Thermodynamics is the branch of science that deals with the study of energy transformations. It specifically focuses on energy changes in macroscopic systems (those involving a large number of particles) rather than microscopic ones. Thermodynamics is concerned with the initial and final states of a system undergoing a change, not the pathway or rate of the change. Its laws apply when a system is in equilibrium or moves between equilibrium states. Macroscopic properties like pressure and temperature are constant for a system in equilibrium.

Key questions addressed by thermodynamics in chemistry include: how to determine energy changes in reactions, whether a reaction will occur spontaneously, what drives a spontaneous process, and to what extent a reaction will proceed.

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Forms Of Energy And Their Interrelation

Molecules store chemical energy, which can be released during reactions (e.g., combustion). Energy can be transformed between various forms, such as chemical, thermal (heat), mechanical (work), or electrical energy.

Thermodynamic Terms

To study energy changes in chemical systems, it's essential to define some fundamental thermodynamic terms.

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The System And The Surroundings

The system is the specific part of the universe being studied, where observations are made. The surroundings are everything else in the universe outside the system. Together, the system and surroundings constitute the universe (Universe = System + Surroundings).

For practical purposes, the surroundings usually refer to the region near the system that can interact with it. The system is separated from the surroundings by a boundary, which can be real (like the walls of a container) or imaginary, allowing us to track matter and energy exchange.

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Types Of The System

Systems are classified based on the exchange of matter and energy with their surroundings:

1. Open System: Exchanges both matter and energy with the surroundings. Example: Reactants in an open beaker.
2. Closed System: Exchanges energy but not matter with the surroundings. Example: Reactants in a sealed container made of a conducting material (like metal).
3. Isolated System: Exchanges neither matter nor energy with the surroundings. Example: Reactants in a sealed, insulated container (like a thermos flask).

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The State Of The System

The state of a thermodynamic system is defined by its measurable or macroscopic properties, such as pressure ($p$), volume ($V$), temperature ($T$), and composition (amount of substance, $n$). These properties are called state variables or state functions because their values depend only on the current state of the system, not on the path taken to reach that state. Once a minimum number of independent state variables are fixed, the values of other properties are also fixed.

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The Internal Energy As A State Function

The total energy of a system, encompassing all forms of energy (chemical, kinetic, potential, etc.), is called its Internal Energy (U). The internal energy is a state function, meaning its value depends only on the system's state and not on the process that brought it to that state. The internal energy of a system can change through the transfer of heat, work done on or by the system, or the transfer of matter.

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Work

One way to change the internal energy of a system is by doing work ($w$) on or by the system. An adiabatic process is one where no heat exchange occurs with the surroundings. For an adiabatic process, the change in internal energy ($\Delta U$) is equal to the adiabatic work ($w_{ad}$) done on the system: $\Delta U = U_{final} - U_{initial} = w_{ad}$. J.P. Joule's experiments showed that the same amount of work, regardless of the method, produced the same change in the state of the system (measured by temperature change), confirming internal energy as a state function. By IUPAC convention, $w$ is positive when work is done on the system (internal energy increases) and negative when work is done by the system (internal energy decreases).

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Heat

Another way to change internal energy is by transferring heat ($q$) across the system boundary due to a temperature difference between the system and surroundings. This occurs through thermally conducting walls. For a system at constant volume where no work is done, the change in internal energy ($\Delta U$) is equal to the heat ($q_V$) transferred: $\Delta U = q_V$. By IUPAC convention, $q$ is positive when heat is transferred to the system (internal energy increases) and negative when heat is transferred from the system (internal energy decreases).

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The General Case

When the state of a system changes due to both heat transfer and work, the total change in internal energy is given by the sum of heat and work:

$$ \Delta U = q + w $$

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 a restatement of the law of conservation of energy: energy cannot be created or destroyed, only transferred or transformed.

For a given change in state, the values of $q$ and $w$ can vary depending on the process (path), but their sum ($\Delta U$) is constant and path-independent because $U$ is a state function.

