Thermodynamics — NEET Notes

Thermal equilibrium & the zeroth law

Thermodynamics is built on a single, almost too-obvious experimental fact: two bodies placed in contact through a heat-conducting wall eventually stop changing. Their pressures and volumes settle into a stable pair of values; nothing more happens. This terminal condition is called thermal equilibrium. The wall that allows heat flow is called diathermic; the wall that forbids it is called adiabatic. In thermodynamics, "equilibrium" does not mean mechanical balance — it means that the macroscopic state variables describing the system have stopped changing in time. A gas sealed inside an insulated rigid container, with fixed pressure, volume, temperature, mass, and composition that do not drift with time, is in thermodynamic equilibrium. The same gas in mid-rush out of an opened valve is not.

The historical road to this idea was long. Before the nineteenth century, heat was imagined as an invisible fluid called caloric that flowed from hot bodies to cold ones, much as water flows between connected tanks until levels equalise. The caloric picture made sense of cooling curves and calorimetry, but it could not explain a now-famous observation: when Benjamin Thompson (Count Rumford) bored a brass cannon in 1798, he found that the work done by the horses produced unlimited quantities of heat — far more than any fluid could plausibly have been "scooped" out. Heat, Rumford concluded, must be a form of motion, a form of energy. James Prescott Joule's paddle-wheel experiments confirmed the equivalence quantitatively. By the 1850s, thermodynamics had its modern macroscopic vocabulary: heat as energy in transit, temperature as a state variable, and equilibrium as the stationary endpoint of every spontaneous process.

From the experimental fact of thermal equilibrium flows the entire concept of temperature. Imagine three bodies, A, B, and C. Bring A and C into thermal contact: they equilibrate. Independently, bring B and C into thermal contact: they too equilibrate. Now bring A and B together. The astonishing experimental finding is that nothing changes — A and B are already in thermal equilibrium with each other. This transitivity is the Zeroth Law of Thermodynamics, formulated by R. H. Fowler in 1931, long after the first and second laws had already been numbered. (The "zeroth" label was Fowler's joke: by the time he stated it, the others were locked into history, but logically this one comes first.) The law reads:

Two systems separately in thermal equilibrium with a third system are in thermal equilibrium with each other.

The zeroth law guarantees that there exists a property — call it temperature — whose equality across two systems is the signature of thermal equilibrium. It is the law that lets thermometers exist. A thermometer is just system C: it is brought into equilibrium with body A, its reading recorded; then brought into equilibrium with body B, its reading recorded again. If the readings are equal, A and B will not exchange heat when placed in contact. The thermometer therefore measures an intensive property whose equality predicts thermal-equilibrium behaviour. Numerical values for temperature are assigned through a thermometric property (mercury column, gas pressure at constant volume, electrical resistance) and a chosen scale, with the triple point of water — 273.16 K — serving as the universal fixed point of the absolute Kelvin scale.

One more distinction matters here. Thermodynamics is a macroscopic science. It describes a system using a handful of variables — pressure, volume, temperature, mass, composition — that can be measured directly with everyday instruments. It deliberately ignores the chaotic motion of the 10²³ individual molecules underneath. Mechanics asks what each particle is doing under what force; thermodynamics asks only what the system as a whole looks like when it has settled down. The two descriptions meet in kinetic theory and statistical mechanics, where the macroscopic variables are recovered as averages over molecular ones — but classical thermodynamics, formulated in the nineteenth century, predated and is independent of that molecular picture.

Internal energy, heat and work

Every bulk system holds a reservoir of energy locked up in the random motions and mutual interactions of its constituent molecules. The sum of all these molecular kinetic and potential energies — measured in the frame in which the centre of mass of the system is at rest — is called the internal energy, denoted U. Internal energy explicitly excludes any bulk kinetic energy of the system as a whole. A bullet flying at 300 m/s is not "hot" because of its speed; its temperature reports only the disordered motion of its atoms. When that bullet pierces a wood block and stops, however, the mechanical kinetic energy of the bullet is converted into the disordered internal energy of the bullet and the surrounding wood — and the temperature of both rises. The bullet has not gained heat in flight; it has converted bulk motion into thermal motion at impact.

