Comprehensive Study Notes on Thermodynamics and Chemical Energetics

Introduction to Thermodynamics

• Thermodynamics is the only physical theory of universal content which, within the framework of the applicability of its basic concepts, will never be overthrown, as stated by Albert Einstein.

• Energy stored by molecules as chemical energy can be released as heat during chemical reactions, such as the burning of methane, cooking gas, or coal in air.

• Chemical energy can be utilized to do mechanical work (in an engine) or to provide electrical energy (through a galvanic cell like a dry cell).

• Thermodynamics is the study of the interrelation and transformation of various forms of energy.

• The laws of thermodynamics deal with macroscopic systems involving a large number of molecules rather than microscopic systems containing only a few molecules.

• Thermodynamics focuses on the initial and final states of a system undergoing change, rather than the specific rate at which energy transformations occur.

• The laws apply only when a system is in equilibrium or moving from one equilibrium state to another.

• In an equilibrium state, macroscopic properties such as pressure and temperature do not change over time.

Thermodynamic Terms: System and Surroundings

• A system refers to the specific part of the universe in which observations are being made.

• The surroundings include everything else in the universe other than the system.

• The Universe is the sum of the system and the surroundings: $\text{Universe} = \text{System} + \text{Surroundings}$.

• In practice, surroundings are the portion of the remaining universe that can interact with the system, typically the immediate neighborhood of the system.

• The boundary is the real or imaginary surface that separates the system from the surroundings, allowing for the control and tracking of matter and energy movements.

Types of Thermodynamic Systems

• Open System: In this system, there is an exchange of both energy and matter between the system and the surroundings. An example is reactants kept in an open beaker.

• Closed System: There is no exchange of matter, but an exchange of energy is possible between the system and its surroundings. An example is reactants in a closed vessel made of conducting material (e.g., copper or steel).

• Isolated System: There is no exchange of either energy or matter between the system and the surroundings. An example is reactants kept in a thermos flask or a closed insulated vessel.

The State of the System and State Functions

• To describe a system, quantitative properties such as pressure ($p$), volume ($V$), temperature ($T$), and composition must be specified.

• Thermodynamics deals with average measurable properties rather than the motion of individual particles.

• State Functions (or State Variables): These are macroscopic properties whose values depend only on the current state of the system and not on the path taken to reach that state. Examples include pressure ($p$), volume ($V$), temperature ($T$), and amount ($n$).

• Fixing a minimum number of macroscopic properties automatically defines the values of the remaining properties.

Internal Energy as a State Function

• Internal Energy ($U$): This represents the total energy of a system, including chemical, electrical, mechanical, or any other type of energy. It is a state function.

• Internal energy changes when:

  • Heat passes into or out of the system.

  • Work is done on or by the system.

  • Matter enters or leaves the system.

Work and Adiabatic Processes

• Adiabatic Process: A process in which there is no transfer of heat between the system and surroundings ($q = 0$). The system is separated from its surroundings by an adiabatic wall.

• Experiments by J. P. Joule (1840–1850) showed that a specific amount of work done on a system produces the same change in internal energy regardless of the method (mechanical or electrical).

• Adiabatic Work ($w_{ad}$): The work required to bring about a change of state is equal to the difference in internal energy: $\Delta U = U_2 - U_1 = w_{ad}$.

• Sign Convention for Work:

  • Work done on the system ($w$) is positive ($+$), leading to an increase in internal energy.

  • Work done by the system ($w$) is negative ($-$), leading to a decrease in internal energy.

Heat and Internal Energy Change

• Heat ($q$): The exchange of energy resulting from a temperature difference between the system and surroundings through thermally conducting walls.

• If no work is done at constant volume, the change in internal energy equals the heat absorbed: $\Delta U = q$.

• Sign Convention for Heat:

  • Heat transferred from surroundings to the system ($q$) is positive ($+$), increasing internal energy.

  • Heat transferred from the system to surroundings ($q$) is negative ($-$), decreasing internal energy.

The First Law of Thermodynamics

• The general case for changing the state of a system involves both work and heat: $\Delta U = q + w$.

• This equation is the mathematical statement of the First Law of Thermodynamics, which states that the energy of an isolated system is constant.

• It is also known as the Law of Conservation of Energy: Energy can neither be created nor destroyed.

• While absolute values of internal energy ($U$) cannot be specified, the changes in internal energy ($\Delta U$) can be accurately measured.

Pressure-Volume Work

• Work done during the compression or expansion of a gas is called mechanical or pressure-volume ($pV$) work.

