Kinetic Theory: NEET notes

1. Why a gas can be explained by moving molecules

NCERT §12.1

• Boyle found his law for gases in 1661. He, Newton and others already guessed that a gas is a swarm of tiny particles, but a working atomic theory arrived more than 150 years later.
• Kinetic theory pictures a gas as atoms or molecules in rapid motion. This works because the forces between atoms are short-ranged: they matter in solids and liquids, where atoms are close, but can be ignored in a gas, where they are far apart.
• Maxwell, Boltzmann and others built the theory in the nineteenth century.
• What it delivers: a molecular meaning for pressure and temperature, agreement with the gas laws and with Avogadro's hypothesis, correct specific heats for many gases, and links between bulk properties (viscosity, heat conduction, diffusion) and molecular ones, which gave the first estimates of molecular sizes and masses.

2. The molecular nature of matter

NCERT §12.2

• Feynman singled out the atomic hypothesis as the idea most worth passing on if all other science were lost: everything is made of atoms that never stop moving, pull on each other when slightly apart and push apart when squeezed together.
• The guess that matter is not continuous is old. Kanada in India (Vaiseshika school) and Democritus in Greece both proposed indivisible building blocks, but these ideas were never tested by measurement, so they did not grow into a science.
• John Dalton is credited with the scientific atomic theory. He used it to explain two laws of chemical combination: a compound always has a fixed proportion of its elements by mass (definite proportions), and when two elements form several compounds, the masses of one that combine with a fixed mass of the other are in small whole-number ratios (multiple proportions).
• Dalton's picture: an element's smallest parts are atoms, identical for one element and different from element to element; a few atoms join to form a molecule of a compound.
• Gay Lussac's law: when gases react to give another gas, their volumes are in small whole-number ratios. Avogadro's law: at the same temperature and pressure, equal volumes of all gases hold the same number of molecules. Avogadro's law together with Dalton's theory explains Gay Lussac's law.
• Scale: an atom is about 1 Å (10⁻¹⁰ m) across. In solids atoms sit about 2 Å apart, and in liquids at about the same spacing but free to move past one another, which is why liquids flow. In gases the spacing is tens of angstroms.
• Mean free path, the average distance a molecule travels between collisions, is of the order of thousands of angstroms in a gas. That freedom is why an unenclosed gas spreads away.
• The force between atoms attracts at a few angstroms and repels at shorter range.
• A gas at rest only looks static. Its molecules keep colliding and changing speed; only the averages stay constant, so the equilibrium is dynamic.
• Atoms are not the end of the story: they have nuclei and electrons, nuclei hold protons and neutrons, and those are made of quarks. This chapter stays with gases (and a little about solids) seen as collections of molecules in ceaseless motion.

3. The ideal-gas equation

NCERT §12.3

• Gases are the easiest state to understand, because their molecules are far apart and interact only during collisions.
• At low pressure, and at temperatures well above the point where they liquefy or solidify, gases nearly obey PV = KT for a given sample, with T in kelvin.
• K grows with the amount of gas: K = Nk, where N is the number of molecules. Experiment shows k is the same for every gas; it is the Boltzmann constant, k_B = 1.38 × 10⁻²³ J K⁻¹.
• So P₁V₁/(N₁T₁) = P₂V₂/(N₂T₂) = k_B. If two gases share P, V and T, they have the same N: this is Avogadro's hypothesis, and kinetic theory backs it up.
• At STP (273 K and 1 atm), 22.4 L of any gas holds 6.02 × 10²³ molecules (Avogadro's number N_A) and has a mass equal to the molecular weight in grams. This amount is one mole.
• Perfect-gas equation: PV = μRT, where μ is the number of moles and R = N_A k_B is universal. With T in kelvin, R = 8.314 J mol⁻¹ K⁻¹.
• μ = M/M₀ = N/N_A, where M is the sample's mass, M₀ the molar mass and N the number of molecules.
• Other forms: PV = k_B N T; P = k_B n T, where n = N/V is the number density; and P = ρRT/M₀, where ρ is the mass density.
• An ideal gas is one that obeys PV = μRT exactly at every pressure and temperature. It is a model; no real gas is truly ideal.

4. Real gases, Boyle's law and Charles' law

NCERT §12.3

• Real gases drift away from ideal behaviour; plots at three temperatures all move towards the ideal line as pressure falls and temperature rises (Fig. 12.1).
• The reason: at low pressure or high temperature the molecules are far apart, their interactions hardly matter, and the gas acts like an ideal one.
• Boyle's law: fix μ and T, and PV = constant. At constant temperature, the pressure of a given mass of gas is inversely proportional to its volume.
• Measured P–V curves for steam at three temperatures follow Boyle's law well at high temperature and low pressure and depart from it elsewhere (Fig. 12.2).
• Charles' law: fix P, and V ∝ T. Hold the pressure steady and a gas's volume grows in step with its absolute temperature (T–V curves for CO₂ in Fig. 12.3).
• Both laws are special cases of PV = μRT with one more variable held fixed.

