Thermodynamics Cheatsheet

Cheatsheet Content

### Introduction to Thermodynamics - **Definition:** Study of energy transformations and its relation to macroscopic properties of matter. - **Scope:** Deals with energy changes of macroscopic systems (large number of molecules), not microscopic. - **Focus:** Initial and final states of a system; not concerned with reaction rate or mechanism. - **Conditions:** Applies when a system is in equilibrium or moves between equilibrium states. - **Key Questions Addressed:** Energy changes in reactions, spontaneity, extent of reactions. ### Thermodynamic Terms #### System and Surroundings - **System:** Part of the universe under observation. - **Surroundings:** Everything else in the universe outside the system that can interact with it. - **Universe:** System + Surroundings. - **Boundary:** Real or imaginary wall separating the system from surroundings, controlling matter and energy exchange. #### Types of Systems 1. **Open System:** Exchanges both energy and matter with surroundings (e.g., open beaker with reactants). 2. **Closed System:** Exchanges energy but NOT matter with surroundings (e.g., closed vessel made of conducting material). 3. **Isolated System:** Exchanges NEITHER energy NOR matter with surroundings (e.g., thermos flask). #### State of the System - **State Variables/Functions:** Properties that depend only on the state of the system, not on how that state was reached (path-independent). Examples: Pressure (p), Volume (V), Temperature (T), Internal Energy (U), Enthalpy (H), Entropy (S), Gibbs Energy (G). - **Macroscopic Properties:** Measurable bulk properties (p, V, T, n). - **Defining State:** A minimum number of independent macroscopic properties are sufficient to define the state; others are fixed. #### Internal Energy (U) - **Definition:** Total energy of the system (sum of all forms of energy: chemical, electrical, mechanical, etc.). - **State Function:** U is a state function; only changes ($\Delta U$) can be measured, not absolute values. - **Changes in U:** Occur when: - Heat (q) passes into or out of the system. - Work (w) is done on or by the system. - Matter enters or leaves the system. #### Work (w) - **Adiabatic Process:** No heat exchange between system and surroundings (q = 0). The wall is adiabatic. - $\Delta U = w_{ad}$ - **Sign Convention (IUPAC):** - $w$ is positive (+) when work is done *on* the system (internal energy increases). - $w$ is negative (-) when work is done *by* the system (internal energy decreases). #### Heat (q) - **Definition:** Energy exchange due to temperature difference. - **Sign Convention (IUPAC):** - $q$ is positive (+) when heat is transferred *to* the system (internal energy increases). - $q$ is negative (-) when heat is transferred *from* the system (internal energy decreases). - **Constant Volume:** If no work is done ($\Delta V = 0$), then $\Delta U = q_V$. #### First Law of Thermodynamics - **Statement:** "The energy of an isolated system is constant" or "Energy can neither be created nor destroyed." - **Mathematical Form:** $\Delta U = q + w$ - For a given change in state, $q$ and $w$ can vary, but their sum ($\Delta U$) is constant and independent of the path. - For an isolated system, $w=0$ and $q=0$, so $\Delta U = 0$. ### Applications of Thermodynamics #### Pressure-Volume Work - **Definition:** Mechanical work done by or on a gas. - **Formula:** $w = -p_{ex} \Delta V = -p_{ex}(V_f - V_i)$ - $p_{ex}$: External pressure. - $\Delta V$: Change in volume. - **Compression:** $V_f V_i$, so $\Delta V$ is positive, $w$ is negative (work done *by* system). #### Reversible vs. Irreversible Processes - **Reversible Process:** Proceeds infinitely slowly through a series of equilibrium states; can be reversed by an infinitesimal change. - $w_{rev} = -\int_{V_i}^{V_f} p_{in} dV$ (where $p_{ex} \approx p_{in}$) - **Irreversible Process:** All natural processes are irreversible. - **Isothermal Reversible Expansion (Ideal Gas):** - $\Delta U = 0$ (for ideal gas, internal energy depends only on T) - $q = -w = nRT \ln \left(\frac{V_f}{V_i}\right) = 2.303 nRT \log \left(\frac{V_f}{V_i}\right)$ - **Free Expansion (against vacuum, $p_{ex}=0$):** - $w = 0$ - For ideal gas: $q = 0$, so $\Delta U = 0$. - **Isothermal Irreversible