For a gas, the molecular motions that contribute to internal energy include not only translational motion (molecules moving across the volume of the container) but also rotational and, at high enough temperatures, vibrational motion. The kinetic theory of gases gives a precise accounting of how each mode contributes to U via the law of equipartition. For the purposes of thermodynamics, however, we treat U as a single macroscopic variable — its molecular origin is set aside, and its value is fixed once the macroscopic state is fixed.

Internal energy is a state function: its value depends only on the current state of the system, specified by macroscopic variables like pressure, volume and temperature. It does not depend on the path that brought the system to that state. This is the single most important fact about U, and it underwrites the entire first law. Two grams of nitrogen at 300 K and 1 atm have a definite internal energy, regardless of whether they reached that state by heating from cold, compressing from a larger volume, or some twisting cyclic detour through a dozen intermediate states.

Internal energy can be changed in only two ways. First, by placing the system in contact with a body at a different temperature — the temperature gradient drives a flow of energy called heat, denoted Q. Second, by performing mechanical work on the system or letting it perform work — for instance, pushing a piston in or letting the gas push it out. Both processes increase or decrease the total energy stored within the molecules of the system, but they do so through different microscopic mechanisms: heat transfer involves chaotic molecular collisions across the boundary, while work transfer involves coherent motion of the boundary itself. Heat and work are not properties of a system; they are modes of energy transfer. It is meaningful to say "the gas has internal energy 500 J"; it is meaningless to say "the gas contains 500 J of heat" or "500 J of work." NCERT phrases this with a sharp warning: heat is energy in transit. Once the energy has crossed the boundary, it no longer is heat — it has become internal energy.

For a gas confined under a movable piston at pressure P, an infinitesimal volume change dV moves the piston against the external force PA, and the work done by the gas on the surroundings is:

W = ∫ P dV   (work done by the gas)

Area under the P-V curve — the indicator diagram

The integral runs from the initial volume V₁ to the final volume V₂. Geometrically, the work done by the gas equals the area under the curve on a P-V (indicator) diagram. The shape of that curve depends on the path — so the work depends on the path even when the endpoints are fixed.

First law of thermodynamics

The first law of thermodynamics is conservation of energy, dressed up for systems that exchange both heat and work. Suppose a gas is sealed in a cylinder with a movable piston. Heat ΔQ is supplied to the gas from a hot body in contact with it. The gas pushes the piston out, doing work ΔW on the surroundings, and its internal energy changes by ΔU. The first law states that the heat supplied must be exactly accounted for by the sum of these two effects:

Although ΔQ and ΔW separately depend on the path taken between two states, their difference ΔQ − ΔW does not — it equals ΔU, which is a state function. This invariance is what makes the first law more than a definition: it is an experimentally tested constraint on every thermodynamic process. If you go from state 1 (P₁, V₁, T₁) to state 2 (P₂, V₂, T₂), you can take infinitely many routes through intermediate states. Along each route, you will measure different values of Q and W. But every route will give the same value of Q − W, equal to ΔU = U₂ − U₁. The first law thus identifies a hidden function of state — the internal energy — by tracking what survives when path-dependent quantities are differenced.

A worked example from NCERT Class XI makes the law concrete. One gram of water at 100 °C absorbs 2256 J to become steam at 100 °C — its latent heat of vaporisation. The vapour occupies 1671 cm³ against atmospheric pressure 1.013 × 10⁵ Pa, while the liquid occupied only 1 cm³. The work done by the expanding steam is W = PΔV = 1.013 × 10⁵ × 1670 × 10⁻⁶ ≈ 169.2 J. Therefore ΔU = ΔQ − W = 2256 − 169.2 = 2086.8 J. The arithmetic is straightforward; the lesson is striking: more than 92 % of the latent heat goes into raising internal energy (breaking intermolecular bonds in the liquid), and less than 8 % goes into the mechanical work of pushing back the atmosphere. The same balance recurs whenever a substance changes phase against atmospheric pressure. NEET 2018 Q.9 used a near-identical setup with 0.1 g of water and asked candidates to compute ΔU; the answer, scaled down by ten, came out to 208.7 J.

The first law also explains a common puzzle. If a gas is taken through a closed cycle and returned to its starting state, ΔU = 0 — because U is a state function. The first law then forces Q_net = W_net: the net heat absorbed equals the net work delivered. There is no "left over" energy stored inside the gas at the end of the cycle. This is precisely what a heat engine does: cycle after cycle, every joule of heat absorbed from the source either becomes work or is dumped to the sink. The first law forbids any other accounting.