5. Partial pressures and the size of a molecule

NCERT §12.3

• A mixture of ideal gases that do not react, μ₁ moles of gas 1, μ₂ of gas 2 and so on, in volume V at temperature T, obeys PV = (μ₁ + μ₂ + …)RT.
• So P = μ₁RT/V + μ₂RT/V + … = P₁ + P₂ + …. Here P₁ = μ₁RT/V is the partial pressure of gas 1: what it would exert alone in the same volume at the same temperature.
• Dalton's law of partial pressures: in a mixture of ideal gases, adding up the partial pressures gives the total pressure.
• Example 12.1: water has density 1000 kg m⁻³; its vapour at 100 °C and 1 atm has 0.6 kg m⁻³. The same mass spreads over 1000/0.6 times the volume, so molecules fill only a fraction 6 × 10⁻⁴ of the vapour's volume.
• Example 12.2: a mole of water is 18 g = 0.018 kg, so one molecule has mass 0.018/(6 × 10²³) = 3 × 10⁻²⁶ kg. Taking the molecule's density as that of liquid water, its volume is 3 × 10⁻²⁶/1000 = 3 × 10⁻²⁹ m³. Setting that equal to (4/3)πr³ gives r ≈ 2 × 10⁻¹⁰ m = 2 Å.
• Example 12.3: vapour gives each molecule about 1.67 × 10³ times the room it had in the liquid. A thousandfold volume means a tenfold length (cube root), so the radius of each molecule's share goes from 2 Å to 20 Å, and the average spacing is 2 × 20 = 40 Å.
• Example 12.4: neon and oxygen share a vessel with partial pressures 3 : 2. With V and T common, P ∝ μ, so the ratio of moles, and hence of molecules, is 3/2.
• Mass densities: ρ₁/ρ₂ = (μ₁M₁)/(μ₂M₂) = (3/2) × (20.2/32.0) = 0.947 for neon (20.2 u) to oxygen (32.0 u).

6. Pressure of an ideal gas from molecular impacts

NCERT §12.4, §12.4.1

• Model: a gas is a very large number of molecules (of order Avogadro's number) in random motion. At ordinary conditions their spacing is ten or more times their size (2 Å), so between encounters they move in straight lines by Newton's first law.
• An encounter where two molecules come close enough for their forces to act is a collision. Molecules collide with each other and with the walls, and every collision is taken as elastic: total kinetic energy is conserved, and momentum is conserved as always.
• Take a cube of side l with axes along its edges. A molecule with velocity (v_x, v_y, v_z) hits the wall parallel to the yz-plane, area A = l². It bounces back as (−v_x, v_y, v_z): only the x-component flips.
• The molecule's momentum changes by −mv_x − mv_x = −2mv_x, so it hands the wall a momentum 2mv_x.
• In time Δt only molecules within v_xΔt of the wall can reach it, i.e. those in a volume Av_xΔt. On average half of them move towards the wall, so ½nAv_xΔt hit it, where n is the number density.
• Momentum delivered in Δt: Q = (2mv_x)(½nAv_xΔt). Pressure is force per unit area, Q/(AΔt) = nmv_x².
• Molecules have a spread of velocities, so each speed group adds its share and P = nm⟨v_x²⟩, with ⟨v_x²⟩ the average of v_x².
• No direction is special (the gas is isotropic), so ⟨v_x²⟩ = ⟨v_y²⟩ = ⟨v_z²⟩ = ⅓⟨v²⟩. Hence P = ⅓nm⟨v²⟩.
• The cube shape does not matter: any small flat patch of any wall gives the same steps, and A and Δt drop out. By Pascal's law the pressure is the same throughout a gas in equilibrium.
• Collisions between molecules were ignored. In a steady state, any molecule knocked out of a velocity group is replaced by another knocked into it, so as long as collisions are brief compared with the time between them the result stands.

