Second law of thermodynamics Knowledge-Based Study

Study notes

1. In thermodynamics, the first law establishes the conservation of energy, stating that energy cannot be created or destroyed. However, the first law does not predict the direction in which a process naturally occurs. For example, a drop of ink spontaneously disperses in water, but dispersed ink never spontaneously gathers back into a single drop, even though energy would be conserved in both directions.

   The Second Law of Thermodynamics provides the fundamental criterion for determining whether a physical or chemical process can occur spontaneously. It states that for any spontaneous (irreversible) process, the total entropy of the universe must increase:

   $$\Delta S_{\text{univ}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}} > 0$$

   Key consequences of this statement include:

   • Spontaneous process: $\Delta S_{\text{univ}} > 0$. The process proceeds naturally without continuous external intervention.
   • Reversible (equilibrium) process: $\Delta S_{\text{univ}} = 0$. The system and its surroundings are in a state of dynamic balance.
   • Non-spontaneous process: $\Delta S_{\text{univ}} < 0$. The process cannot occur spontaneously under the specified conditions; rather, the reverse process is spontaneous.

   While the entropy change of the system ($\Delta S_{\text{sys}}$) can be calculated from standard molar entropy values ($S^\circ$), determining the direction of spontaneity requires evaluating the entropy change of the surroundings ($\Delta S_{\text{surr}}$).

   When a system undergoes a chemical or physical change at constant temperature ($T$) and pressure, it exchanges heat ($q_{\text{sys}} = \Delta H_{\text{sys}}$) with its surroundings. The heat absorbed by the surroundings is $-q_{\text{sys}}$. Therefore, the entropy change of the surroundings is defined quantitatively as:

   $$\Delta S_{\text{surr}} = -\frac{\Delta H_{\text{sys}}}{T}$$

   This relationship highlights two important principles:

   • Exothermic processes ($\Delta H_{\text{sys}} < 0$): Heat flows into the surroundings, causing $\Delta S_{\text{surr}} > 0$. This positive contribution can drive a reaction spontaneously even if the system becomes more ordered ($\Delta S_{\text{sys}} < 0$), such as the freezing of water below 0 °C.
   • Endothermic processes ($\Delta H_{\text{sys}} > 0$): Heat flows out of the surroundings into the system, causing $\Delta S_{\text{surr}} < 0$. Such processes can only be spontaneous if the increase in system entropy ($\Delta S_{\text{sys}} > 0$) is large enough to outweigh this negative contribution.
   • Temperature dependence: Transferring a given quantity of heat causes a greater fractional change in entropy at lower temperatures than at higher temperatures, as reflected by $T$ in the denominator.

   Directly monitoring the entropy of the entire universe is impractical in laboratory conditions. To focus exclusively on state functions of the system, we can substitute the expression for $\Delta S_{\text{surr}}$ into the second law inequality:

   $$\Delta S_{\text{univ}} = \Delta S_{\text{sys}} - \frac{\Delta H_{\text{sys}}}{T} > 0$$

   Multiplying both sides of the inequality by $-T$ (recalling that absolute temperature $T$ in kelvins is always positive, which reverses the inequality sign) gives:

   $$-T\Delta S_{\text{univ}} = \Delta H_{\text{sys}} - T\Delta S_{\text{sys}} < 0$$

   The quantity $\Delta H_{\text{sys}} - T\Delta S_{\text{sys}}$ defines the change in Gibbs free energy ($\Delta G_{\text{sys}}$) at constant temperature and pressure:

   $$\Delta G_{\text{sys}} = -T\Delta S_{\text{univ}}$$

   Because $\Delta G_{\text{sys}}$ has the opposite algebraic sign of $\Delta S_{\text{univ}}$, the criteria for spontaneity under constant temperature and pressure translate directly:

   • $\Delta G_{\text{sys}} < 0$: The process is spontaneous in the forward direction ($\Delta S_{\text{univ}} > 0$).
   • $\Delta G_{\text{sys}} = 0$: The system is at equilibrium ($\Delta S_{\text{univ}} = 0$).
   • $\Delta G_{\text{sys}} > 0$: The forward process is non-spontaneous; work must be done on the system to drive it ($\Delta S_{\text{univ}} < 0$).

   Thus, Gibbs free energy provides a convenient, system-focused expression of the Second Law of Thermodynamics.