• For a gas in a cylinder with a frictionless piston and cross-sectional area ($A$), if the piston moves a distance ($l$) under external pressure ($p_{ex}$):

  • $\text{Work } (w) = \text{force} \times \text{distance} = p_{ex} \times A \times l$

  • $\Delta V = (V_f - V_i)$. Since compression involves a decrease in volume, the expression is: $w = -p_{ex} \times (V_f - V_i) = -p_{ex} \Delta V$.

• If volume decreases (compression), $\Delta V$ is negative, so $w$ is positive (work done on the system).

• If volume increases (expansion), $\Delta V$ is positive, so $w$ is negative (work done by the system).

Reversible and Irreversible Processes

• Reversible Process: A process that can be reversed at any moment by an infinitesimal change. It proceeds infinitely slowly through a series of equilibrium states where the system and surroundings are always in near equilibrium.

• Irreversible Process: Any process that is not reversible.

• For a reversible process, the work done is: $w_{rev} = - \int_{V_i}^{V_f} p_{in} dV$.

• For an ideal gas at constant temperature (isothermal process):

  • $pV = nRT$

  • $w_{rev} = -2.303 nRT \log\left(\frac{V_f}{V_i}\right)$.

• Free Expansion: The expansion of a gas into a vacuum ($p_{ex} = 0$). Here, $w = 0$. If it is also isothermal, $\Delta U = 0$ and $q = 0$.

Enthalpy: A New State Function

• Most reactions occur at constant pressure rather than constant volume. We define Enthalpy ($H$) as: $H = U + pV$.

• Change in enthalpy at constant pressure is equal to the heat absorbed ($q_p$):

  • $\Delta H = q_p$

  • For finite changes at constant pressure: $\Delta H = \Delta U + p \Delta V$.

• Relationship for reactions involving gases:

  • Using the ideal gas law ($pV = nRT$) at constant temperature and pressure: $p \Delta V = \Delta n_g RT$.

  • Hence, $\Delta H = \Delta U + \Delta n_g RT$, where $\Delta n_g$ is the difference between moles of gaseous products and gaseous reactants.

• Exothermic reactions evolve heat and have a negative $\Delta H$. Endothermic reactions absorb heat and have a positive $\Delta H$.

Extensive and Intensive Properties

• Extensive Property: A property whose value depends on the quantity or size of matter in the system. Examples: mass, volume, internal energy, enthalpy, heat capacity.

• Intensive Property: A property that does not depend on the quantity or size of matter. Examples: temperature, density, pressure.

• Molar Property ($\chi_m$): The value of an extensive property for 1 mole of substance, which is an intensive property ($\chi_m = \frac{\chi}{n}$).

Heat Capacity and Its Specific Forms

• Heat Capacity ($C$): The coefficient relating heat absorbed to temperature rise ($q = C \Delta T$). It is an extensive property.

• Molar Heat Capacity ($C_m$): The heat required to raise the temperature of one mole of a substance by $1^{\circ} C$ (or $1\,K$).

• Specific Heat Property ($c$): The heat required to raise the temperature of one unit mass of a substance by $1^{\circ} C$ (or $1\,K$).

• $q = c \times m \times \Delta T = C \Delta T$.

• Relationship between $C_p$ and $C_V$ for an Ideal Gas:

  • At constant volume: $q_V = C_V \Delta T = \Delta U$.

  • At constant pressure: $q_p = C_p \Delta T = \Delta H$.

  • For one mole of an ideal gas: $\Delta H = \Delta U + \Delta(RT) = \Delta U + R \Delta T$.

  • Substituting heat capacities: $C_p \Delta T = C_V \Delta T + R \Delta T$.

  • Thus: $C_p - C_V = R$.

Calorimetry and Measuring Energy Changes

• Calorimetry is an experimental technique to measure energy changes in a vessel called a calorimeter submerged in a liquid.

• $\Delta U$ Measurements: Carried out in a bomb calorimeter. A combustible substance is burned in oxygen inside a sealed steel vessel (bomb). Because volume is constant ($\Delta V = 0$), no work is done. The temperature change is converted to $q_V$ using the heat capacity of the calorimeter.

• $\Delta H$ Measurements: Carried out at constant pressure. The heat change measured ($q_p$) is the enthalpy of reaction ($\Delta_r H$).

Enthalpy of Reaction ($\Delta_r H$)

• $\Delta_r H = (\sum \text{enthalpies of products}) - (\sum \text{enthalpies of reactants})$.

• Mathematically: $\Delta_r H = \sum_{i} a_i H_{products} - \sum_{i} b_i H_{reactants}$, where $a_i$ and $b_i$ are stoichiometric coefficients.

