Thermodynamics — Class 11 Chapter 5 · Complete NCERT Exercise Solutions

What Is a Thermodynamic State Function?

In thermodynamics, we study the properties of a system — its pressure, temperature, volume, internal energy, enthalpy, and so on. These properties can be divided into two types depending on how their values are determined.

A state function (also called a state variable or state property) is a thermodynamic quantity whose value depends only on the current state of the system, and not on the path or history by which the system arrived at that state. In other words, if we know the conditions (temperature, pressure, volume, composition) at any instant, we can determine the value of every state function completely — it does not matter how many steps were taken, what route was followed, or how much work or heat was involved along the way.

Mathematically, if a system moves from an initial state (1) to a final state (2), the change in any state function \(X\) is always given by:

This expression shows clearly that \(\Delta X\) depends only on \(X_2\) (the final value) and \(X_1\) (the initial value). It is completely independent of the path taken between the two states. No matter how many steps, reversals, or detours occur between state 1 and state 2, \(\Delta X\) always comes out the same.

Common examples of state functions include: internal energy (U), enthalpy (H), entropy (S), Gibbs free energy (G), temperature (T), pressure (p), and volume (V).

In contrast, path functions are quantities whose values do depend on the specific route or process followed. Heat \(q\) and work \(w\) are the most important path functions in thermodynamics. The same initial and final states can be connected by different processes, and the heat absorbed and work done will differ depending on which process is chosen.

Classification of Thermodynamic Processes

Thermodynamic processes are classified according to which variable is held constant during the change. Understanding these classifications is fundamental to applying the First Law correctly in different situations.

The First Law of Thermodynamics expresses the principle of conservation of energy for thermodynamic systems. It states that energy can neither be created nor destroyed, but it can be transformed from one form to another. For any thermodynamic process, the change in internal energy of the system equals the heat absorbed by the system plus the work done on the system:

Here \(\Delta U\) is the change in internal energy, \(q\) is the heat exchanged with the surroundings (positive when absorbed, negative when released), and \(w\) is the work done (positive when done on the system, negative when done by the system).

The four standard process types are:

An adiabatic process is one in which the system is thermally isolated from its surroundings — no heat enters or leaves during the process. This can be achieved either by perfect thermal insulation (for slow processes) or by carrying out the process so rapidly that there is not enough time for heat transfer to occur.

Standard State and Reference Enthalpy

The standard state of a substance is its most stable physical form at a specified temperature (usually 298 K) and at 1 bar pressure. For example, the standard state of carbon is graphite (not diamond), the standard state of oxygen is O₂(g), and the standard state of sodium is Na(s).

The standard enthalpy of formation, \(\Delta_f H^\Theta\), is defined as the enthalpy change when one mole of a compound is formed from its constituent elements, with each element in its standard state. This quantity serves as the thermochemical reference for all enthalpy calculations.

A reference point must be established for all thermodynamic calculations, just as sea level is used as the reference (zero) for altitude measurements. In thermochemistry, the agreed-upon convention is to assign a value of exactly zero to the standard enthalpy of formation of every element in its standard state. This is not measured — it is a defined convention that makes all thermochemical calculations consistent and universally comparable.

Some examples of this convention:

Relationship Between ΔH and ΔU for Reactions Involving Gases

Internal energy change (\(\Delta U\)) represents the total energy change of a system at constant volume — this is what a bomb calorimeter measures.

Enthalpy change (\(\Delta H\)) represents the heat change at constant pressure — this is what is measured in most open-system reactions (since they occur at atmospheric pressure).

For reactions in which gases are produced or consumed, the two quantities differ because the system must do extra work to push back the atmosphere when gas volume changes. The relationship between them is derived from the definition of enthalpy \(H = U + pV\):

For ideal gases, \(pV = nRT\), so \(\Delta(pV) = \Delta n_g \cdot RT\), giving:

where \(\Delta n_g = n_g(\text{products}) - n_g(\text{reactants})\) counts only gaseous species (solids and liquids have negligible \(\Delta(pV)\) contribution and are ignored).

