Thermodynamics

The state of a system refers to the condition of the system at a particular moment, as described by its measurable physical properties.

A system is said to be in a definite state when its macroscopic properties — such as pressure \((P)\), volume \((V)\), temperature \((T)\), and composition — have fixed and well-defined values. These properties are known as state variables because any change in their values leads to a change in the state of the system.

For example, if a gas is confined in a container at a certain temperature, pressure, and volume, it is considered to be in a specific thermodynamic state. If any one of these variables changes, the system moves to a new state.

The state of a system is completely specified when all its relevant thermodynamic variables are known, and it does not depend on how the system reached that condition.

🔹 State Variables

State variables are the measurable properties of a system that describe its physical condition at a given moment, including pressure \((P)\), volume \((V)\), temperature \((T)\), and amount or composition of the substance.

State variables depend only on the present condition of the system and not on the path or process by which the system reached that condition.

🔹 State Functions

State functions are those thermodynamic properties whose values depend only on the present state of the system and not on the path by which the system reached that state.

Common examples of state functions include internal energy \(\mathrm{(U)}\), enthalpy \(\mathrm{(H)}\), pressure \(\mathrm{(P)}\), volume \(\mathrm{(V)}\), temperature \(\mathrm{(T)}\), entropy \(\mathrm{(S)}\), and Gibbs free energy \(\mathrm{(G)}\).

Thus, state functions are properties that are completely defined by the state of the system itself and are independent of the process path.

Internal energy \((U)\) is described as a state function because its value depends only on the present state of the system and not on the path followed to reach that state.

Internal energy represents the total energy contained within a system. It includes the kinetic energy of molecules due to their motion (translation, rotation, vibration) and the potential energy arising from intermolecular forces and chemical bonds.

When a system undergoes a change from an initial state to a final state, the change in internal energy is expressed as:

This change remains the same regardless of whether the process occurs in one step or through multiple intermediate steps. The first law of thermodynamics gives:

Therefore, internal energy is called a state function because its change is independent of the process path and is determined solely by the state of the system.

Work is defined as the mode of energy transfer that occurs when a force causes displacement against an opposing force. In thermodynamics, work is associated mainly with changes in volume of a system, especially in the case of gases.

When a gas expands or contracts inside a container fitted with a movable piston, work is done. If the gas expands against an external pressure, it pushes the piston outward and performs work on the surroundings. Conversely, if the surroundings compress the gas, work is done on the system.

Mathematically, when a gas expands or compresses against a constant external pressure:

where \(P_{\mathrm{ext}}\) is the external pressure and \(\Delta V\) is the change in volume. The negative sign follows the sign convention used in chemistry:

• When the system does work on the surroundings (expansion), \(w\) is negative.
• When work is done on the system (compression), \(w\) is positive.

Work is considered a path function because its value depends on how the process is carried out, not just on the initial and final states.

Work is defined as the energy transferred when a system undergoes displacement against an opposing force. In thermodynamics, we mainly deal with pressure–volume (P–V) work, which occurs when a gas expands or is compressed.

🔹 Expression for Work at Constant External Pressure

Consider a cylinder containing one mole of an ideal gas fitted with a frictionless piston. We know: \(\mathrm{Pressure = Force / Area}\), so \(\mathrm{Force} = P_{\mathrm{ext}} \times A\). If the piston moves by distance \(l\):

Since \(\Delta V = A \times l\), and applying the chemical sign convention:

• Compression \(\Rightarrow \Delta V < 0 \Rightarrow w > 0\) (work done on system)
• Expansion \(\Rightarrow \Delta V > 0 \Rightarrow w < 0\) (work done by system)

🔹 Work When Pressure Changes Continuously

If the external pressure changes gradually in infinitesimally small steps:

Integrating between initial volume \(V_i\) and final volume \(V_f\):

🔹 Reversible Process

In a reversible process, the external pressure differs from internal pressure by an infinitesimal amount \(P_{\mathrm{ext}} = P_{\mathrm{in}} \pm dP\). Since \(dP \cdot dV\) is extremely small:

🔹 Work in Isothermal Reversible Expansion of Ideal Gas

For an ideal gas \(P = nRT/V\). Substituting and integrating since \(nRT\) is constant at constant temperature:

Or in common logarithm form:

🔹 Free Expansion

If gas expands into vacuum, \(P_{\mathrm{ext}} = 0\), therefore \(w = 0\). In free expansion, no work is done.