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Applications

Chemical reactions often involve energy changes in the form of heat and work (especially if gases are produced or consumed). Thermodynamics helps quantify these changes.

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Work

In chemical thermodynamics, the most common type of work is pressure-volume (PV) work, which occurs when a system expands or contracts against an external pressure. For a gas expanding or being compressed in a cylinder with a piston:

Work done on the system ($w$) = $-p_{ex} \Delta V = -p_{ex} (V_f - V_i)$, where $p_{ex}$ is the external pressure, $V_i$ is the initial volume, and $V_f$ is the final volume.

According to the sign convention, compression ($V_f < V_i$, $\Delta V$ is negative) results in positive work (work done on the system). Expansion ($V_f > V_i$, $\Delta V$ is positive) results in negative work (work done by the system).

If the process involves multiple steps or occurs reversibly (where the external pressure is always infinitesimally close to the internal pressure, $p_{ex} \approx p_{in}$), the work must be calculated by integration:

$$ w = -\int_{V_i}^{V_f} p_{ex} dV $$

For a reversible process of an ideal gas at constant temperature (isothermal): $p_{ex} = p_{in} = \frac{nRT}{V}$.

$$ w_{rev} = -\int_{V_i}^{V_f} \frac{nRT}{V} dV = -nRT \int_{V_i}^{V_f} \frac{1}{V} dV = -nRT [\ln V]_{V_i}^{V_f} = -nRT \ln \left(\frac{V_f}{V_i}\right) $$ $$ w_{rev} = -2.303 nRT \log \left(\frac{V_f}{V_i}\right) $$

Free expansion is expansion against zero external pressure ($p_{ex}=0$), which occurs in a vacuum. In free expansion, $w=0$. For an ideal gas, free expansion is also isothermal ($\Delta U=0$).

Reversible Process: A process that can be reversed at any point by an infinitesimal change in conditions, proceeding infinitely slowly through a series of equilibrium states.

Irreversible Process: Any process that is not reversible, typically occurring rapidly or in a single step against a constant external pressure.

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Enthalpy, H

Most chemical reactions are carried out at constant atmospheric pressure, not constant volume. To quantify heat changes under constant pressure conditions, a new state function called Enthalpy (H) is defined.

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A Useful New State Function

Enthalpy (H) is defined as the sum of internal energy (U) and the product of pressure (p) and volume (V):

$$ H = U + pV $$

Since U, p, and V are state functions, H is also a state function. For a process occurring at constant pressure ($\Delta p = 0$), the change in enthalpy ($\Delta H$) is given by:

$$ \Delta H = \Delta U + p\Delta V $$

Combining this with the First Law ($\Delta U = q + w$) and the definition of PV work ($w = -p\Delta V$) when only PV work is involved:

$\Delta H = (q + w) + p\Delta V = (q - p\Delta V) + p\Delta V = q$

So, for a process at constant pressure, the heat absorbed ($q_p$) is equal to the change in enthalpy:

$$ \Delta H = q_p $$

This is a very useful relationship in chemistry. If heat is absorbed by the system at constant pressure (endothermic reaction), $\Delta H$ is positive. If heat is released by the system (exothermic reaction), $\Delta H$ is negative. For reactions involving only solids or liquids, volume changes are usually small, so $p\Delta V$ is negligible, and $\Delta H \approx \Delta U$. The difference is significant when gases are involved.

For reactions involving gases at constant temperature and pressure, the volume change ($\Delta V$) is related to the change in the number of moles of gas ($\Delta n_g$) by the ideal gas law: $p\Delta V = \Delta n_g RT$. Here, $\Delta n_g = (\text{moles of gaseous products}) - (\text{moles of gaseous reactants})$.

Substituting this into the enthalpy equation:

$$ \Delta H = \Delta U + \Delta n_g RT $$

This equation allows interconversion between $\Delta H$ and $\Delta U$ for reactions involving gases.

