Thermal Properties, Thermodynamics & Kinetic Theory

Heat, the behaviour of gases and the laws that govern energy · Class 11 Physics

1The big idea: what heat really is

Every object is a crowd of jiggling molecules. Temperature is a measure of the average kinetic energy of that jiggling — how vigorously the molecules move — and nothing more. It does not measure how much energy is present; a spark at 1000 °C carries far less energy than a bathtub of warm water, because the bathtub has vastly more molecules.

Heat is different. Heat is energy in transit — energy that flows from a hotter body to a colder one purely because of their temperature difference. Once it has arrived, it is no longer "heat"; it has become part of the internal energy of the receiving body. So it is wrong to say a hot object "contains heat"; it contains internal energy, and heat is the name we give that energy only while it is crossing a boundary.

This one distinction — temperature as average energy per molecule, heat as energy in flow — underlies the whole chapter. It is why we measure temperature in kelvin (an absolute scale where T = 0 means all molecular motion has ceased), with T(K) = T(°C) + 273.15.

2Specific heat & latent heat

Pour the same flame under a pan of water and an equal mass of oil: the oil heats up far faster. Every substance has its own resistance to a change in temperature, captured by its specific heat capacity c — the heat needed to raise 1 kg of it by 1 K.

Water's specific heat is unusually large, c ≈ 4180 J kg⁻¹ K⁻¹. That is why coastal climates are mild (the sea soaks up and releases heat slowly) and why water is the coolant of choice. Sometimes we quote heat capacity per mole instead — the molar specific heat C — which is what we will need for gases later.

Now push a solid all the way to its melting point and keep heating. The temperature stops rising even though heat keeps flowing in. That energy is going into breaking the bonds that hold the solid together — a change of phase, not of temperature. The heat needed to change the phase of 1 kg with no temperature change is the latent heat L:

For water, the latent heat of fusion (melting) is L_f ≈ 3.34 × 10⁵ J kg⁻¹ and of vaporisation (boiling) is a huge L_v ≈ 2.26 × 10⁶ J kg⁻¹. That enormous L_v is why steam burns are so severe and why sweating cools you so effectively — evaporating sweat drags a great deal of energy off your skin.

3Thermal expansion

Heat a solid and its molecules jiggle harder, sitting slightly farther apart on average — so the object grows. For a rod of length L, the change is proportional to both the length and the temperature rise:

where α is the coefficient of linear expansion (about 1.2 × 10⁻⁵ K⁻¹ for steel). Area expands with a coefficient β ≈ 2α and volume with γ ≈ 3α — the factors of 2 and 3 come simply from a cube growing in two or three directions at once. This is why railway tracks are laid with gaps, why bridges sit on expansion joints, and why a tight metal lid loosens under hot water. Water plays its famous trick here too: between 0 °C and 4 °C it contracts on heating, so ice floats and ponds freeze top-down.

4Heat transfer: three ways

Heat moves from hot to cold by three mechanisms — know which one dominates where.

• Conduction — energy passes molecule-to-molecule through a material with no bulk movement. This is heat travelling up a metal spoon left in tea. The rate through a slab of area A, thickness d and conductivity k is Qt = k A ΔTd. Metals conduct well; air, wool and foam conduct badly (which is exactly what makes them insulators).
• Convection — heat carried by the bulk motion of a fluid. Warm fluid expands, becomes less dense, and rises; cooler fluid sinks to replace it, setting up a circulating current. This drives boiling water, sea breezes and the weather itself. Convection needs a fluid — it cannot happen in a solid or a vacuum.
• Radiation — energy carried by electromagnetic waves, needing no medium at all. This is how the Sun's warmth crosses empty space to reach us, and how any warm body glows. Every object radiates; the hotter it is, the more it pours out.

5Radiation laws — Stefan & Wien

A perfect radiator (a "black body") pours out energy at a rate that depends staggeringly steeply on its absolute temperature. Stefan's law (Stefan–Boltzmann) says the power radiated per unit area is

with σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴. The fourth power is the headline: double the absolute temperature and the radiated power leaps by 2⁴ = 16 times. This is why a filament that is only a few times hotter than its surroundings glows so brilliantly, and it lets astronomers gauge a star's luminosity from its temperature.

Wien's displacement law tells you the colour: the wavelength at which the radiation peaks shifts inversely with temperature,

so hotter bodies glow at shorter wavelengths — a heating iron runs from dull red, through orange and yellow, to blue-white as it climbs. The Sun peaks in the visible; your own body, at 310 K, peaks in the infrared, which is exactly why thermal cameras can see you in the dark.

6Kinetic theory: pressure from molecular chaos

Now we zoom in on a gas and derive its behaviour from nothing but bouncing molecules. The kinetic theory makes a few clean assumptions: molecules are tiny points in ceaseless random motion, they collide elastically with the walls and each other, and between collisions they fly free (no forces). From this alone we can explain pressure.

Where pressure comes from. Pressure is simply the drumbeat of countless molecules hammering the walls. Consider one molecule of mass m moving with speed component v_x toward a wall in a cubic box of side L. It hits, bounces back, and its momentum changes by 2 m v_x. It returns to hit again after travelling 2L, i.e. every 2L / v_x seconds. So the force from one molecule is momentum-change over time, m v_x² / L. Sum over all N molecules, split the motion equally three ways (⟨v_x²⟩ = 13⟨v²⟩), and divide by the wall area to get the pressure. The result, with volume V = L³, is the central equation of kinetic theory:

Connecting to temperature. Compare this with the experimental ideal gas law P V = n R T = N k T (here n is the number of moles, R = 8.314 J mol⁻¹ K⁻¹ is the gas constant, and k = R / N_A = 1.38 × 10⁻²³ J K⁻¹ is Boltzmann's constant). Matching the two expressions for PV gives 13 N m ⟨v²⟩ = N k T, so the average translational kinetic energy of a single molecule is

This is the promise made in Section 1, now proved: temperature really is average molecular kinetic energy. Notice it depends on T alone — at a given temperature every gas, heavy or light, has the same average molecular energy. Rearranging for the typical molecular speed gives the root-mean-square speed:

where M is the molar mass. Two things to carry away: v_rms ∝ √T (quadruple the absolute temperature and molecules move only twice as fast), and v_rms ∝ 1/√M (light molecules are faster — hydrogen zips along, which is why it escapes the atmosphere and heavy gases linger).