🔹 Relation with First Law of Thermodynamics

Substituting \(w = -P_{\mathrm{ext}}\Delta V\) into the first law \(\Delta U = q + w\):

At constant volume \((\Delta V = 0)\):

Heat is defined as the form of energy that is transferred between a system and its surroundings due to a temperature difference.

Heat flows spontaneously from a body at higher temperature to one at lower temperature until thermal equilibrium is established. It is important to understand that heat is not a substance contained within a system; rather, it is energy in transit.

In thermodynamics, heat is represented by the symbol \(q\). According to the sign convention used in chemistry:

• \(q\) is positive when heat is absorbed by the system from the surroundings.
• \(q\) is negative when heat is released by the system to the surroundings.

Heat is classified as a path function because the amount of heat exchanged depends on the specific process by which the system changes from its initial state to its final state.

In thermodynamics, most chemical reactions are performed in open containers where the pressure remains constant at atmospheric value. At constant volume, the heat absorbed equals the change in internal energy: \(\Delta U = q_V\). However, since laboratory reactions generally occur at constant pressure, it becomes necessary to define a new thermodynamic quantity — enthalpy.

Starting from the first law at constant pressure and denoting initial state as 1 and final state as 2:

Rearranging:

This suggests defining enthalpy (H):

Therefore:

Thus, at constant pressure, the heat absorbed by a system equals the change in enthalpy. For finite changes at constant pressure:

• For exothermic reactions, heat is evolved and \(\Delta H\) is negative.
• For endothermic reactions, heat is absorbed and \(\Delta H\) is positive.

At constant volume, \(\Delta V = 0\), so \(\Delta H = \Delta U = q_V\). Enthalpy is a state function defined as \(H = U + pV\), and under constant pressure conditions, the heat absorbed or released equals the enthalpy change.

🔹 Extensive Properties

An extensive property is one whose value changes with the quantity or size of the system. If the amount of substance increases or decreases, the value of an extensive property also changes. Examples: mass, volume, internal energy, enthalpy, and heat capacity.

• Depend on the amount of matter in the system.
• Additive in nature (values of parts add up to give the total).
• Change when the system is divided or combined.
• Their molar values (property per mole) become independent of quantity.

🔹 Intensive Properties

An intensive property is one whose value does not depend on the quantity or size of the system. It remains the same regardless of how much substance is present. Examples: temperature, pressure, and density.

• Independent of the amount of matter.
• Not additive in nature.
• Remain unchanged when the system is divided.
• Useful for identifying the state of a system.

🔹 Molar Property

A molar property \((\chi_m)\) is obtained by dividing an extensive property \((\chi)\) by the number of moles \((n)\):

Since it is expressed per mole, a molar property becomes independent of the amount of substance. Examples: molar volume \((V_m)\) and molar heat capacity \((C_m)\).

Heat capacity \((C)\) tells us how much heat energy a system needs to increase its temperature by a certain amount. The two are directly related:

Here \(q\) = heat supplied, \(C\) = heat capacity, \(\Delta T\) = change in temperature. The bigger the heat capacity, the more heat is required to raise the temperature.

• the size of the system
• the amount of substance
• the nature and composition of the material

For example, water has a very high heat capacity — it takes a lot of energy to warm water, but once warm, it also cools slowly. Heat capacity is an extensive property.

🔹 Molar Heat Capacity

When heat capacity is expressed per mole, we call it molar heat capacity \((C_m)\). It is the amount of heat required to raise the temperature of one mole of a substance by 1°C (or 1 K).

🔹 Specific Heat Capacity

Specific heat capacity \((c)\) is the heat required to raise the temperature of one unit mass of a substance by 1°C (or 1 K).

Heat capacity at constant volume is denoted by \(C_v\) and at constant pressure by \(C_p\).

For one mole of an ideal gas, the difference between heat capacities at constant pressure and constant volume is equal to the gas constant \(R\) — this is known as Mayer's relation:

🔹 Enthalpy Change — Definition and Derivation

In any chemical reaction, reactants are transformed into products: \(\mathrm{Reactants \rightarrow Products}\). Whenever this transformation occurs, heat may be absorbed or released at constant pressure. The heat change associated with a chemical reaction at constant pressure is called the enthalpy change of reaction, denoted by \(\Delta_rH\).