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Extensive And Intensive Properties

Thermodynamic properties can be classified based on their dependence on the amount of substance:

• Extensive Properties: Properties that depend on the quantity or size of matter in the system. Examples: mass, volume, internal energy (U), enthalpy (H), heat capacity (C). If you double the amount of substance, the value of the extensive property typically doubles.
• Intensive Properties: Properties that do not depend on the quantity or size of matter. Examples: temperature (T), pressure (p), density, concentration, molar volume ($V_m$, volume per mole), molar heat capacity ($C_m$, heat capacity per mole). These properties are characteristic of the substance itself, regardless of how much is present.

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Heat Capacity

When heat is transferred to a system, its temperature usually changes. The increase in temperature ($\Delta T$) is proportional to the heat transferred ($q$). The proportionality constant is the heat capacity (C):

$$ q = C \Delta T $$

Heat capacity is an extensive property; it depends on the amount of substance and its composition. A larger heat capacity means more heat is needed to cause a given temperature rise.

• Molar heat capacity ($C_m$): Heat capacity per mole of substance ($C_m = C/n$). This is an intensive property, representing the heat needed to raise the temperature of one mole by one degree Celsius (or Kelvin).
• Specific heat (or specific heat capacity, $c$): Heat capacity per unit mass of substance ($c = C/m$). This is also an intensive property, representing the heat needed to raise the temperature of one gram by one degree Celsius (or Kelvin). $q = c \times m \times \Delta T$.

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The Relationship Between Cp And CV For An Ideal Gas

Heat capacity can be measured under constant volume ($C_V$) or constant pressure ($C_p$) conditions.

• At constant volume: $\Delta U = q_V = C_V \Delta T$.
• At constant pressure: $\Delta H = q_p = C_p \Delta T$.

For an ideal gas, the relationship between $C_p$ and $C_V$ is:

$$ C_p - C_V = R $$

where R is the ideal gas constant. This difference arises because, at constant pressure, some of the heat added is used to do expansion work, whereas at constant volume, all the heat added increases the internal energy (and temperature).

Measurement Of ΔU And ΔH: Calorimetry

Calorimetry is the experimental technique used to measure the heat changes ($q$) associated with chemical or physical processes. A calorimeter is a device where the process is carried out, typically immersed in a liquid of known heat capacity. By measuring the temperature change of the liquid and the calorimeter, the heat absorbed or evolved can be determined using the heat capacity formula ($q = C \Delta T$). Measurements are usually done under constant volume or constant pressure conditions.

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ΔU Measurements

Heat changes at constant volume ($q_V = \Delta U$) are measured using a bomb calorimeter. This consists of a sealed steel vessel (the "bomb") where the reaction (often combustion) takes place, immersed in a water bath within an insulated container (the calorimeter). Since the bomb is sealed, the volume is constant ($\Delta V = 0$), so no PV work is done ($w=0$). The heat released or absorbed by the reaction changes the temperature of the bomb and the surrounding water. By knowing the total heat capacity of the calorimeter (bomb + water), the heat of reaction at constant volume ($q_V = \Delta U$) is calculated from the measured temperature change.

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ΔH Measurements

Heat changes at constant pressure ($q_p = \Delta H$) are measured using a different type of calorimeter, often a simple insulated container open to the atmosphere (or maintained at constant pressure). This is suitable for reactions in solution or those not involving significant gas volume changes at constant pressure. The heat absorbed or evolved ($q_p = \Delta H$) is calculated from the temperature change of the solution and the calorimeter, using their heat capacities.

For an exothermic reaction, the system releases heat, $q_p$ is negative, and $\Delta H$ is negative. For an endothermic reaction, the system absorbs heat, $q_p$ is positive, and $\Delta H$ is positive.

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Enthalpy Change, ΔrH Of A Reaction – Reaction Enthalpy

The enthalpy change of a chemical reaction, denoted as $\Delta_r H$, is the difference between the sum of the enthalpies of the products and the sum of the enthalpies of the reactants, considering their stoichiometry.