Specific heat capacity & Mayer's relation

If an amount of heat ΔQ raises the temperature of a substance by ΔT, the heat capacity of that particular sample is S = ΔQ/ΔT. Heat capacity scales with mass; dividing by mass gives the specific heat capacity s = ΔQ / (m ΔT), measured in J kg⁻¹ K⁻¹. Dividing instead by the number of moles µ gives the molar specific heat C = ΔQ / (µ ΔT), measured in J mol⁻¹ K⁻¹. Specific heat is a material property characteristic of the substance, not of the particular sample. For water, s = 4186 J kg⁻¹ K⁻¹ — the SI value behind the older calorie definition (1 cal = 4.186 J was originally defined as the heat needed to raise the temperature of 1 g of water from 14.5 °C to 15.5 °C). For solids the law of equipartition predicts a molar heat capacity of about 3R = 25 J mol⁻¹ K⁻¹ at ordinary temperatures, which agrees with measurement for most metals (Dulong-Petit law) and breaks down at low temperature where quantum effects dominate.

For gases, the specific heat depends on the conditions under which heat is supplied — there is no single "specific heat of nitrogen." Two operationally important cases stand out. Heating at constant volume (V fixed) forbids the gas from doing any work, because the piston does not move. ΔW = 0, so every joule of heat goes into raising internal energy and therefore temperature. The molar specific heat in this regime, C_v, satisfies ΔQ = µ C_v ΔT = ΔU. Heating at constant pressure (P fixed) lets the gas expand as it warms, doing work PΔV on the surroundings. For the same temperature rise, you must now supply not only ΔU but also the PΔV that goes out as work. C_p is therefore strictly larger than C_v. For one mole of an ideal gas, applying the first law and the equation of state PV = RT yields a famously clean result:

C_p − C_v = R

This is Mayer's relation, derived by Julius Robert von Mayer in 1842. The extra heat supplied per kelvin at constant pressure equals exactly PΔV/ΔT = R per mole. R is the universal gas constant, 8.314 J mol⁻¹ K⁻¹, and it sets a fixed gap between C_p and C_v for any ideal gas regardless of its complexity. The ratio γ = C_p/C_v shows up everywhere adiabatic processes are involved. For monatomic ideal gases (helium, argon) γ = 5/3 ≈ 1.67; for diatomic gases (nitrogen, oxygen, hydrogen at room temperature) γ = 7/5 = 1.4; for polyatomic gases γ ≈ 4/3. NEET 2018 Q.34 turned on this idea precisely: given a process where V ∝ T (which makes the process isobaric, since PV = µRT implies P constant if V/T is constant), they asked for W/Q for a monatomic gas. Substituting Mayer's relation and γ = 5/3 gives W/Q = R/C_p = (γ − 1)/γ = 2/5.

Crucially, internal energy of an ideal gas is a function of temperature alone: U = µ C_v T plus a constant. Volume and pressure do not enter independently. This means that for an ideal gas, ΔU = µ C_v ΔT regardless of process — isothermal, adiabatic, isobaric, isochoric, all use the same formula for ΔU. The four processes differ in how ΔU is divided between Q and W, not in how it is calculated.

State variables & the equation of state

An equilibrium state of a thermodynamic system is completely described by a handful of macroscopic variables — pressure P, volume V, temperature T, mass m, internal energy U, and entropy S — called state variables. The defining feature of a state variable is that its value depends only on the current state, never on the history that produced it. Pressure, temperature, volume, and internal energy are state variables. Heat and work are not.

State variables come in two flavours. Extensive variables — volume, mass, internal energy — scale with the size of the system: cut the system in half and they halve. Intensive variables — pressure, temperature, density — are unchanged by such division. A relation that connects state variables at equilibrium is called an equation of state. For an ideal gas the equation of state is PV = µRT, where µ is the number of moles. This single relation reduces the number of independent variables from three (P, V, T) to two — once you fix two, the third is determined.

Crucially, a thermodynamic state needs the system to be in equilibrium. A gas in mid-explosion, or in mid-free-expansion, has no well-defined single pressure or temperature, and so cannot be marked as a single point on a P-V diagram.