Enthalpy change of a reaction is the difference between the total enthalpy of the products and the total enthalpy of the reactants:

If \(\Delta_rH < 0 \Rightarrow\) reaction is exothermic (heat released). If \(\Delta_rH > 0 \Rightarrow\) reaction is endothermic (heat absorbed).

General Mathematical Expression

The enthalpy change of a reaction depends on the conditions under which it takes place. Because of this, chemists use standard conditions so that results from different experiments can be compared easily.

The standard enthalpy of reaction is the enthalpy change measured when all reactants and products are in their standard states, written as \(\Delta H^\circ\).

Standard State:

• Substance in pure form
• Pressure = 1 bar
• Usually temperature = 298 K

🔹 Enthalpy of Fusion

🔹 Enthalpy of Vaporisation

🔹 Enthalpy of Sublimation

Sometimes a solid changes directly into vapour without becoming liquid first. This process is called sublimation and is associated with \(\Delta_{\mathrm{sub}}H^\circ\).

The standard enthalpy change for the formation of one mole of a compound from its elements in their most stable states of aggregation is called standard molar enthalpy of formation, symbol \(\Delta_f H^\circ\).

Reference state examples:

• Hydrogen → H₂(g)
• Oxygen → O₂(g)
• Carbon → C(graphite)
• Sulphur → S(rhombic)

Formation of Water

Formation of Methane

Formation of Ethanol

A thermochemical equation is a balanced chemical equation written together with the enthalpy change \((\Delta_rH^\circ)\) associated with the reaction. It not only tells us what substances react and form, but also how much heat is released or absorbed during the process.

Because energy changes depend on conditions and physical forms, a proper thermochemical equation always specifies the physical state (solid, liquid, gas, or aqueous) and even the allotropic form of the substances involved.

🔹 Standard Example

🔹 Important Conventions

Hess's Law is one of the most powerful ideas in thermochemistry. It arises from a simple but profound fact: enthalpy is a state function. This means the enthalpy change of a reaction depends only on the initial and final states, not on the path taken between them.

🔹 Statement of Hess's Law

If a chemical reaction occurs in several steps, the standard enthalpy change of the overall reaction equals the algebraic sum of the standard enthalpy changes of the individual steps, provided the temperature is the same.

🔹 Understanding Through an Example

Consider the reaction \(C_{\text{graphite}}(s) + \frac{1}{2}O_2(g) \rightarrow CO(g)\). This is difficult to measure directly because some CO₂ always forms alongside CO.

Given Reactions

Step 1: Reverse Reaction (ii)

Step 2: Add the Equations

🔹 Key Rules to Remember

• 🔹 Enthalpy is additive
• 🔹 Reverse reaction → change sign
• 🔹 Multiply equation → multiply ΔH
• 🔹 Physical states must match

The standard enthalpy of combustion is the enthalpy change when one mole of a substance undergoes complete combustion in oxygen, with all reactants and products in their standard states at 298 K and 1 bar.

• Measured per mole of the substance burned
• Combustion must be complete (no CO or soot)
• \(\Delta_cH^\circ\) is usually negative (exothermic)
• Units are typically kJ mol⁻¹

🔹 Example 1: Combustion of Butane

🔹 Example 2: Combustion of Glucose

This high energy output is the reason glucose is a primary energy source in living organisms. The human body extracts energy from glucose through cellular respiration — a controlled multistep equivalent of combustion — where energy is released gradually and stored as ATP rather than all at once as heat.

The standard enthalpy of atomization is the enthalpy change when one mole of gaseous atoms is formed from a substance in its standard state by completely breaking all the bonds present. Since breaking bonds always requires energy, atomization enthalpy is positive in most cases.

🔹 Example 1: Dihydrogen

🔹 Example 2: Methane

All four C–H bonds are broken and all atoms are produced in the gaseous state. This large positive value shows that methane has strong covalent bonds requiring significant energy to break.

🔹 Example 3: Sodium (Metal)

• For metals: Enthalpy of atomization = Enthalpy of sublimation

Bond enthalpy is the enthalpy change required to break one mole of a specific covalent bond in gaseous molecules to form gaseous atoms or fragments. Because bond breaking needs energy, bond enthalpy values are generally positive.

🔹 Bond Dissociation Enthalpy (Diatomic Molecules)

For diatomic molecules, bond dissociation enthalpy is straightforward because only one bond exists in each molecule.