For a general reaction: $aA + bB \rightarrow cC + dD$

$$ \Delta_r H = \sum (\text{enthalpies of products}) - \sum (\text{enthalpies of reactants}) $$ $$ \Delta_r H = [c H_m(C) + d H_m(D)] - [a H_m(A) + b H_m(B)] $$

where $H_m$ is the molar enthalpy of the substance, and $a, b, c, d$ are the stoichiometric coefficients.

$\Delta_r H$ is negative for exothermic reactions (heat released) and positive for endothermic reactions (heat absorbed).

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Standard Enthalpy Of Reactions

The standard enthalpy of reaction ($\Delta_r H^\ominus$) is the enthalpy change for a reaction carried out under standard conditions. The standard state of a substance at a specified temperature (usually 298 K) is its pure form at 1 bar pressure. Standard state is indicated by the superscript $\ominus$ (e.g., $\Delta H^\ominus$).

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Enthalpy Changes During Phase Transformations

Changes in the physical state of a substance (phase transformations) also involve enthalpy changes. These typically occur at constant temperature and pressure.

• Standard enthalpy of fusion ($\Delta_{fus} H^\ominus$): Enthalpy change when one mole of a solid melts at its melting point under standard pressure. Melting is endothermic, so $\Delta_{fus} H^\ominus$ is always positive. (e.g., H₂O(s) $\rightarrow$ H₂O(l); $\Delta_{fus} H^\ominus = +6.00$ kJ mol⁻¹ at 273 K). Freezing is the reverse process with $\Delta_{freeze} H^\ominus = -\Delta_{fus} H^\ominus$.
• Standard enthalpy of vaporization ($\Delta_{vap} H^\ominus$): Enthalpy change when one mole of a liquid vaporizes at its boiling point under standard pressure. Vaporization is endothermic, so $\Delta_{vap} H^\ominus$ is always positive. (e.g., H₂O(l) $\rightarrow$ H₂O(g); $\Delta_{vap} H^\ominus = +40.79$ kJ mol⁻¹ at 373 K). Condensation is the reverse process with $\Delta_{cond} H^\ominus = -\Delta_{vap} H^\ominus$.
• Standard enthalpy of sublimation ($\Delta_{sub} H^\ominus$): Enthalpy change when one mole of a solid directly converts to its vapor at a constant temperature and standard pressure. Sublimation is endothermic, so $\Delta_{sub} H^\ominus$ is always positive. (e.g., CO₂(s) $\rightarrow$ CO₂(g); $\Delta_{sub} H^\ominus = +25.2$ kJ mol⁻¹ at 195 K).

These enthalpy changes reflect the strength of intermolecular forces; substances with stronger forces have higher enthalpies of fusion and vaporization.

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Standard Enthalpy Of Formation (Symbol : ΔfH)

The standard molar enthalpy of formation ($\Delta_f H^\ominus$) of a compound is the enthalpy change that occurs when one mole of the compound is formed from its constituent elements in their most stable states of aggregation (standard states) at a specified temperature (usually 298 K) and 1 bar pressure.

The most stable state of aggregation for an element at 298 K and 1 bar is its standard state. For example:

• Hydrogen: H₂(g)
• Oxygen: O₂(g)
• Carbon: C(graphite, s)
• Sulfur: S(rhombic, s)
• Bromine: Br₂(l)
• Iodine: I₂(s)
• Metals: Solid state (e.g., Na(s), Fe(s))

By definition, the standard enthalpy of formation of an element in its most stable standard state is taken as zero ($\Delta_f H^\ominus \text{ (element in standard state)} = 0$).