Expansion:** - $q = -w = p_{ex}(V_f - V_i)$ #### Enthalpy (H) - **Definition:** $H = U + pV$ (Heat content at constant pressure). - **State Function:** H is a state function. - **Constant Pressure:** At constant pressure, $\Delta H = q_p$. - $\Delta H = \Delta U + p\Delta V$ - **Relationship between $\Delta H$ and $\Delta U$ (for reactions involving gases):** - $\Delta H = \Delta U + \Delta n_g RT$ - $\Delta n_g$: (moles of gaseous products) - (moles of gaseous reactants). - **Exothermic Reaction:** $\Delta H 0$ (heat absorbed). #### Heat Capacity (C) - **Definition:** Amount of heat required to raise the temperature of a substance by $1^\circ C$ (or $1 K$). - **Formula:** $q = C \Delta T$ - **Molar Heat Capacity ($C_m$):** Heat capacity per mole ($C_m = C/n$). - **Specific Heat Capacity (c):** Heat capacity per unit mass ($q = c \times m \times \Delta T$). - **Heat Capacity at Constant Volume ($C_V$):** - $C_V = \left(\frac{\partial U}{\partial T}\right)_V = \frac{q_V}{\Delta T}$ - **Heat Capacity at Constant Pressure ($C_P$):** - $C_P = \left(\frac{\partial H}{\partial T}\right)_P = \frac{q_P}{\Delta T}$ - **Relationship for Ideal Gas:** $C_P - C_V = R$ #### Extensive and Intensive Properties - **Extensive Properties:** Depend on the quantity of matter (e.g., mass, volume, internal energy, enthalpy, heat capacity). - **Intensive Properties:** Independent of the quantity of matter (e.g., temperature, density, pressure, molar heat capacity, molar volume). #### Calorimetry - **Definition:** Experimental technique to measure energy changes (heat) associated with chemical or physical processes. - **Bomb Calorimeter (Constant Volume):** Measures $\Delta U = q_V$. - Sealed steel vessel (bomb) immersed in water bath. - $q_{reaction} = -C_{calorimeter} \times \Delta T$ - **Calorimeter at Constant Pressure:** Measures $\Delta H = q_P$. - Open to atmosphere, typically simpler design. #### Reaction Enthalpy ($\Delta_r H$) - **Definition:** Enthalpy change accompanying a chemical reaction. - **Formula:** $\Delta_r H = \sum (\text{enthalpies of products}) - \sum (\text{enthalpies of reactants})$ - $\Delta_r H = \sum a_i H_{products} - \sum b_i H_{reactants}$ ($a_i, b_i$ are stoichiometric coefficients). #### Standard Enthalpy of Reactions ($\Delta_r H^\circ$) - **Standard State:** Pure form of a substance at 1 bar pressure and specified temperature (usually 298 K). - **Notation:** Superscript '$\circ$' denotes standard conditions. #### Enthalpy Changes during Phase Transformations - **Standard Enthalpy of Fusion ($\Delta_{fus} H^\circ$):** Heat absorbed to melt one mole of solid at its melting point and 1 bar. (Always positive). - H$_2$O(s) $\rightarrow$ H$_2$O(l); $\Delta_{fus} H^\circ = +6.00 \text{ kJ mol}^{-1}$ - **Standard Enthalpy of Vaporization ($\Delta_{vap} H^\circ$):** Heat absorbed to vaporize one mole of liquid at its boiling point and 1 bar. (Always positive). - H$_2$O(l) $\rightarrow$ H$_2$O(g); $\Delta_{vap} H^\circ = +40.79 \text{ kJ mol}^{-1}$ - **Standard Enthalpy of Sublimation ($\Delta_{sub} H^\circ$):** Heat absorbed to directly convert one mole of solid to gas at 1 bar. - $\Delta_{sub} H^\circ = \Delta_{fus} H^\circ + \Delta_{vap} H^\circ$ #### Standard Enthalpy of Formation ($\Delta_f H^\circ$) - **Definition:** Enthalpy change when one mole of a compound is formed from its elements in their most stable states of aggregation (reference states) at 1 bar and specified temperature. - **Reference State:** Most stable state of an element at 25°C and 1 bar (e.g., H$_2$(g), O$_2$(g), C(graphite)). - **Convention:** $\Delta_f H^\circ$ of an element in its reference state is zero. - **Calculation of $\Delta_r H^\circ$ from $\Delta_f H^\circ$:** - $\Delta_r H^\circ = \sum (\Delta_f H^\circ \text{ products}) - \sum (\Delta_f H^\circ \text{ reactants})$ #### Thermochemical Equations - **Definition:** Balanced chemical equation including the physical states (and allotropic states) of reactants and products, and the $\Delta_r H$ value. - **Conventions:** 1. Coefficients refer to moles. 2. $\Delta_r H^\circ$ unit is kJ mol$^{-1}$ (per mole of reaction as written). 3. Reversing equation reverses sign of $\Delta_r H^\circ$. 