🔹 Mean (Average) Bond Enthalpy for Polyatomic Molecules

In polyatomic molecules, even identical bonds have slightly different dissociation energies when broken stepwise. For methane \((CH_4)\), the four stepwise C–H energies are 427, 439, 452, and 347 kJ mol⁻¹. Therefore, we use an average:

🔹 Predicting Reaction Enthalpy

Example: Formation of HCl

Bonds broken = 435 + 242 = 677 kJ. Bonds formed = 2 × 431 = 862 kJ.

Lattice enthalpy \((\Delta_{\text{lattice}}H^\circ)\) is defined as the enthalpy change when one mole of an ionic solid dissociates completely into its constituent gaseous ions under standard conditions.

🔹 Born–Haber Cycle for NaCl

The Born–Haber cycle breaks the formation of NaCl into hypothetical steps whose enthalpies are known experimentally:

• Step 1 — Sublimation of Na: \(Na(s) \rightarrow Na(g) \quad \Delta_{\text{sub}}H^\circ = 108.4\ \text{kJ mol}^{-1}\)
• Step 2 — Ionization of Na: \(Na(g) \rightarrow Na^+(g) + e^- \quad \Delta_i H^\circ = 496\ \text{kJ mol}^{-1}\)
• Step 3 — Dissociation of Cl₂: \(\frac{1}{2}Cl_2(g) \rightarrow Cl(g) \quad \Delta_{\text{bond}}H^\circ = 121\ \text{kJ mol}^{-1}\)
• Step 4 — Electron gain by Cl: \(Cl(g) + e^- \rightarrow Cl^-(g) \quad \Delta_{\text{eg}}H^\circ = -348.6\ \text{kJ mol}^{-1}\)

🔹 Enthalpy of Solution Relation

For NaCl: \(\Delta_{\text{sol}}H^\circ = 788 - 784 = +4\ \text{kJ mol}^{-1}\). This very small value explains why dissolving common salt produces almost no noticeable temperature change.

The enthalpy of solution is the enthalpy change when one mole of a substance dissolves in a specified amount of solvent under standard conditions. It may be positive (endothermic) or negative (exothermic).

🔹 What Happens When an Ionic Solid Dissolves?

• Step 1 — Breaking the Crystal Lattice: \(AB(s) \rightarrow A^+(g) + B^-(g)\) — Endothermic; energy absorbed.
• Step 2 — Hydration (Solvation) of Ions: \(A^+(g) \rightarrow A^+(aq)\) and \(B^-(g) \rightarrow B^-(aq)\) — Exothermic; energy released.

🔹 Why Fluorides Are Less Soluble Than Chlorides

Fluoride ion (F⁻) is much smaller than chloride ion (Cl⁻), so ionic attraction in fluorides is stronger, making their lattice enthalpy very high. Although hydration of F⁻ is also strong, the increase in lattice enthalpy is greater, so many fluorides are less soluble: \(MgF_2\) is less soluble than \(MgCl_2\).

🔹 Example: Ammonium Nitrate

Here \(\Delta_{\text{sol}}H^\circ\) is strongly positive — highly endothermic. This is why ammonium nitrate is used in instant cold packs.

The enthalpy of dilution is the enthalpy change that accompanies the addition of extra solvent to an already prepared solution at constant temperature and pressure.

🔹 Understanding Through HCl

🔹 Calculating Enthalpy of Dilution

Subtracting the 25 aq. result from the 40 aq. result gives the heat change on further dilution:

In thermodynamics, one of the most fundamental questions is not just how much energy changes, but whether a process will occur on its own. The first law of thermodynamics deals with energy conservation but does not tell us the direction in which a process will naturally proceed.

🔹 Natural Direction of Processes

• Heat always flows from higher temperature to lower temperature when left undisturbed.
• A gas released in a container spreads out to fill the entire volume.
• Carbon burns in oxygen to form carbon dioxide.

🔹 What is a Spontaneous Process?

A spontaneous process is one that has the tendency or potential to occur on its own under given conditions, without the continuous assistance of external energy.

🔹 Example: Hydrogen and Oxygen

H₂ and O₂ can be mixed and kept together at room temperature for years without noticeable change, yet the reaction is thermodynamically spontaneous. Without a spark, the activation barrier prevents the reaction from proceeding.