Example formation reactions:

• H₂(g) + ½O₂(g) $\rightarrow$ H₂O(l); $\Delta_f H^\ominus(\text{H}_2\text{O, l})$
• C(graphite, s) + 2H₂(g) $\rightarrow$ CH₄(g); $\Delta_f H^\ominus(\text{CH}_4\text{, g})$

$\Delta_f H^\ominus$ is a specific type of reaction enthalpy where the reactants are defined as elements in their standard states, and the product is one mole of the compound in its standard state.

Standard enthalpies of formation are used to calculate the standard enthalpy change of any reaction using the formula derived from Hess's Law:

$$ \Delta_r H^\ominus = \sum (\text{stoich. coeff. of products}) \times \Delta_f H^\ominus (\text{products}) - \sum (\text{stoich. coeff. of reactants}) \times \Delta_f H^\ominus (\text{reactants}) $$

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Thermochemical Equations

A thermochemical equation is a balanced chemical equation that includes the physical states of all reactants and products and the enthalpy change ($\Delta_r H$) for the reaction as written.

Conventions for thermochemical equations:

1. Coefficients refer to the number of moles of substances.
2. Physical states (s, l, g, aq) and allotropic forms (e.g., graphite, diamond for carbon) are specified.
3. The sign of $\Delta_r H$ indicates whether the reaction is exothermic (-) or endothermic (+).
4. If the equation is reversed, the sign of $\Delta_r H$ is reversed.
5. If the stoichiometric coefficients are multiplied by a factor, $\Delta_r H$ is also multiplied by the same factor (enthalpy is an extensive property).

Example: C₂H₅OH(l) + 3O₂(g) $\rightarrow$ 2CO₂(g) + 3H₂O(l); $\Delta_r H^\ominus = -1367$ kJ mol⁻¹

This indicates that when 1 mole of liquid ethanol reacts completely with 3 moles of oxygen gas to produce 2 moles of carbon dioxide gas and 3 moles of liquid water at standard conditions, 1367 kJ of heat is released.

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Hess’s Law Of Constant Heat Summation

Hess's Law states that the total enthalpy change for a reaction is the same whether the reaction occurs in one step or in a series of steps, provided the initial and final states are the same. This is a direct consequence of enthalpy being a state function.

If a reaction A $\rightarrow$ B has an enthalpy change $\Delta_r H$, and this reaction can also occur through a series of steps A $\rightarrow$ C $\rightarrow$ D $\rightarrow$ B with enthalpy changes $\Delta H_1, \Delta H_2, \Delta H_3$, respectively, then:

$$ \Delta_r H = \Delta H_1 + \Delta H_2 + \Delta H_3 $$

Hess's law is extremely useful for calculating the enthalpy changes of reactions that are difficult or impossible to measure directly by calorimetry (e.g., formation of carbon monoxide from graphite).

Enthalpies For Different Types Of Reactions

Various types of reactions have specific enthalpy changes associated with them, and these are given specific names.

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Standard Enthalpy Of Combustion (Symbol : ΔcH)

The standard enthalpy of combustion ($\Delta_c H^\ominus$) is the enthalpy change when one mole of a substance undergoes complete combustion (typically reacting with excess oxygen) in its standard state to form products in their standard states under standard conditions. Combustion reactions are usually highly exothermic ($\Delta_c H^\ominus$ is negative).

Example: C₄H₁₀(g) + ¹³/₂ O₂(g) $\rightarrow$ 4CO₂(g) + 5H₂O(l); $\Delta_c H^\ominus = -2658.0$ kJ mol⁻¹ (Combustion of butane)

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Enthalpy Of Atomization (Symbol: ΔaH)

The enthalpy of atomization ($\Delta_a H^\ominus$) is the enthalpy change when one mole of a substance is completely converted into gaseous atoms in their standard states. This involves breaking all bonds in the substance.

Example: H₂(g) $\rightarrow$ 2H(g); $\Delta_a H^\ominus = +435.0$ kJ mol⁻¹

Example: CH₄(g) $\rightarrow$ C(g) + 4H(g); $\Delta_a H^\ominus = +1665$ kJ mol⁻¹

For diatomic molecules, $\Delta_a H^\ominus$ is equal to the bond dissociation enthalpy.