Enthalpy of Formation and Hess's Law

The standard enthalpy of formation, \(\Delta_f H^\Theta\), of a compound is the enthalpy change when one mole of that compound is formed from its constituent elements, each in their standard states at 298 K and 1 bar.

Direct measurement of \(\Delta_f H^\Theta\) is often impossible (e.g., we cannot simply mix carbon and hydrogen in a flask and get methane). Instead, we use Hess's Law, which states that the total enthalpy change of a chemical reaction is independent of the route taken — it depends only on the initial and final states. We can therefore add and subtract known thermochemical equations to derive an unknown one.

The formation reaction we want to establish is:

The three combustion reactions we are given are:

Gibbs Free Energy and Spontaneity

The spontaneity of a chemical reaction at constant temperature and pressure is determined by the Gibbs free energy change, \(\Delta G\). This quantity elegantly combines the two thermodynamic driving forces — the tendency to minimise enthalpy (energy) and the tendency to maximise entropy (disorder).

The spontaneity rules are:

The four possible combinations of \(\Delta H\) and \(\Delta S\) signs, and their spontaneity consequences, are summarised below — this table is essential for MCQs:

First Law of Thermodynamics — IUPAC Sign Convention

The First Law of Thermodynamics states that the total energy of an isolated system is constant. For any process, the change in internal energy equals the algebraic sum of heat and work:

The sign convention used in modern chemistry (IUPAC convention) defines everything from the system's perspective:

Bomb Calorimeter — Why ΔU ≠ ΔH for Gas-Phase Reactions

A bomb calorimeter is a sealed, rigid container — the volume is constant during the reaction. Since the volume cannot change, no PV expansion work is done, and the heat measured equals the change in internal energy (\(\Delta U\)) rather than enthalpy (\(\Delta H\)).

Most chemical reactions, however, occur in open flasks at constant atmospheric pressure, where any volume change due to gas production or consumption involves work done against (or by) the atmosphere. The heat measured at constant pressure is the enthalpy change \(\Delta H\).

To convert between the two, we use the relationship derived from \(H = U + pV\):

where \(\Delta n_g = n_g(\text{gaseous products}) - n_g(\text{gaseous reactants})\), \(R = 8.314 \times 10^{-3}\ \mathrm{kJ\ K^{-1}\ mol^{-1}}\), and \(T\) is in Kelvin.

Molar Heat Capacity and Sensible Heat

When the temperature of a substance is raised without any phase change occurring, the heat absorbed is called sensible heat. The amount of heat required depends on three factors: the number of moles of substance, the temperature change, and the molar heat capacity of the substance.

The molar heat capacity \(C_m\) is defined as the amount of heat required to raise the temperature of one mole of a substance by one kelvin (or one degree Celsius, since degree sizes are equal). The formula relating heat, moles, heat capacity, and temperature change is:

where \(n\) = number of moles, \(C_m\) = molar heat capacity in \(\mathrm{J\ mol^{-1}\ K^{-1}}\), and \(\Delta T\) = temperature change in K (or °C — the size of one degree is identical).

Enthalpy Changes Involving Temperature Change and Phase Change

When a substance undergoes a change that involves both a temperature change (within a single phase) and a phase change (e.g., freezing, melting), the total enthalpy change must be calculated in separate steps and then summed — this is an application of Hess's Law.

For a temperature change within one phase (sensible heat), the formula is:

For a phase change (latent heat — no temperature change), heat is absorbed or released at constant temperature. For melting (fusion), heat is absorbed: \(\Delta H = +\Delta_\text{fus}H\). For freezing (the reverse of melting), heat is released: \(\Delta H = -\Delta_\text{fus}H\).

