Chemical Thermodynamics (JEE)

Cheatsheet Content

1. Basic Concepts System: Part of the universe under observation. Surroundings: Everything else interacting with the system. Boundary: Separates system from surroundings. Types of Systems: Open: Exchanges both matter and energy. Closed: Exchanges energy but not matter. Isolated: Exchanges neither matter nor energy. State Functions: Properties depending only on the initial and final states ($P, V, T, U, H, S, G$). Path Functions: Properties depending on the path taken ($W, Q$). Extensive Properties: Depend on the amount of substance (mass, volume, energy). Intensive Properties: Independent of the amount of substance (temperature, pressure, density, specific heat). 2. Thermodynamic Processes Isothermal: $T = \text{constant}$, $dT = 0$, $\Delta U = 0$ (for ideal gas). Adiabatic: $Q = 0$ (no heat exchange). Isobaric: $P = \text{constant}$, $dP = 0$. Isochoric: $V = \text{constant}$, $dV = 0$. Cyclic: Initial and final states are the same, $\Delta U = 0$, $\Delta H = 0$, etc. Reversible: Process occurs infinitesimally slowly, equilibrium maintained at each step. Irreversible: Process occurs rapidly, equilibrium not maintained. 3. First Law of Thermodynamics Statement: Energy can neither be created nor destroyed. $\Delta U = Q + W$ $\Delta U$: Change in internal energy. $Q$: Heat absorbed by the system (positive) or released (negative). $W$: Work done on the system (positive) or by the system (negative). Work Done ($W$): Mechanical work: $W = -P_{ext} \Delta V$ (irreversible, constant $P_{ext}$). Reversible isothermal expansion (ideal gas): $W = -nRT \ln \left(\frac{V_2}{V_1}\right) = -nRT \ln \left(\frac{P_1}{P_2}\right)$. Isochoric process: $W = 0$ (since $\Delta V = 0$). Free expansion ($P_{ext}=0$): $W = 0$. Heat Capacity: $C = \frac{Q}{\Delta T}$ $C_V = \left(\frac{\partial U}{\partial T}\right)_V$ (at constant volume) $C_P = \left(\frac{\partial H}{\partial T}\right)_P$ (at constant pressure) Relation: $C_P - C_V = R$ (for ideal gas). Ratio of heat capacities: $\gamma = \frac{C_P}{C_V}$. 4. Enthalpy ($H$) Definition: $H = U + PV$. Change in Enthalpy: $\Delta H = \Delta U + \Delta (PV)$. For constant pressure: $\Delta H = Q_P$. Relation between $\Delta H$ and $\Delta U$: $\Delta H = \Delta U + \Delta n_g RT$. ($\Delta n_g$ = moles of gaseous products - moles of gaseous reactants). Standard Enthalpies: Formation ($\Delta H_f^\circ$): Enthalpy change when 1 mole of a compound is formed from its elements in their standard states. $\Delta H_f^\circ (\text{element}) = 0$. Combustion ($\Delta H_c^\circ$): Enthalpy change when 1 mole of a substance undergoes complete combustion. Neutralization ($\Delta H_{neut}^\circ$): Enthalpy change when 1 mole of water is formed from $H^+$ and $OH^-$. For strong acid-strong base, it's approx. $-57.3 \text{ kJ/mol}$. Phase Transition: Fusion ($\Delta H_{fus}$), Vaporization ($\Delta H_{vap}$), Sublimation ($\Delta H_{sub}$). Hess's Law: Enthalpy change for a reaction is the sum of enthalpy changes for individual steps, regardless of the path. Bond Enthalpy: Energy required to break one mole of a specific bond. $\Delta H_{rxn}^\circ = \sum (\text{Bond energies of reactants}) - \sum (\text{Bond energies of products})$. 5. Second Law of Thermodynamics Statement: The entropy of an isolated system always increases in a spontaneous process. Clausius Statement: Heat cannot spontaneously flow from a colder body to a hotter body. Kelvin-Planck Statement: It is impossible to construct a device which operates in a cycle and produces no effect other than the extraction of heat from a reservoir and the performance of an equivalent amount of work. 6. Entropy ($S$) Definition: A measure of randomness or disorder. Change in Entropy: $\Delta S = \frac{Q_{rev}}{T}$. For a spontaneous process: $\Delta S_{total} = \Delta S_{sys} + \Delta S_{surr} > 0$. Standard Entropy Change: $\Delta S^\circ = \sum S^\circ (\text{products}) - \sum S^\circ (\text{reactants})$. Factors increasing entropy: Increase in temperature. Phase transitions (Solid $\to$ Liquid $\to$ Gas). Increase in number of moles of gas. Mixing of gases. 7. Third Law of Thermodynamics Statement: The entropy of a perfectly crystalline substance at absolute zero (0 K) is taken as zero. 8. Gibbs Free Energy ($G$) Definition: $G = H - TS$. Change in Gibbs Free Energy: $\Delta G = \Delta H - T \Delta S$. Criteria for Spontaneity: (at constant $T, P$) $\Delta G $\Delta G > 0$: Non-spontaneous $\Delta G = 0$: Equilibrium Effect of $\Delta H$ and $\Delta S$ on Spontaneity: $\Delta H$ $\Delta S$ $\Delta G = \Delta H - T\Delta S$ Spontaneity $-$ $+$ Always $-$ Spontaneous at all T $+$ $-$ Always $+$ Non-spontaneous at all T $-$ $-$ $-$ at low T, $+$ at high T Spontaneous at low T $+$ $+$ $+$ at low T, $-$ at high T Spontaneous at high T Relationship with Equilibrium Constant ($K_{eq}$): $\Delta G^\circ = -RT \ln K_{eq}$ $\Delta G = \Delta G^\circ + RT \ln Q$ (where $Q$ is reaction quotient) Temperature at Equilibrium: $T_{eq} = \frac{\Delta H}{\Delta S}$ (when $\Delta G=0$). 9. Important Relations & Formulas For Ideal Gas: $U = \frac{f}{2} nRT$ (where $f$ is degrees of freedom) $C_V = \frac{f}{2} R$ $C_P = \left(\frac{f}{2} + 1\right) R$ Adiabatic Process (Ideal Gas): $PV^\gamma = \text{constant}$ $T V^{\gamma-1} = \text{constant}$ $T^\gamma P^{1-\gamma} = \text{constant}$ $W = \frac{nR(T_1 - T_2)}{\gamma - 1}$