For solids like metals, atomization is the conversion of the solid into gaseous atoms:

Na(s) $\rightarrow$ Na(g); $\Delta_a H^\ominus = +108.4$ kJ mol⁻¹ (This is also enthalpy of sublimation for metals)

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Bond Enthalpy (Symbol: ΔbondH)

Chemical reactions involve breaking bonds in reactants and forming new bonds in products. Energy is absorbed to break bonds and released when bonds are formed. Bond enthalpy relates to the strength of chemical bonds.

• Bond dissociation enthalpy: The enthalpy change required to break one specific bond in a gaseous molecule. For diatomic molecules, this is the same as enthalpy of atomization (e.g., H₂ $\rightarrow$ 2H, $\Delta H = \Delta_{H-H} H^\ominus$). For polyatomic molecules, the energy to break the same type of bond can vary depending on the environment (e.g., breaking the first C-H bond in CH₄ is different from breaking the fourth).
• Mean bond enthalpy: In polyatomic molecules with multiple bonds of the same type (e.g., C-H bonds in CH₄), the mean bond enthalpy is the average energy required to break that type of bond. It is calculated as the total atomization enthalpy of the molecule divided by the number of bonds of that type. For CH₄, mean C-H bond enthalpy = $\Delta_a H^\ominus(\text{CH}_4) / 4 = 1665 / 4 = 416$ kJ mol⁻¹.

Mean bond enthalpies are approximately constant for a given bond type across different molecules and can be used to estimate the enthalpy of reaction for gas-phase reactions:

$$ \Delta_r H^\ominus \approx \sum (\text{mean bond enthalpies of reactants}) - \sum (\text{mean bond enthalpies of products}) $$

This is an approximation because mean bond enthalpies are averages and may differ slightly from actual bond dissociation energies in specific molecules.

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Lattice Enthalpy

The Lattice Enthalpy ($\Delta_{lattice} H^\ominus$) of an ionic compound is the enthalpy change when one mole of the solid ionic compound separates into its constituent gaseous ions. This process is highly endothermic ($\Delta_{lattice} H^\ominus$ is positive). It's the reverse of the formation of the ionic lattice from gaseous ions.

Example: NaCl(s) $\rightarrow$ Na⁺(g) + Cl⁻(g); $\Delta_{lattice} H^\ominus = +788$ kJ mol⁻¹.

Lattice enthalpy cannot be measured directly. It is calculated indirectly using Hess's Law in a series of steps called the Born-Haber Cycle. A Born-Haber cycle links the standard enthalpy of formation of an ionic compound to the enthalpy changes for sublimation (or vaporization), ionization, dissociation, electron gain, and lattice formation of the constituent elements and ions.

Applying Hess's Law to the cycle (sum of enthalpy changes around the cycle is zero) allows calculation of the unknown lattice enthalpy.

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Enthalpy Of Solution (Symbol : ΔsolH )

The enthalpy of solution ($\Delta_{sol} H^\ominus$) is the enthalpy change when one mole of a substance dissolves in a specified amount of solvent. For ionic compounds, dissolution involves breaking the crystal lattice (requiring energy, related to lattice enthalpy) and hydrating/solvating the ions (releasing energy, enthalpy of hydration/solvation, $\Delta_{hyd} H^\ominus$).

For an ionic compound AB(s) dissolving in water:

AB(s) $\rightarrow$ A⁺(g) + B⁻(g); $\Delta H = \Delta_{lattice} H^\ominus$ (Lattice dissociation)

A⁺(g) + B⁻(g) + aq $\rightarrow$ A⁺(aq) + B⁻(aq); $\Delta H = \Delta_{hyd} H^\ominus$ (Hydration of ions)

Overall: AB(s) + aq $\rightarrow$ A⁺(aq) + B⁻(aq); $\Delta_{sol} H^\ominus = \Delta_{lattice} H^\ominus + \Delta_{hyd} H^\ominus$

If $\Delta_{sol} H^\ominus$ is positive, dissolution is endothermic (solubility often increases with temperature). If it's negative, dissolution is exothermic. High lattice enthalpy generally leads to lower solubility.