7. Kinetic interpretation of temperature

NCERT §12.4.2

• Rewrite P = ⅓nm⟨v²⟩ with N = nV molecules: PV = ⅔N(½m⟨v²⟩). The bracket is the average translational kinetic energy of one molecule.
• For an ideal gas the internal energy is purely kinetic: E = N × ½m⟨v²⟩, so PV = ⅔E. (E here is the translational part; other degrees of freedom come in §12.5.)
• Compare with PV = Nk_BT: E = (3/2)Nk_BT, i.e. ½m⟨v²⟩ = (3/2)k_BT.
• So a molecule's mean kinetic energy rises in direct proportion to the kelvin temperature, whatever the pressure, the volume or the kind of ideal gas. The Boltzmann constant is the bridge between a measured temperature and a molecular energy.
• Side result: the internal energy of an ideal gas depends only on T, not on P or V. With this reading of temperature, kinetic theory agrees fully with the ideal-gas equation and the gas laws.
• Mixtures: P = ⅓[n₁m₁⟨v₁²⟩ + n₂m₂⟨v₂²⟩ + …]. In equilibrium every species has the same average kinetic energy (3/2)k_BT, so P = (n₁ + n₂ + …)k_BT, which is Dalton's law again.
• Root mean square speed: v_rms = √⟨v²⟩ = √(3k_BT/m).
• Nitrogen at 300 K: m = M/N_A = 28/(6.02 × 10²⁶) = 4.65 × 10⁻²⁶ kg, so ⟨v²⟩ = 3k_BT/m = (516)² m² s⁻² and v_rms = 516 m s⁻¹, comparable to how fast sound travels in air.
• At the same temperature, lighter molecules have larger rms speeds.

8. Worked cases: speeds, isotopes and a moving bat

NCERT §12.4.2

• Example 12.5: argon and chlorine (2 : 1 by mass) at 27 °C. Average kinetic energy per molecule depends only on T, so the ratio is 1 : 1.
• Their rms speeds: (v_rms,Ar/v_rms,Cl)² = M_Cl/M_Ar = 70.9/39.9 = 1.77, so v_rms,Ar/v_rms,Cl = 1.33. The 2 : 1 mass ratio of the mixture plays no part.
• Example 12.6: uranium hexafluoride made with ²³⁵U has molecular mass 235 + 6 × 19 = 349 u; with ²³⁸U it is 352 u. The lighter molecule is faster: v₃₄₉/v₃₅₂ = (352/349)^½ = 1.0044, a difference of 0.44%.
• That small gap is used for enrichment. The gas is held inside a thick, narrow porous cylinder; molecules wander through the long pores one at a time, the faster ²³⁵U ones leak out slightly more, and the step is repeated many times (Fig. 12.5).
• The same reasoning shows why the rate of diffusion of a gas is inversely proportional to the square root of its molecular mass.
• Example 12.7: a ball approaching at speed u meets a massive bat moving towards it at V. Relative to the bat the ball arrives at V + u and leaves at V + u, so relative to the ground it leaves at 2V + u: it speeds up.
• Map it onto a gas: piston → bat, cylinder → ground, molecule → ball. A piston pushed in makes molecules rebound faster, so compressing a gas raises its temperature; a piston moving out makes them rebound slower, so an expanding gas cools.
• If the bat is not massive the rebound speed is less than u. For a cricketer, a heavier bat sends the ball back faster.

9. Degrees of freedom and equipartition of energy

NCERT §12.5

• A molecule's translational kinetic energy is ε_t = ½mv_x² + ½mv_y² + ½mv_z². At temperature T its average is (3/2)k_BT, and with no preferred direction each term averages ½k_BT.
• Degrees of freedom count the independent coordinates needed to locate something: one for motion along a line, two in a plane, three in space. Moving freely in three dimensions, a molecule therefore has three translational degrees of freedom, each contributing one squared term.
• A monatomic gas such as argon has only these three.
• A diatomic molecule such as O₂ or N₂ can also rotate about two axes perpendicular to the line joining its atoms (Fig. 12.6), adding ½I₁ω₁² + ½I₂ω₂². Spin about the bond line itself has almost no moment of inertia and, for quantum-mechanical reasons, stores no energy here.
• Treating O₂ as a rigid rotator (no vibration) works at moderate temperatures. Molecules such as CO vibrate even then: the atoms oscillate along their axis and add ε_v = ½m(dy/dt)² + ½ky², where k is the force constant and y the vibrational coordinate.
• Each translational or rotational degree of freedom adds one squared term; a vibrational mode adds two, one kinetic and one potential.
• Law of equipartition of energy (first proved by Maxwell): in thermal equilibrium the energy is shared equally among all the squared terms, each averaging ½k_BT.
• So each translational and each rotational degree of freedom gets ½k_BT, and each vibrational frequency gets 2 × ½k_BT = k_BT.
• The proof is beyond this book; the law is used here to predict specific heats.