In this problem, liquid water at 10°C is converted to ice at −10°C. This requires three distinct steps:

Enthalpy of Combustion — Per Mole Basis

The enthalpy of combustion is always quoted on a per mole basis — it represents the heat released when exactly one mole of the substance is completely burned in excess oxygen. The balanced equation for the combustion of carbon is:

This tells us that the formation of 1 mole of CO₂ releases 393.5 kJ of heat. For any other quantity, we scale proportionally using the number of moles actually formed.

Calculating Reaction Enthalpy from Enthalpies of Formation

The most general and powerful application of Hess's Law is the formula that expresses the standard enthalpy change of any reaction in terms of the standard enthalpies of formation of all reactants and products:

where \(\nu\) are the stoichiometric coefficients in the balanced equation. In words: sum of (coefficient × ΔfH°) for all products minus sum of (coefficient × ΔfH°) for all reactants.

The logic is: we "decompose" all reactants into elements (costing/releasing ΔfH° each), then "form" all products from those elements (releasing/costing ΔfH° each). The elements themselves all have ΔfH° = 0 and cancel out, leaving only the net reaction.

Standard Enthalpy of Formation — The Per-Mole Concept

The standard enthalpy of formation \(\Delta_f H^\Theta\) of a compound is defined as the enthalpy change when exactly one mole of that compound is formed from its constituent elements in their standard states. This one-mole basis is crucial.

The given reaction \(\mathrm{N_2 + 3H_2 \rightarrow 2NH_3}\) produces two moles of NH₃, not one. Therefore, the given \(\Delta_r H^\Theta = -92.4\ \mathrm{kJ\ mol^{-1}}\) corresponds to the formation of 2 mol of NH₃. The enthalpy of formation of one mole of NH₃ will be half of this value.

We can verify this using the general formula. Since N₂ and H₂ are elements in their standard states, their \(\Delta_f H^\Theta = 0\):

Deriving ΔfH° from Combustion Data Using Hess's Law

The standard enthalpy of formation of methanol is the enthalpy change for:

This reaction cannot be carried out directly. However, using the combustion data provided and Hess's Law, we can combine the three given reactions algebraically to derive it. Alternatively, we can use the standard formula involving enthalpies of formation directly.

Bond Enthalpy and the Atomisation Cycle

The bond enthalpy of a bond X–Y is the enthalpy required to break one mole of that bond in the gaseous phase to produce gaseous atoms:

For CCl₄, there are four equivalent C–Cl bonds. When all four are broken:

Since we cannot measure this directly, we use a Hess's Law cycle. The strategy is to construct two paths from the same elements in standard states to the same gaseous atoms:

Isolated Systems and the Second Law of Thermodynamics

An isolated system is one that exchanges neither matter nor energy (neither heat nor work) with its surroundings. Because of this total isolation:

The internal energy of an isolated system is therefore constant. The universe as a whole can be considered an isolated system.

However, the constancy of internal energy (\(\Delta U = 0\)) does not tell us whether a spontaneous process can occur or in which direction it proceeds. This is governed by the Second Law of Thermodynamics, which deals with entropy.

The Second Law states that for any spontaneous process occurring in an isolated system, the total entropy of the system increases. The entropy can never decrease spontaneously in an isolated system. In its most compact form:

This inequality has two cases:

Entropy \(S\) is a measure of the disorder, randomness, or dispersal of energy in a system. Natural processes — heat flowing from hot to cold, gases expanding to fill a container, crystal dissolving in a solvent — all involve an increase in entropy of the isolated system. Nature always moves spontaneously toward greater disorder.

Finding the Temperature of Spontaneity

The spontaneity of a reaction is determined by the sign of the Gibbs free energy change:

At a given temperature, a reaction is: spontaneous if \(\Delta G < 0\), at equilibrium if \(\Delta G = 0\), and non-spontaneous if \(\Delta G > 0\).

In this question, \(\Delta H = +400\ \mathrm{kJ\ mol^{-1}}\) (positive — endothermic) and \(\Delta S = +0.2\ \mathrm{kJ\ K^{-1}\ mol^{-1}}\) (positive). Looking at the spontaneity table, a reaction with \(\Delta H > 0\) and \(\Delta S > 0\) is spontaneous only at high temperatures, because the \(T\Delta S\) term must be large enough to overcome the unfavourable positive \(\Delta H\).