🔹 Important Characteristics

• Can occur without continuous external help
• Proceeds in one preferred direction
• Is generally irreversible
• Its rate may be fast or slow
• Depends on thermodynamic feasibility, not speed

At first glance, it seems natural to believe that a process will occur spontaneously if it releases energy. Many familiar chemical reactions are exothermic and occur spontaneously:

🔹 But the Rule Fails in Some Cases

There are several reactions that are endothermic yet still spontaneous:

🔹 Classic Everyday Examples

• Melting of ice above 0°C: endothermic but spontaneous
• Dissolution of ammonium nitrate: solution becomes cold yet occurs spontaneously
• Evaporation of water at room temperature: requires heat but happens naturally

Since decrease in enthalpy alone cannot always explain why a process occurs on its own, thermodynamics introduces another powerful concept — entropy (S) — which accounts for the role of disorder in determining spontaneity.

🔹 A Case Where ΔH ≈ 0 but the Process is Spontaneous

Consider a closed, isolated container divided into two compartments — left side contains gas A, right side contains gas B. When the partition is removed, the gases diffuse and form a uniform mixture spontaneously even though there may be no significant enthalpy change \((\Delta H \approx 0)\). Increased randomness is the key driving factor.

🔹 Introducing Entropy (S)

Entropy is the thermodynamic quantity that measures the degree of disorder or randomness in a system:

🔹 Entropy in Physical States

🔹 Quantitative Definition of Entropy

Experimentally, for a reversible process:

🔹 Entropy Criterion for Spontaneity

• ΔSₜₒₜₐₗ > 0 → spontaneous
• ΔSₜₒₜₐₗ = 0 → equilibrium
• ΔSₜₒₜₐₗ < 0 → non-spontaneous

Most real chemical reactions occur in open or closed systems where both enthalpy and entropy change simultaneously. In such cases, neither \(\Delta H\) alone nor \(\Delta S\) alone can reliably predict spontaneity. This leads to the concept of Gibbs energy (G).

🔹 Definition of Gibbs Energy

where \(H\) = enthalpy, \(T\) = absolute temperature (K), \(S\) = entropy. Like enthalpy and entropy, Gibbs energy is a state function and an extensive property.

🔹 Gibbs Equation

For a process at constant temperature:

🔹 Criteria for Spontaneity at Constant T and P

• ✅ If \(\Delta G < 0\) → spontaneous
• ❌ If \(\Delta G > 0\) → non-spontaneous
• ⚖️ If \(\Delta G = 0\) → equilibrium

🌡️ Role of Temperature in Spontaneity

The first law tells us that energy is conserved, but it does not explain why natural processes prefer one direction over another. To understand the natural direction of change, we turn to the second law of thermodynamics.

🔹 Statement of the Second Law

🔥 Example: Combustion

Heat is evolved, surrounding molecules move more randomly, total entropy increases — hence the reaction is spontaneous.

🔹 Why the Second Law is Fundamental

• Explains direction of natural processes
• Predicts feasibility of reactions
• Explains irreversibility
• Forms the basis of Gibbs free energy
• Governs engines, refrigerators, and biological systems

As temperature approaches absolute zero (0 K), molecular motion gradually diminishes. In the limit of absolute zero, an ideal perfectly ordered crystal would have no translational or rotational motion, minimum vibrational motion, and perfectly fixed particle arrangement — so entropy approaches its lowest possible value.

📜 Statement of the Third Law

⚠️ Limitation

The third law strictly applies only to pure crystalline solids. Solutions, glasses, supercooled liquids, and imperfect crystals may retain some residual entropy even near 0 K.

🧮 Standard Entropy and Hess-Type Relation

Once absolute entropies are known, entropy change of reactions can be calculated using:

Gibbs free energy helps us answer two very practical questions: Will a reaction occur on its own? And if it occurs, how far will it proceed?

🔹 What Happens to Gibbs Energy at Equilibrium?

At equilibrium, the system has no tendency to change in either direction. Thermodynamically, this means the Gibbs energy of the system is at its minimum:

🔹 Standard Gibbs Energy and the Equilibrium Constant

Or in base-10 form:

🔹 Combined Form

This equation shows that equilibrium position depends on both enthalpy and entropy changes.

📊 What the Value of K Tells Us

🔹 Final Insight

Gibbs free energy provides the bridge between spontaneity and equilibrium. While a negative \(\Delta G\) drives reactions forward, equilibrium is reached when free energy becomes minimum and \(\Delta G = 0\). The elegant relation between \(\Delta_r G^\circ\) and the equilibrium constant allows chemists to predict how far reactions will proceed and how temperature influences product yield. Thus, Gibbs energy stands at the heart of chemical thermodynamics, governing both the direction and the extent of chemical change.