Four thermodynamic processes

NEET concentrates almost obsessively on four quasi-static processes — those slow enough that the system stays in equilibrium with its surroundings throughout. Each is named for the variable held constant:

The isothermal process is one in which the system is kept in thermal contact with a large reservoir at temperature T throughout. The ideal gas equation reduces to Boyle's law, PV = constant. Because internal energy of an ideal gas depends only on temperature, ΔU = 0, and the first law gives Q = W. The work done in expanding from V₁ to V₂ is W = µRT ln(V₂/V₁). In an isothermal expansion, the gas absorbs heat from the reservoir and converts it entirely into work; in an isothermal compression, work is done on the gas and an equal quantity of heat is dumped to the reservoir.

The adiabatic process is one in which no heat is exchanged — the system is insulated, or the process is too fast for heat to escape. With Q = 0 the first law collapses to ΔU = −W: any work done by the gas comes entirely from its internal energy, which is why an adiabatic expansion cools the gas and an adiabatic compression heats it. The state equation is PV^γ = constant, and the work is W = µR(T₁ − T₂) / (γ − 1). The slope of an adiabatic curve at any point is steeper than that of an isothermal through the same point by a factor of γ — this is the geometric basis of NEET 2022's curve-identification question.

The isobaric process holds pressure constant — typically a gas heated in an open vessel against the atmosphere. Volume rises linearly with temperature (V/T = constant, Charles' law). Work is simply W = P(V₂ − V₁) = µR(T₂ − T₁). Heat absorbed is µC_pΔT — more than µC_vΔT, because some of it must be paid out as work. The isochoric process holds volume constant; no work is done; all the heat goes to changing internal energy: Q = ΔU = µC_vΔT.

A fifth case — the cyclic process — returns the system to its starting state after a finite sequence of operations. Since U is a state function, ΔU = 0 around any closed loop. The first law then gives Q_net = W_net: in a complete cycle, all net heat absorbed is converted to net work done. This is the operating principle of every heat engine.

Heat engines & efficiency

A heat engine is any device that, operating in a cycle, absorbs heat from a high-temperature reservoir (the source), converts part of it into mechanical work, and rejects the remainder to a low-temperature reservoir (the sink). Steam engines, internal-combustion engines, jet engines and power-plant turbines are all variations on this basic scheme. The working substance — typically a gas or vapour — undergoes a cyclic sequence of expansions and compressions, returning each time to its original state.

Conservation of energy demands that across one full cycle, Q₁ = W + Q₂, where Q₁ is heat drawn from the source, Q₂ is heat dumped to the sink, and W is the net work delivered. The efficiency of the engine is defined as the fraction of input heat converted to useful work:

For a real engine, η is typically between 20% and 40%. The rest is dumped to the environment — to a river, a cooling tower, or the air. The second law of thermodynamics asserts, and Carnot proved, that even for the best possible engine working between two temperatures, η is bounded — and the bound is severe.

Refrigerators & coefficient of performance

A refrigerator is a heat engine run in reverse. Work W is done on the system; heat Q₂ is extracted from a cold reservoir (the inside of the fridge); and a larger amount Q₁ = Q₂ + W is delivered to a hot reservoir (typically the kitchen air via the radiator coils on the back). The same energy balance applies — only the direction of every arrow is flipped. A heat pump is the same device, used for the heat it delivers to the warm side rather than the cooling it provides to the cold side.

For a refrigerator we do not speak of efficiency — the more useful figure is the coefficient of performance, α (sometimes written COP or β):

The COP α = Q₂ / W tells you how many joules of heat you can pull out of the cold space for every joule of electrical work you pay for. For a domestic refrigerator it is typically between 2 and 5. The reversible (Carnot) refrigerator achieves the maximum possible α = T₂ / (T₁ − T₂), where T₁ and T₂ are the hot and cold reservoir temperatures in kelvin. NEET 2016 Q.178 (working between 4 °C and 30 °C, extracting 600 cal/s) and NEET 2017 Q.153 (Carnot engine with 10% efficiency used as a refrigerator) both turned on this relation.

