaneous (also called endergonic) in the forward direction. The reverse process would be spontaneous.

  • If $\Delta G = 0$: The system is at equilibrium.

• Conditions for Spontaneity based on $\Delta H$ and $\Delta S$:

  • If $\Delta H < 0$ (exothermic) and $\Delta S > 0$ (increasing entropy), then $\Delta G$ will always be negative, making the process spontaneous at all temperatures.

  • If $\Delta H > 0$ (endothermic) and $\Delta S < 0$ (decreasing entropy), then $\Delta G$ will always be positive, making the process non-spontaneous at all temperatures.

  • If $\Delta H < 0$ and $\Delta S < 0$: Spontaneous at low temperatures (when $|\Delta H| > |T\Delta S|$).

  • If $\Delta H > 0$ and $\Delta S > 0$: Spontaneous at high temperatures (when $|T\Delta S| > |\Delta H|$).

Units and Calculations:

• Temperature ($T$) must always be in Kelvin. To convert Celsius to Kelvin, add $273.15$ (or approximately $273$).

• Units of $\Delta H$ are typically in kilojoules ($ext{kJ}$) or kilojoules per mole ($ext{kJ/mol}$).

• Units of $\Delta S$ are typically in joules per Kelvin ($ext{J/K}$) or joules per Kelvin per mole ($ext{J/K \cdot mol}$).

• Crucial Unit Conversion: Before adding/subtracting $\Delta H$ and $T\Delta S$, their units must be consistent. If $\Delta H$ is in kilojoules, $\Delta S$ (and thus $T\Delta S$) must also be converted to kilojoules. For example, convert joules per Kelvin to kilojoules per Kelvin ($1 \text{ kJ} = 1000 \text{ J}$).

Reaction Progress Diagrams:

• Illustrate the change in free energy (or enthalpy) as a reaction proceeds.

• Reactants: Initial free energy level.

• Products: Final free energy level.

• Transition State: The highest energy point on the path, where bonds are partially broken and formed. This state is related to activation energy.

• Activation Energy ($E<em>a$): The energy difference between the reactants and the transition state. It is related to the rate of the reaction (lower $E</em>a$ = faster rate).

• Calculating $\Delta G$ from Diagram: $\Delta G = G<em>{\text{products}} - G</em>{\text{reactants}}$ (final minus initial).

  • If $G<em>{\text{products}} < G</em>{\text{reactants}}$, then $\Delta G$ is negative (spontaneous/exergonic).

  • If $G<em>{\text{products}} > G</em>{\text{reactants}}$, then $\Delta G$ is positive (non-spontaneous/endergonic).

Catalysts:

• A catalyst speeds up a reaction by lowering the activation energy ($E_a$) of both the forward and reverse reactions.

• A catalyst does not affect the free energy change ($\Delta G$) of the reaction.

• A catalyst does not affect the spontaneity of the reaction or the equilibrium position; it only helps the reaction reach equilibrium faster.

Chemical Equilibrium (A $\rightleftharpoons$ B):

• Definition: A state where the rate of the forward reaction equals the rate of the reverse reaction.

• Misconception: Equilibrium does not mean that the concentrations of reactants and products are equal. It means their rates of interconversion are equal.

• Equilibrium Constant ($K<em>{eq}$): At equilibrium, $\Delta G = 0$. Therefore, using the non-standard state equation, we can derive: $\Delta G^0 = -RT \ln K</em>{eq}$ This equation links the standard free energy change to the equilibrium constant.

Standard State vs. Non-Standard State Conditions:

• Standard State Free Energy Change ($\Delta G^0$):

  • Refers to specific standard conditions:

    • For solutes: $1$ M concentration.

    • For gases: $1$ atmosphere ($1 \text{ atm}$) partial pressure.

    • Temperature is usually specified (e.g., $298 \text{ K}$ or $25^{\circ}\text{C}$).

• Non-Standard State Free Energy Change ($\Delta G$):

  • Describes processes under actual, non-standard conditions.

  • Calculated using the equation: $\Delta G = \Delta G^0 + RT \ln Q$ where $Q$ is the reaction quotient.

  • Reaction Quotient ($Q$): An expression with the same form as the equilibrium constant but uses non-equilibrium (current) concentrations or partial pressures. For a reversible reaction $aA + bB \rightleftharpoons cC + dD$, $Q = \frac{[C]^c [D]^d}{[A]^a [B]^b}$.

  • If current concentrations are plugged into $Q$ and they are at equilibrium, then $Q = K_{eq}$.

• Biological Standard State ($\Delta G^{0'}$):

  • A modification of the standard state to better reflect physiological conditions.

  • The key difference is that the hydrogen ion concentration ($[\text{H}^+]$) is set to $10^{-7} \text{ M}$ (which corresponds to $pH = 7$), rather than $1 \text{ M}$ ($pH = 0$) in the chemical standard state.

  • Most biological processes occur around a physiological $pH$ of $7$ to $7.2$.

  • Example of very low $pH$ in the body: stomach, with hydrochloric acid concentration around $2 \text{ M}$.

State Functions:

• A property whose value depends only on the current state of the system, not on the path taken to reach that state.

• Examples: Altitude, free energy ($\Delta G$), enthalpy ($\Delta H$).

• Not examples: Work, heat.

Coupling Reactions: Cells often couple an unfavorable (endergonic, $\Delta G > 0$) reaction with a highly favorable (exergonic, $\Delta G < 0$) reaction to drive overall processes.

• Example: The phosphorylation of glucose to glucose-6-phosphate (an endergonic process, $\Delta G^0 = +18 \text{ kJ/mol}$) is coupled with the hydrolysis of ATP to ADP and phosphate (an exergonic process, $\Delta G^0 = -30.5 \text{ kJ/mol}$).

• The overall reaction (Glucose + ATP $\rightarrow$ Glucose-6-Phosphate + ADP) has a net negative $\Delta G^0$ (approx. $-12.5 \text{ kJ/mol}$), making it spontaneous.