7Degrees of freedom & C_p − C_v = R

A molecule can store energy in more ways than just flying about. Each independent way is a degree of freedom, and the equipartition theorem hands each one an average energy of 12 k T per molecule. A single atom (like helium) only moves in three directions — 3 translational degrees, so its energy is 32 k T, matching what we derived. A diatomic molecule (like O₂ or N₂) can also tumble about two axes — 2 rotational degrees — giving f = 5 and energy 52 k T at ordinary temperatures.

Two ways to heat a gas. Heat a gas while pinning its volume, and every joule you add goes into internal energy (the gas does no work — it cannot push its walls out). Heat it at constant pressure and it expands, spending some of your heat pushing the surroundings back. So it takes more heat per degree at constant pressure: C_p > C_v. The gap is exactly the work of expansion for one mole per kelvin, which turns out to be R:

The ratio of the two, γ = C_p / C_v, is set purely by the degrees of freedom: C_v = f2 R, so γ = 1 + 2f. That gives γ = 5/3 ≈ 1.67 for a monatomic gas (f = 3) and γ = 7/5 = 1.40 for a diatomic gas (f = 5). This γ will govern how a gas behaves when squeezed suddenly, just ahead.

8Thermodynamics & the first law

Thermodynamics is the bookkeeping of energy for a gas (or any system). Its central quantity is the internal energy U — the total microscopic energy of all the molecules. For an ideal gas U depends on temperature alone (U = f2 n R T); change the temperature and you change U, full stop.

You can change a system's internal energy two ways: pour heat Q into it, or do work on it (or let it do work). Energy conservation then reads:

The work done by a gas as it expands is W = ∫ P dV — the area under the process curve on a pressure–volume graph. Two special cases follow immediately: at constant volume W = 0 (so ΔU = Q), and at constant pressure W = P ΔV.

9Isothermal vs adiabatic

Two ways of expanding a gas sit at opposite extremes, and telling them apart is a favourite exam theme.

Isothermal ("same temperature") — the gas is squeezed or expanded slowly, in good thermal contact with a reservoir, so it always has time to exchange heat and hold T constant. Since T is fixed, ΔU = 0, and the first law collapses to Q = W — all the heat absorbed comes straight back out as work. The path obeys Boyle's law, P V = constant.

Adiabatic ("no heat crosses") — the gas is changed so fast, or is so well insulated, that Q = 0. Now the first law gives ΔU = −W: if the gas expands it must pay for that work out of its own internal energy, so it cools; compress it and it heats. This is why a bicycle pump warms up and why rising air cools to form clouds. The path is steeper than an isotherm and follows

with the very same γ = C_p/C_v from Section 7. Because γ > 1, the adiabatic curve always falls off more sharply on a P–V diagram than the isothermal one through the same point.

10Second law & the Carnot engine

The first law says energy is conserved, but it does not say which way processes go. Heat never flows uphill from cold to hot on its own; a smashed cup never reassembles. The second law of thermodynamics captures this one-way street. One statement (Kelvin–Planck) is blunt: no engine can turn heat entirely into work — some heat must always be dumped to a colder reservoir. Equivalently (Clausius), heat cannot flow from cold to hot without work being done, which is precisely what a refrigerator's compressor is for.

A heat engine takes heat Q₁ from a hot source, converts part to work W, and rejects Q₂ to a cold sink; its efficiency is η = W/Q₁ = 1 − Q₂/Q₁. The second law forbids η = 1. The very best possible engine, running the ideal reversible Carnot cycle between temperatures T₁ (hot) and T₂ (cold), reaches

No real engine can beat this. It tells you efficiency improves only by making the source hotter or the sink colder, and that 100% efficiency would need an impossible sink at absolute zero. It is the ceiling every power station and car engine works beneath.

11Common confusions to clear up

• Heat is not temperature. Temperature is average energy per molecule; heat is total energy in transit. A lit match is hotter than a warm bath but carries far less energy.
• An object does not "contain heat". It contains internal energy. Heat is only the name for that energy while it flows across a boundary because of a temperature difference.
• During a phase change the temperature does not rise. Heat added at the melting or boiling point goes into mL (breaking bonds), not mcΔT, so a thermometer sits still until the change is complete.
• rms speed grows as √T, not T. To double the molecular speeds you must quadruple the absolute temperature. And it must be absolute temperature (kelvin) — never Celsius — in every kinetic-theory formula.
• Isothermal ≠ adiabatic. Isothermal holds temperature fixed by exchanging heat (ΔU = 0); adiabatic exchanges no heat (Q = 0) so the temperature must change. The adiabatic curve is the steeper one.
• Mind the first-law signs. W is the work done by the gas. When the gas is compressed, work is done on it, so W is negative and ΔU = Q − W makes the internal energy rise.

12Check yourself

Class 11 Physics · Thermal Properties, Thermodynamics & Kinetic Theory · aligned to NCERT Chapters 11–13 · SmartStudy.School