• Standard Enthalpy of Reaction ($\Delta_r H^{\small o}$): The enthalpy change when all participating substances are in their standard states.

• Standard State: The pure form of a substance at $1\,bar$ pressure and a specified temperature (usually $298\,K$).

Phase Transformations and Their Enthalpies

• Standard Enthalpy of Fusion ($\Delta_{fus} H^{\small o}$): Enthalpy change for melting one mole of a solid substance in its standard state. It is always positive.

• Standard Enthalpy of Vaporization ($\Delta_{vap} H^{\small o}$): Heat required to vaporize one mole of a liquid at constant temperature and standard pressure.

• Standard Enthalpy of Sublimation ($\Delta_{sub} H^{\small o}$): Enthalpy change when one mole of a solid sublimes at constant temperature and standard pressure.

• The magnitude of these changes depends on intermolecular interactions (e.g., hydrogen bonding in water vs. dipole-dipole in acetone).

Standard Molar Enthalpy of Formation ($\Delta_f H^{\small o}$)

• This is the enthalpy change for the formation of one mole of a compound from its constituent elements in their most stable states of aggregation (reference states).

• By convention, $\Delta_f H^{\small o}$ of an element in its reference state is taken as zero.

• Reference states at $25^{\circ} C$ and $1\,bar$: $\text{H}_2\,(g)$, $\text{O}_2\,(g)$, $\text{C}_{graphite}\,(s)$, $\text{S}_{rhombic}\,(s)$.

• $\Delta_r H^{\small o} = \sum a_i \Delta_f H^{\small o}(products) - \sum b_i \Delta_f H^{\small o}(reactants)$.

Hess’s Law of Constant Heat Summation

• Since enthalpy is a state function, the change in enthalpy for a reaction is independent of the path.

• Hess’s Law: If a reaction takes place in several steps, the standard reaction enthalpy is the sum of the standard enthalpies of the intermediate reactions.

• Example: Enthalpy for $\text{C}(graphite) + \frac{1}{2} \text{O}_2(g) \rightarrow \text{CO}(g)$ can be calculated by combining enthalpies of forming $\text{CO}_2$ from carbon and then reacting $\text{CO}$ with oxygen to form $\text{CO}_2$.

Bond Enthalpy and Enthalpy of Atomization

• Enthalpy of Atomization ($\Delta_a H^{\small o}$): Enthalpy change on breaking one mole of bonds completely to obtain gaseous atoms.

• Bond Dissociation Enthalpy: For diatomic molecules, this is the enthalpy required to break one mole of covalent bonds in the gas phase. It is identical to the enthalpy of atomization for diatomic species.

• Mean Bond Enthalpy: In polyatomic molecules like $\text{CH}_4$, breaking successive $\text{C-H}$ bonds requires different amounts of energy. The mean bond enthalpy is the average of these values ($\frac{1}{4} \Delta_a H^{\small o}$ for $\text{CH}_4$).

• $\Delta_r H^{\small o} = \sum (\text{bond enthalpies of reactants}) - \sum (\text{bond enthalpies of products})$.

Lattice Enthalpy and Born-Haber Cycle

• Lattice Enthalpy ($\Delta_{lattice} H^{\small o}$): The enthalpy change when one mole of an ionic compound dissociates into its gaseous ions.

• Lattice enthalpies cannot be measured directly. They are determined via the Born-Haber Cycle, which involves steps such as:

  1. Sublimation of metal ($\Delta_{sub} H^{\small o}$).

  2. Ionization of metal atoms ($\Delta_i H^{\small o}$).

  3. Dissociation of non-metal molecules ($\frac{1}{2} \Delta_{bond} H^{\small o}$).

  4. Electron gain by non-metal atoms ($\Delta_{eg} H^{\small o}$).

  5. Formation of lattice ($\Delta_{lattice} H^{\small o}$).

• Example for $\text{NaCl}(s)$: $\Delta_{lattice} H^{\small o} = +788\,kJ/mol$.

Enthalpies of Solution and Dilution

• Enthalpy of Solution ($\Delta_{sol} H^{\small o}$): The enthalpy change when one mole of a substance dissolves in a specified amount of solvent.

• Infinite Dilution: Enthalpy change when interactions between ions/solute molecules are negligible as they are in an infinite amount of solvent.

• $\Delta_{sol} H^{\small o} = \Delta_{lattice} H^{\small o} + \Delta_{hyd} H^{\small o}$, where $\Delta_{hyd} H^{\small o}$ is the enthalpy of hydration.