Second law of thermodynamics

The first law forbids the impossible feat of creating energy from nothing — but it does not forbid many things that nevertheless never happen. A book lying on a table never spontaneously cools its surroundings and leaps into the air. A pond at 20 °C never spontaneously divides into a hot half and a cold half. These processes would conserve energy perfectly. They are forbidden by an additional principle of nature called the second law of thermodynamics, which has two equivalent classical statements.

The Kelvin-Planck statement, articulated independently by Lord Kelvin and Max Planck, denies the possibility of a perfect heat engine: No process is possible whose sole result is the absorption of heat from a reservoir and the complete conversion of the heat into work. You cannot build an engine that takes heat from the ocean and turns all of it into shaft work without also dumping some heat somewhere colder.

The Clausius statement, by Rudolf Clausius, denies the possibility of a perfect refrigerator: No process is possible whose sole result is the transfer of heat from a colder object to a hotter object. Heat does not flow uphill without external work.

No heat engine can have η = 1. No refrigerator can have α = ∞.

The second law, in its sharpest form

The two statements look different but are mathematically equivalent: a violation of one would imply a violation of the other. Together they impose a fundamental, irreducible cost on energy conversion. Heat is not just energy — it is energy with a quality, a directionality, that work does not have. Some heat must always be wasted; some heat will always flow downhill. The second law is what makes time irreversible.

Reversible and irreversible processes

A process is reversible if it can be reversed step by step so that both the system and its surroundings return exactly to their initial states, with no trace left anywhere else in the universe. Reversibility is an idealisation; no real process is truly reversible. To be reversible a process must satisfy two stringent conditions: it must be quasi-static (infinitely slow, so the system stays in equilibrium throughout), and it must be non-dissipative (no friction, no viscosity, no resistive heating). Every real process violates one or both, sometimes badly.

Irreversibility shows up everywhere. Heat flowing across a finite temperature gap, free expansion of a gas into vacuum, combustion, diffusion, the stirring of a liquid, even the dropping of a coin on a tabletop — every one of these is irreversible. The arrow of time in thermodynamics is the arrow of increasing disorder, formalised by the concept of entropy (beyond the NEET syllabus, but worth knowing it exists).

Why does reversibility matter for an exam? Because the second law of thermodynamics, applied carefully, shows that a heat engine operating between two reservoir temperatures has the maximum possible efficiency if and only if every step of its cycle is reversible. Real engines, riddled with friction and turbulence and finite temperature gradients, fall short. The Carnot engine is the only fully reversible engine that uses just two reservoirs — and so it sets the ceiling.

Carnot engine

Sadi Carnot, a young French engineer, asked in 1824 the question that defined classical thermodynamics: what is the maximum efficiency possible for any heat engine operating between two fixed temperatures? His answer, derived before the first law was even properly stated, has survived two centuries of scrutiny intact.

Carnot's reasoning was geometric. To be reversible, every step must be quasi-static and non-dissipative. Heat exchange with a reservoir must therefore occur isothermally (no finite temperature gap), and temperature changes from one reservoir level to the other must occur adiabatically (no heat exchange at all). The unique cycle satisfying both constraints — using two isotherms and two adiabats — is the Carnot cycle:

Net work over one complete cycle equals the area enclosed by the loop on the P-V diagram, and (since the two adiabatic works cancel in magnitude) reduces to W = Q₁ − Q₂. Using the adiabatic relations for the two adiabats one shows that V₂/V₁ = V₃/V₄, so the ratio Q₂/Q₁ = T₂/T₁. The efficiency becomes:

Two profound consequences follow. First, Carnot's theorem: no engine operating between two given temperatures can exceed the Carnot efficiency, and the efficiency of a Carnot engine is independent of the working substance. The proof is by contradiction — if some engine I were more efficient than a Carnot engine R, coupling them so that I drives R as a refrigerator would extract net work from a single reservoir, violating Kelvin-Planck. Second, the universal relation Q₁/Q₂ = T₁/T₂ provides a definition of thermodynamic temperature independent of any particular substance — the foundation of the absolute Kelvin scale.

NEET 2023 Q.18, NEET 2018 Q.36 and NEET 2017 Q.153 all reduce to plugging numbers into η = 1 − T₂/T₁ — the most-tested formula in the chapter. With source at 100 °C (373 K) and sink at 0 °C (273 K), the Carnot efficiency is 1 − 273/373 = 26.8 % — the upper limit on every steam engine ever built.