The critical temperature — the threshold above which the reaction becomes spontaneous — is found by setting \(\Delta G = 0\):

For \(T > T_\text{critical}\): \(\Delta G < 0\) → spontaneous.
For \(T < T_\text{critical}\): \(\Delta G > 0\) → non-spontaneous.

Enthalpy and Entropy of Bond Formation

To answer this question, we need to reason about what happens at the molecular level when two chlorine atoms combine to form a chlorine molecule.

Enthalpy (\(\Delta H\)): When a chemical bond forms between two atoms, the resulting molecule is more stable (lower energy) than the two separated atoms. Energy must be released to the surroundings for the system to reach this lower-energy bonded state. Bond formation is therefore always exothermic (\(\Delta H < 0\)). (Conversely, bond breaking is always endothermic.)

Entropy (\(\Delta S\)): Entropy measures the degree of randomness, disorder, or dispersal in a system. The number of particles is a key contributor to entropy: more particles means more ways to arrange their positions and momenta, hence more disorder. In this reaction, two separate gaseous particles (2 Cl atoms) combine into one particle (Cl₂ molecule). The number of moles of gas decreases from 2 to 1.

A decrease in the number of gas particles means a decrease in the randomness/disorder of the system, so entropy decreases. Therefore \(\Delta S < 0\).

Complete Workflow: ΔU → ΔH → ΔG

This problem requires a two-step conversion:

The reason we cannot directly use ΔU in the Gibbs equation is that the Gibbs equation is defined at constant pressure, where the relevant energy quantity is enthalpy (\(H = U + pV\)), not internal energy (\(U\)).

The Fundamental Link Between ΔG° and the Equilibrium Constant K

One of the most profound results in physical chemistry is the relationship connecting the standard Gibbs free energy change (\(\Delta G^\Theta\)) to the equilibrium constant (\(K\)) of a reaction. This equation bridges thermodynamics and kinetics/equilibrium:

or equivalently, using base-10 logarithms:

The physical interpretation is: if \(K > 1\) (products favoured at equilibrium), then \(\ln K > 0\), so \(\Delta G^\Theta < 0\) — the reaction is thermodynamically downhill in the forward direction under standard conditions. If \(K < 1\) (reactants favoured), \(\Delta G^\Theta > 0\). If \(K = 1\), \(\Delta G^\Theta = 0\).

Thermodynamic Stability — What It Means

The thermodynamic stability of a compound refers to its stability with respect to its constituent elements (or other possible compounds) as judged by energy considerations. A compound is said to be thermodynamically stable if it has lower energy than its constituent elements — i.e., it forms spontaneously from those elements. Conversely, if forming the compound requires absorption of energy (positive \(\Delta_f H^\Theta\)), the compound has higher energy than the elements and is thermodynamically unstable.

We assess stability by examining:

Additionally, if the compound readily reacts with surrounding molecules (like O₂ in air) to form even more stable products (with negative \(\Delta_r H^\Theta\)), the compound is described as thermodynamically unstable with respect to those products too.

Entropy Change of the Surroundings

The total entropy change of the universe is the sum of the entropy change of the system and the entropy change of the surroundings:

For a process occurring at constant temperature \(T\) and constant pressure, the surroundings exchange heat with the system. If the system releases heat (exothermic, \(\Delta H_\text{sys} < 0\)), the surroundings absorb that heat and their entropy increases. If the system absorbs heat (endothermic), the surroundings lose heat and their entropy decreases.

The entropy change of the surroundings is given by:

The negative sign arises because when the system releases heat (\(\Delta H_\text{sys} < 0\)), the surroundings absorb it (\(q_\text{surr} = -\Delta H_\text{sys} > 0\)), increasing their entropy. The formula correctly gives a positive \(\Delta S_\text{surr}\) when \(\Delta H_\text{sys} < 0\).