JEE Thermodynamics

Cheatsheet Content

1. Basic Concepts System: Part of the universe chosen for study. Open: Exchanges both matter and energy. Closed: Exchanges energy but not matter. Isolated: Exchanges neither matter nor energy. Surroundings: Everything else in the universe. Boundary: Separates system from surroundings. State Functions: Properties depending only on the initial and final states, not path (e.g., $P, V, T, U, H, S, G$). Path Functions: Properties depending on the path taken (e.g., $q$ (heat), $w$ (work)). Intensive Properties: Independent of amount of matter (e.g., $T, P, \rho$). Extensive Properties: Dependent on amount of matter (e.g., $V, U, H, S, G$). 2. Sign Conventions (JEE Chemistry) Heat ($q$): $q > 0$ (positive): Heat absorbed by the system (endothermic). $q Work ($w$): $w > 0$ (positive): Work done on the system by surroundings. $w Internal Energy ($\Delta U$): $\Delta U > 0$: Internal energy of system increases. $\Delta U Enthalpy ($\Delta H$): $\Delta H > 0$: Endothermic process (heat absorbed). $\Delta H 3. First Law of Thermodynamics Statement: Energy can neither be created nor destroyed, only transformed. Mathematical Form: $\Delta U = q + w$ $\Delta U$: Change in internal energy of the system. $q$: Heat exchanged by the system. $w$: Work done. Work Done by Gas: For expansion/compression against constant external pressure ($P_{ext}$): $w = -P_{ext}\Delta V$ For reversible isothermal expansion/compression of an ideal gas: $w = -nRT \ln \left(\frac{V_2}{V_1}\right) = -nRT \ln \left(\frac{P_1}{P_2}\right)$ For reversible adiabatic expansion/compression of an ideal gas: $w = nC_V(T_2 - T_1)$ or $w = \frac{P_2V_2 - P_1V_1}{\gamma - 1}$ For free expansion ($P_{ext}=0$): $w = 0$ 4. Enthalpy ($H$) Definition: $H = U + PV$ Change in Enthalpy: $\Delta H = \Delta U + \Delta (PV)$ For constant pressure process: $\Delta H = q_P$ (Heat exchanged at constant pressure) Relationship between $\Delta H$ and $\Delta U$: $\Delta H = \Delta U + \Delta n_g RT$ (for reactions involving gases) $\Delta n_g = (\text{moles of gaseous products}) - (\text{moles of gaseous reactants})$ Heat Capacity: $C = \frac{q}{\Delta T}$ Specific Heat Capacity ($c$): Heat required to raise temperature of 1 unit mass by $1^\circ C$. $q = mc\Delta T$ Molar Heat Capacity ($C_m$): Heat required to raise temperature of 1 mole by $1^\circ C$. $q = nC_m\Delta T$ At constant volume: $C_V = \left(\frac{\partial U}{\partial T}\right)_V$. $\Delta U = nC_V\Delta T$ At constant pressure: $C_P = \left(\frac{\partial H}{\partial T}\right)_P$. $\Delta H = nC_P\Delta T$ Mayer's Relation (for ideal gas): $C_P - C_V = R$ Ratio of heat capacities: $\gamma = \frac{C_P}{C_V}$ Standard Enthalpy Changes: Standard Enthalpy of Formation ($\Delta H_f^\circ$): Enthalpy change when 1 mole of a compound is formed from its elements in their standard states. ($\Delta H_f^\circ$ of an element in its standard state is 0). Standard Enthalpy of Reaction ($\Delta H_r^\circ$): $\Delta H_r^\circ = \sum \nu_P \Delta H_f^\circ(\text{products}) - \sum \nu_R \Delta H_f^\circ(\text{reactants})$ Standard Enthalpy of Combustion ($\Delta H_c^\circ$): Enthalpy change when 1 mole of a substance is completely burnt in oxygen. Standard Enthalpy of Neutralization ($\Delta H_{neut}^\circ$): For strong acid-strong base, approx. $-57.3 \text{ kJ/mol}$. Hess's Law of Constant Heat Summation: The total enthalpy change for a reaction is independent of the pathway. Bond Enthalpy (Bond Energy): Energy required to break one mole of a particular type of bond. ($\Delta H = \sum BE(\text{reactants}) - \sum BE(\text{products})$) 5. Second Law of Thermodynamics Statement: The entropy of an isolated system always increases in the course of a spontaneous change. Clausius Statement: Heat cannot flow spontaneously 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. Entropy ($S$): Measure of randomness or disorder. State function. Change in Entropy: $\Delta S = \frac{q_{rev}}{T}$ For a spontaneous process in an isolated system: $\Delta S_{total} = \Delta S_{sys} + \Delta S_{surr} > 0$ For a reversible process: $\Delta S_{total} = 0$ For an irreversible process: $\Delta S_{total} > 0$ Third Law of Thermodynamics: The entropy of a perfectly crystalline substance at absolute zero (0 K) is taken to be zero. Standard Entropy of Reaction: $\Delta S_r^\circ = \sum \nu_P S^\circ(\text{products}) - \sum \nu_R S^\circ(\text{reactants})$ 6. Gibbs Free Energy ($G$) Definition: $G = H - TS$ Change in Gibbs Free Energy: $\Delta G = \Delta H - T\Delta S$ (at constant T) Criterion for Spontaneity (at constant T, P): $\Delta G $\Delta G > 0$: Non-spontaneous process (reverse is spontaneous). $\Delta G = 0$: System is at equilibrium. Effect of $\Delta H$ and $\Delta S$ on $\Delta G$: $\Delta H$ $\Delta S$ $\Delta G = \Delta H - T\Delta S$ 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 ($K$): $\Delta G^\circ = -RT \ln K$ (at standard conditions) $\Delta G = \Delta G^\circ + RT \ln Q$ (at non-standard conditions, $Q$ is reaction quotient) At equilibrium, $\Delta G = 0$, so $\Delta G^\circ = -RT \ln K$ 7. Important Processes and Their Characteristics Isothermal Process: Temperature ($T$) is constant. $\Delta T = 0$. For ideal gas: $\Delta U = 0$, $\Delta H = 0$. $q = -w$. Adiabatic Process: No heat exchange ($q=0$). $\Delta U = w$. $PV^\gamma = \text{constant}$. $T_1V_1^{\gamma-1} = T_2V_2^{\gamma-1}$. $P_1^{1-\gamma}T_1^\gamma = P_2^{1-\gamma}T_2^\gamma$. Isobaric Process: Pressure ($P$) is constant. $\Delta H = q_P$. Isochoric Process: Volume ($V$) is constant. $\Delta V = 0$. $w = 0$. $\Delta U = q_V$. Cyclic Process: Returns to initial state. $\Delta U = 0$, $\Delta H = 0$, $\Delta S = 0$, $\Delta G = 0$. $q = -w$. 8. Carnot Cycle (Ideal Heat Engine) Efficiency ($\eta$): $\eta = \frac{W_{net}}{q_{hot}} = 1 - \frac{T_{cold}}{T_{hot}}$ $T_{hot}$ is the temperature of the hot reservoir, $T_{cold}$ is the temperature of the cold reservoir.