10. Specific heats of gases

NCERT §12.6.1–§12.6.3

• Monatomic gas: three translational degrees of freedom, so (3/2)k_BT per molecule and U = (3/2)k_BT × N_A = (3/2)RT per mole. Then C_v = dU/dT = (3/2)R.
• For any ideal gas C_p − C_v = R, so a monatomic gas has C_p = (5/2)R and γ = C_p/C_v = 5/3.
• Rigid diatomic gas (a dumbbell): 3 translational + 2 rotational = 5 degrees of freedom, U = (5/2)RT, C_v = (5/2)R, C_p = (7/2)R, γ = 7/5.
• Diatomic gas that also vibrates: U = (5/2 k_BT + k_BT)N_A = (7/2)RT, so C_v = (7/2)R, C_p = (9/2)R, γ = 9/7.
• Polyatomic gas with 3 translational, 3 rotational degrees of freedom and f vibrational modes: U = ((3/2)k_BT + (3/2)k_BT + f k_BT)N_A, so C_v = (3 + f)R, C_p = (4 + f)R and γ = (4 + f)/(3 + f).
• C_p − C_v = R holds for every ideal gas, monatomic, diatomic or polyatomic.
• Table 12.1 lists the predictions with vibration left out (J mol⁻¹ K⁻¹): a monatomic gas has C_v = 12.5, C_p = 20.8, γ = 1.67; a diatomic gas C_v = 20.8, C_p = 29.1, γ = 1.40; a triatomic gas C_v = 24.93, C_p = 33.24, γ = 1.33. In every row C_p − C_v = 8.31.
• Table 12.2, measured values (same order): He 12.5, 20.8, 8.30, 1.66; Ne 12.7, 20.8, 8.12, 1.64; Ar 12.5, 20.8, 8.30, 1.67; H₂ 20.4, 28.8, 8.45, 1.41; O₂ 21.0, 29.3, 8.32, 1.40; N₂ 20.8, 29.1, 8.32, 1.40; H₂O 27.0, 35.4, 8.35, 1.31; CH₄ 27.1, 35.4, 8.36, 1.31.
• Prediction and measurement agree well for these gases, so equipartition is well confirmed at ordinary temperatures. For gases such as Cl₂ and C₂H₆ the measured values are usually higher than Table 12.1, and including vibrational modes brings them closer.
• Example 12.8: a fixed 44.8 L cylinder of helium at STP holds 2 mol (1 mol fills 22.4 L at 273 K and 1 atm). Fixed volume means C_v = (3/2)R applies, so heating it by 15.0 °C needs 2 × 1.5R × 15.0 = 45R = 45 × 8.31 = 374 J.

11. Specific heat of solids

NCERT §12.6.4

• Picture a solid as N atoms, each oscillating about a fixed point in its lattice.
• A one-dimensional oscillation has average energy 2 × ½k_BT = k_BT; in three dimensions an atom has 3k_BT.
• For one mole (N = N_A) the energy is U = 3k_BT × N_A = 3RT.
• At constant pressure ΔQ = ΔU + PΔV ≈ ΔU, because a solid's volume change is negligible. So C = ΔQ/ΔT = ΔU/ΔT = 3R.
• Table 12.3 (molar specific heats, J mol⁻¹ K⁻¹, at room temperature and atmospheric pressure): aluminium 24.4, carbon 6.1, copper 24.5, lead 26.5, silver 25.5, tungsten 24.9. The specific heats in J kg⁻¹ K⁻¹ are 900.0, 506.5, 386.4, 127.7, 236.1 and 134.4.
• 3R ≈ 24.9 J mol⁻¹ K⁻¹ matches these values at ordinary temperatures; carbon is the exception.

12. Mean free path

NCERT §12.7

• Gas molecules move about as fast as sound, yet gas leaking from a kitchen cylinder takes a long time to reach the far corners, and the top of a smoke cloud can hold together for hours. The reason is collisions: molecules have a small but finite size, so their paths keep being deflected.
• Model molecules as spheres of diameter d. A chosen molecule with average speed ⟨v⟩ hits any other whose centre comes within d of its own. In time Δt it sweeps a cylinder of volume πd²⟨v⟩Δt (Fig. 12.7).
• With n molecules per unit volume it makes nπd²⟨v⟩Δt collisions in Δt, so the average time between collisions is τ = 1/(nπ⟨v⟩d²).
• Mean free path, the average distance between successive collisions: l = ⟨v⟩τ = 1/(nπd²).
• That picture freezes the other molecules. They move too, so the relative speed should be used; a more exact treatment gives l = 1/(√2 nπd²).
• Air at STP: ⟨v⟩ = 485 m s⁻¹, n = (0.02 × 10²³)/(22.4 × 10⁻³) = 2.7 × 10²⁵ m⁻³ and d = 2 × 10⁻¹⁰ m give τ = 6.1 × 10⁻¹⁰ s and l = 2.9 × 10⁻⁷ m ≈ 1500d.
• l falls as the number density or the molecular size grows. In a well-evacuated tube n is so small that l can reach the length of the tube.
• Example 12.9: water vapour at 373 K, with the same d as air. n scales as 1/T: n = 2.7 × 10²⁵ × 273/373 ≈ 2 × 10²⁵ m⁻³, so l ≈ 4 × 10⁻⁷ m.
• That is about 100 times the 40 Å (4 × 10⁻⁹ m) spacing found in Example 12.3. A mean free path this large is what makes a gas behave like a gas: it cannot be kept without a container.
• Relations between bulk properties (viscosity, heat conduction, diffusion) and molecular size, worked out from kinetic theory, gave the first estimates of the size of molecules.

