Thermodynamics and Kinetics

• Thermodynamics predicts spontaneity, while kinetics predicts rate.
• Thermodynamics considers initial and final states, whereas kinetics considers the path between them.
• Thermodynamics determines equilibrium composition, but kinetics does not predict it.
• Kinetics predicts the timescale to reach equilibrium, which thermodynamics doesn't.
• State functions depend only on the current state of the system.
• Examples of state functions include internal energy ($U$), enthalpy ($H$), entropy ($S$), and Gibbs free energy ($G$).
• Path functions, such as heat ($q$) and work ($w$), depend on the path taken.

First Law of Thermodynamics

• The First Law is represented by the equation $\Delta U = q + w$, where $\Delta U$ is the change in internal energy, $q$ is heat added, and $w$ is work done on the system.
• Expansion work is calculated by $w = -P\Delta V$, where $P$ is pressure and $\Delta V$ is the change in volume.
• Shaft work includes electrical work and stirring.

Enthalpy

• Enthalpy ($H$) is defined as $H = U + PV$.
• At constant pressure, the change in enthalpy is $\Delta H = \Delta U + P\Delta V = q_p$, where $q_p$ is the heat added.

Heat Capacity

• Heat capacity ($C$) is defined as $C = \frac{dq}{dT}$, where $dq$ is heat required for temperature change $dT$.
• Constant volume heat capacity: $C_v = (\frac{\partial U}{\partial T})_V$.
• Constant pressure heat capacity: $C_p = (\frac{\partial H}{\partial T})_P$.
• For ideal gases: $C_p = C_v + nR$, where $n$ is the number of moles and $R$ is the ideal gas constant.

Second Law of Thermodynamics

• Entropy ($S$) is a measure of the disorder of a system
• For a reversible process: $\Delta S = \frac{q_{rev}}{T}$.
• For an irreversible process: $\Delta S > \frac{q}{T}$.
• The Second Law states: $\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings} \ge 0$

The Third Law of Thermodynamics

• The entropy of a perfect crystal at absolute zero (0 K) is zero: $S = 0 \text{ at } T = 0 \text{ K}$.

Gibbs Free Energy

• Gibbs Free Energy ($G$) is defined as $G = H - TS$.
• $\Delta G = \Delta H - T\Delta S$ at constant temperature.
• Spontaneity is determined as follows: $\Delta G < 0$ (spontaneous), $\Delta G > 0$ (nonspontaneous), $\Delta G = 0$ (equilibrium).
• Temperature Dependence: $\dfrac{\Delta G}{T} = - \dfrac{\Delta H}{T^2}$ (Gibbs-Helmholtz equation).

Chemical Potential

• Chemical potential ($\mu_i$) is defined as $\mu_i = (\frac{\partial G}{\partial n_i}){T, P, n{j\neq i}}$.
• Chemical potential determines equilibrium conditions and substances move from regions of high to low chemical potential.

Phase Equilibria

• Clausius-Clapeyron Equation: $\dfrac{d\ln P}{dT} = \dfrac{\Delta H_{vap}}{RT^2}$, where $P$ is vapor pressure, $T$ is temperature, $\Delta H_{vap}$ is the enthalpy of vaporization and $R$ is the ideal gas constant.
• Phase Rule: $F = C - P + 2$, where $F$ is degrees of freedom, $C$ is number of components, and $P$ is number of phases.

Chemical Reactions

• For the reaction $aA + bB \rightleftharpoons cC + dD$, the equilibrium constant is $K = \dfrac{[C]^c [D]^d}{[A]^a [B]^b}$.
• The relationship between $\Delta G^\circ$ and $K$ is $\Delta G^\circ = -RT\ln K$, where $\Delta G^\circ$ is the standard Gibbs free energy change.
• Van't Hoff Equation: $\dfrac{d\ln K}{dT} = \dfrac{\Delta H^\circ}{RT^2}$, where $\Delta H^\circ$ is the standard enthalpy change.