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Enthalpy Of Dilution

The enthalpy of dilution is the enthalpy change that occurs when an additional amount of solvent is added to a solution of a given concentration. It represents the heat absorbed or released when the solution is diluted. The enthalpy of dilution depends on the initial concentration and the amount of solvent added.

Spontaneity

The First Law of Thermodynamics deals with energy conservation but doesn't predict whether a process will occur spontaneously or the direction of a spontaneous change. Many natural processes occur spontaneously in one direction only (e.g., gas expansion, heat flow from hot to cold, a stone falling downhill).

A spontaneous process is one that has the potential to proceed without continuous external assistance. It does not imply the process is fast. Spontaneous processes are generally irreversible; they can only be reversed by intervention from the surroundings.

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Is Decrease In Enthalpy A Criterion For Spontaneity ?

Many spontaneous processes are exothermic (release heat, $\Delta H < 0$), leading to a decrease in enthalpy (e.g., combustion, neutralization). This suggests that a decrease in energy might be a driving force for spontaneity.

However, some spontaneous processes are endothermic (absorb heat, $\Delta H > 0$), like the dissolution of some salts (e.g., NH₄NO₃ in water makes the water colder) or the reaction N₂(g) + O₂(g) $\rightarrow$ 2NO(g) ($\Delta H$ is positive). This shows that while a decrease in enthalpy is often associated with spontaneity, it is not the sole criterion.

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Entropy And Spontaneity

If not just enthalpy, what else drives spontaneity? Consider processes with $\Delta H = 0$, like the mixing of two gases by diffusion in an isolated container. This process is spontaneous but involves no enthalpy change.

Mixing increases the disorder or randomness of the system. Before mixing, you know which gas molecule is in which compartment. After mixing, the molecules are distributed throughout, and the system is less ordered and more unpredictable. This suggests that a driving force for spontaneity is the tendency towards increased disorder or randomness.

A thermodynamic function called Entropy (S) is introduced as a measure of the degree of randomness or disorder in a system. Entropy is a state function; its change ($\Delta S$) depends only on the initial and final states. Solids have the lowest entropy (most ordered), and gases have the highest entropy (most disordered). An increase in the number of particles during a reaction also generally increases entropy.

Entropy change ($\Delta S$) is related to heat transfer ($q$) and temperature ($T$). For a reversible process, the entropy change of the system is given by:

$$ \Delta S_{sys} = \frac{q_{rev}}{T} $$

For a spontaneous (irreversible) process, the total entropy change of the universe (system + surroundings) must be positive:

$$ \Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings} > 0 $$

For a system at equilibrium, $\Delta S_{total} = 0$.

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Gibbs Energy And Spontaneity

For processes occurring at constant temperature and pressure (common in chemistry labs), a more convenient criterion for spontaneity is the Gibbs energy (G), also known as Gibbs function or free energy. It combines enthalpy and entropy and is defined as:

$$ G = H - TS $$

Gibbs energy is a state function and an extensive property. For a process at constant temperature, the change in Gibbs energy ($\Delta G$) is related to the changes in enthalpy ($\Delta H$) and entropy ($\Delta S$) of the system by the Gibbs equation:

$$ \Delta G = \Delta H - T\Delta S $$

This equation is derived from the second law of thermodynamics applied to constant temperature and pressure conditions. The sign of $\Delta G$ provides a criterion for spontaneity:

• If $\Delta G < 0$ (negative), the process is spontaneous (proceeds in the forward direction).
• If $\Delta G > 0$ (positive), the process is non-spontaneous (spontaneous in the reverse direction).
• If $\Delta G = 0$, the system is at equilibrium (no net change).