• Enthalpy of Dilution: The heat withdrawn from surroundings when additional solvent is added to a solution. It depends on original concentration and the amount of solvent added.

Spontaneity and Entropy

• Spontaneous Process: An irreversible process that proceeds without external assistance. It has the potential to occur but may occur slowly.

• Decrease in enthalpy ($\Delta H < 0$) is a contributory factor to spontaneity (exothermic reactions), but not the sole criterion (some endothermic reactions are spontaneous).

• Entropy ($S$): A thermodynamic function measuring the degree of randomness or disorder in a system. For an isolated system, the tendency to become more disordered drives spontaneous change.

• Entropy Change ($\Delta S$): For a reversible process, $\Delta S = \frac{q_{rev}}{T}$.

• Entropy order: $\text{Solid} < \text{Liquid} < \text{Gas}$.

• For a spontaneous process in an isolated system, $\Delta S_{total} > 0$, where $\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings}$.

The Second and Third Laws of Thermodynamics

• Second Law of Thermodynamics: The entropy of the universe (or an isolated system) increases during a spontaneous process.

• In exothermic reactions, heat released increases the disorder of the surroundings, making $\Delta S_{total}$ positive.

• Third Law of Thermodynamics: The entropy of any pure crystalline substance approaches zero as the temperature approaches absolute zero ($0\,K$).

• This law allows for the calculation of absolute entropy values for pure substances.

Gibbs Energy Change ($\Delta G$)

• Gibbs Function ($G$): Defined as $G = H - TS$. It is an extensive property and a state function.

• Gibbs Equation: at constant temperature, $\Delta G = \Delta H - T \Delta S$.

• Criteria for Spontaneity:

  • If $\Delta G < 0$ (negative), the process is spontaneous.

  • If $\Delta G > 0$ (positive), the process is non-spontaneous.

  • If $\Delta G = 0$, the system is in equilibrium.

• $\Delta G$ represents the net energy available to do useful work, also known as the "free energy" of the reaction.

Gibbs Energy and Equilibrium

• For a reversible chemical reaction $\text{A} + \text{B} \rightleftharpoons \text{C} + \text{D}$, equilibrium occurs when Gibbs energy is at a minimum ($\Delta_r G = 0$).

• Relationship between standard Gibbs energy ($\Delta_r G^{\small o}$) and the equilibrium constant ($K$):

  • $\Delta_r G^{\small o} = -RT \ln K = -2.303 RT \log K$.

• Strongly exothermic reactions ($\Delta_r H^{\small o}$ large and negative) usually have negative $\Delta_r G^{\small o}$ and large $K$, going near completion.

• Strongly endothermic reactions ($\Delta_r H^{\small o}$ large and positive) usually have much smaller $K$ values, forming little product unless compensated by large entropy increases at high temperatures.

Questions and Discussion

• Problem 5.1: Expressing internal energy change.

  • (i) No heat absorbed ($q=0$), work done on system ($w$): $\Delta U = w_{ad}$, wall is adiabatic.

  • (ii) No work done ($w=0$), heat taken out ($-q$): $\Delta U = -q$, walls are thermally conducting.

  • (iii) Work done by system ($w$) and heat supplied ($q$): $\Delta U = q - w$, system is closed.

• Problem 5.2: Two litres of ideal gas expand isothermally into a vacuum. $\text{Work } (w) = -p_{ex}\Delta V = 0(8) = 0$. $\text{Heat } (q) = -w = 0$.

• Problem 5.5: Internal energy change for vaporizing $1\,mol of water at $1\,bar and $100^{\circ} C$:

  • $\Delta H = 41\,kJ/mol$, $\Delta n_g = 1$.

  • $\Delta U = \Delta H - \Delta n_g RT = 41.00\,kJ/mol - (1 \times 8.314 \times 10^{-3} \times 373) = 37.904\,kJ/mol$.

• Problem 5.10: Entropy trends.

  • (i) Liquid crystallizes: Entropy decreases (more ordered).

  • (ii) Crystalline solid 0 K to 115 K: Entropy increases (more disorder/oscillation).

  • (iv) $\text{H}_2(g) \rightarrow 2\text{H}(g)$: Entropy increases (more particles).

• Problem 5.14: $\text{N}_2\text{O}_4$ dissociation at $60^{\circ} C$ ($333\,K$). If $50\% dissociated, mole fractions $x_{\text{N}_2\text{O}_4} = 0.33 and $x_{\text{NO}_2} = 0.67$. Equilibrium constant$K_p = 1.33\,atm$. Standard free energy change$\Delta_r G^{\small o} = -RT \ln K_p = -763.8\,kJ/mol$$.