4. Multiplying equation by a factor multiplies $\Delta_r H^\circ$ by the same factor (enthalpy is extensive). #### Hess's Law of Constant Heat Summation - **Statement:** The total enthalpy change for a reaction is the same regardless of whether the reaction occurs in one step or in several steps. (Enthalpy is a state function). - **Application:** If a reaction can be expressed as the sum of several steps, then $\Delta_r H$ for the overall reaction is the sum of the $\Delta_r H$ values for the individual steps. #### Standard Enthalpy of Combustion ($\Delta_c H^\circ$) - **Definition:** Enthalpy change per mole of a substance when it undergoes complete combustion with all reactants and products in their standard states. (Combustion reactions are typically exothermic, $\Delta_c H^\circ ### Spontaneity #### Criteria for Spontaneity - **First Law Limitation:** The first law explains energy conservation but not the direction of spontaneous processes. - **Spontaneous Process:** Occurs without external assistance. Does not imply speed. Irreversible. - **Driving Force:** Not solely decrease in enthalpy (some endothermic reactions are spontaneous). #### Entropy (S) - **Definition:** A measure of the degree of randomness or disorder in a system. - **State Function:** S is a state function. - **Factors Affecting Entropy:** - **Physical State:** $S_{gas} > S_{liquid} > S_{solid}$ (due to increasing disorder). - **Temperature:** Increasing temperature increases entropy (more vigorous molecular motion). - **Volume (for gases):** Increasing volume increases entropy (more space for disorder). - **Number of Particles:** Increasing number of gaseous particles increases entropy. - **Second Law of Thermodynamics:** For a spontaneous process in an isolated system, the total entropy change ($\Delta S_{total}$) is always positive. - $\Delta S_{total} = \Delta S_{sys} + \Delta S_{surr} > 0$ - **Equilibrium:** At equilibrium, $\Delta S_{total} = 0$, and entropy is maximum. - **Quantifying Entropy Change (Reversible Process):** $\Delta S_{sys} = \frac{q_{rev}}{T}$ - **Entropy Change of Surroundings:** For an exothermic reaction, heat released to surroundings increases $S_{surr}$. For an endothermic reaction, heat absorbed from surroundings decreases $S_{surr}$. - $\Delta S_{surr} = -\frac{\Delta H_{sys}}{T}$ #### Gibbs Energy (G) - **Definition:** A thermodynamic potential that measures the "useful" or process-initiating work obtainable from an isothermal, isobaric thermodynamic system. - **Formula:** $G = H - TS$ - **State Function:** G is a state function and an extensive property. - **Gibbs Energy Change ($\Delta G$):** For a process at constant temperature and pressure: - $\Delta G = \Delta H - T\Delta S$ (Gibbs-Helmholtz Equation) - **Criteria for Spontaneity (Constant T, P):** - **$\Delta G 0$:** Process is non-spontaneous (endergonic). - **$\Delta G = 0$:** System is at equilibrium. - **Interpretation of $\Delta G$:** Represents the net energy available to do useful work. $T\Delta S$ is the energy unavailable for useful work. #### Effect of Temperature on Spontaneity | $\Delta H$ | $\Delta S$ | $\Delta G = \Delta H - T\Delta S$ | Spontaneity | |:----------:|:----------:|:---------------------------------:|:------------| | $-$ | $+$ | $-$ | Spontaneous at all T | | $-$ | $-$ | $-$ (low T) | Spontaneous at low T | | $-$ | $-$ | $+$ (high T) | Non-spontaneous at high T | | $+$ | $+$ | $+$ (low T) | Non-spontaneous at low T | | $+$ | $+$ | $-$ (high T) | Spontaneous at high T | | $+$ | $-$ | $+$ | Non-spontaneous at all T | #### Third Law of Thermodynamics - **Statement:** "The entropy of any pure crystalline substance approaches zero as the temperature approaches absolute zero (0 K)." - **Significance:** Allows calculation of absolute entropy values for pure substances from thermal data. Entropy of solutions and supercooled liquids is not zero at 0 K. ### Gibbs Energy Change and Equilibrium - **Equilibrium Condition:** For a reversible reaction at equilibrium, $\Delta_r G = 0$. - **Standard Gibbs Energy Change ($\Delta_r G^\circ$) and Equilibrium Constant (K):** - $\Delta_r G^\circ = -RT \ln K$ - $\Delta_r G^\circ = -2.303 RT \log K$ - Where R is the gas constant, T is the absolute temperature. - **Relationship between $\Delta_r G^\circ$ and K:** - If $\Delta_r G^\circ 1$ (products favored at equilibrium). - If $\Delta_r G^\circ > 0$, then $K