$\Delta G$ represents the maximum amount of non-expansion work that can be extracted from a process at constant temperature and pressure. It is often referred to as "free energy" because it's the energy available to do useful work.

The sign of $\Delta G$ depends on the signs of $\Delta H$ and $\Delta S$ and the temperature (T > 0 K). The effect of $\Delta H$, $\Delta S$, and T on spontaneity is summarized in the table below:

$\Delta_r H$	$\Delta_r S$	$\Delta_r G = \Delta_r H - T\Delta_r S$	Spontaneity
Negative (-)	Positive (+)	Negative (-) at all T	Spontaneous at all temperatures
Negative (-)	Negative (-)	Negative (-) at low T Positive (+) at high T	Spontaneous at low temperatures Non-spontaneous at high temperatures
Positive (+)	Positive (+)	Positive (+) at low T Negative (-) at high T	Non-spontaneous at low temperatures Spontaneous at high temperatures
Positive (+)	Negative (-)	Positive (+) at all T	Non-spontaneous at all temperatures

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Entropy And Second Law Of Thermodynamics

The Second Law of Thermodynamics can be stated in several ways, but a key implication is that for any spontaneous process occurring in an isolated system, the entropy of the system increases ($\Delta S_{system} > 0$). Since an isolated system has no energy or matter exchange, $\Delta S_{total} = \Delta S_{system}$, so $\Delta S_{total} > 0$ for spontaneous processes in isolated systems. This law explains why natural processes tend towards states of greater disorder.

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Absolute Entropy And Third Law Of Thermodynamics

The Third Law of Thermodynamics states that the entropy of any pure crystalline substance approaches zero as the temperature approaches absolute zero (0 K). At absolute zero, a perfect crystal has perfect order, and the motion of its constituent particles (translational, rotational, vibrational) is minimal or ceases, resulting in minimum entropy.

This law allows for the determination of absolute values of entropy for pure substances based on heat capacity measurements from 0 K up to a desired temperature. Standard molar entropies ($S^\ominus$) are the absolute entropies of substances in their standard states at a specified temperature (usually 298 K). Standard entropy changes for reactions ($\Delta_r S^\ominus$) can be calculated using a similar formula to Hess's law, summing standard entropies of products and reactants.

Gibbs Energy Change And Equilibrium

The Gibbs energy change ($\Delta G$) is also related to the equilibrium state of a reversible reaction. At equilibrium, the net change in Gibbs energy is zero ($\Delta G = 0$), meaning there is no longer a driving force for the reaction to proceed in either direction.

For a reaction A + B $\rightleftharpoons$ C + D, at equilibrium, $\Delta_r G = 0$.

The standard Gibbs energy change ($\Delta_r G^\ominus$), which is the Gibbs energy change when reactants and products are in their standard states, is related to the equilibrium constant (K) of the reaction by the following equation:

$$ \Delta_r G^\ominus = -RT \ln K $$ $$ \Delta_r G^\ominus = -2.303 RT \log K $$

where R is the ideal gas constant and T is the absolute temperature.

This fundamental equation connects thermodynamics (energy changes) to chemical equilibrium (the extent of reaction). It shows:

• If $\Delta_r G^\ominus < 0$, $\ln K > 0$, so $K > 1$. Products are favored at equilibrium.
• If $\Delta_r G^\ominus > 0$, $\ln K < 0$, so $K < 1$. Reactants are favored at equilibrium.
• If $\Delta_r G^\ominus = 0$, $\ln K = 0$, so $K = 1$. Equilibrium lies roughly in the middle.

Since $\Delta_r G^\ominus = \Delta_r H^\ominus - T \Delta_r S^\ominus$, the equilibrium constant K is affected by both the standard enthalpy and entropy changes of the reaction, as well as the temperature.

This relationship allows us to calculate K from thermodynamic data ($\Delta_r H^\ominus$, $\Delta_r S^\ominus$) or calculate $\Delta_r G^\ominus$ from experimental equilibrium data (K).

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