Must-know facts

1. Atom ≈ 1 Å; spacing ≈ 2 Å in solids and liquids, tens of Å in gases; mean free path thousands of Å.
2. Avogadro: equal volumes at the same T and P hold equal numbers of molecules; 22.4 L at STP holds 6.02 × 10²³.
3. PV = μRT = k_BNT, R = N_Ak_B = 8.314 J mol⁻¹ K⁻¹, k_B = 1.38 × 10⁻²³ J K⁻¹.
4. Real gases approach ideal behaviour at low pressure and high temperature.
5. Dalton: total pressure is the sum of partial pressures.
6. Water molecule: mass 3 × 10⁻²⁶ kg, radius ≈ 2 Å; spacing in vapour ≈ 40 Å.
7. Each wall hit hands over momentum 2mv_x; P = ⅓nm⟨v²⟩.
8. PV = ⅔E; average KE per molecule = (3/2)k_BT, independent of the gas.
9. Internal energy of an ideal gas depends only on T.
10. v_rms = √(3k_BT/m); nitrogen at 300 K: 516 m s⁻¹.
11. Speed ratio at the same T: v₁/v₂ = √(M₂/M₁); Ar : Cl₂ = 1.33, UF₆ isotopes differ by 0.44%.
12. Compressing a gas with a piston speeds up the molecules (2V + u), so it heats.
13. Equipartition: ½k_BT per translational or rotational degree of freedom, k_BT per vibrational mode.
14. Monatomic γ = 5/3, rigid diatomic 7/5, vibrating diatomic 9/7, polyatomic (4 + f)/(3 + f).
15. 44.8 L of helium at STP, heated by 15 °C at fixed volume: 374 J.
16. Solids: C = 3R ≈ 24.9 J mol⁻¹ K⁻¹; carbon is the exception.
17. Mean free path l = 1/(√2nπd²); air at STP: τ = 6.1 × 10⁻¹⁰ s, l = 2.9 × 10⁻⁷ m.
18. Water vapour at 373 K: l ≈ 4 × 10⁻⁷ m, about 100 times the molecular spacing.
19. ⟨v²⟩ is not in general equal to ⟨v⟩².
20. Air does not settle to the floor because mgh for ordinary heights is far below the molecules' kinetic energy.

Common traps

Formulas

Key terms

Atomic hypothesis

All matter is made of atoms in constant motion that attract when slightly apart and repel when pressed together.

Avogadro's hypothesis

Equal volumes of all gases at the same temperature and pressure contain the same number of molecules.

Avogadro number

6.02 × 10²³, the number of molecules in one mole; one mole of gas fills 22.4 L at STP.

Boltzmann constant

k_B = 1.38 × 10⁻²³ J K⁻¹, the gas constant per molecule, R/N_A.

Ideal gas

A model gas that obeys PV = μRT exactly at all pressures and temperatures.

Partial pressure

The pressure one gas of a mixture would exert if it alone filled the vessel at the same temperature.

Number density

Number of molecules per unit volume, n = N/V.

Elastic collision

A collision in which total kinetic energy is conserved as well as momentum.

rms speed

The square root of the mean of the squared molecular speeds, √(3k_BT/m).

Degree of freedom

An independent way a molecule can move or store energy; each squared energy term counts once.

Law of equipartition of energy

In thermal equilibrium each squared energy term averages ½k_BT.

Rigid rotator

A molecule that rotates but does not vibrate, like a dumbbell.

Ratio of specific heats (γ)

C_p/C_v, 5/3 for a monatomic gas and 7/5 for a rigid diatomic gas.

Mean free path

Average distance a molecule travels between successive collisions.

Dynamic equilibrium

A steady state in which molecules keep moving and colliding while the averages stay fixed.