Chapter 13 - Kinetic Theory

1661 me Boyle ne law diya PV=const. Newton aur dusron ne socha gas tiny atomic particles se bani hai. Actual atomic theory 150 saal baad establish hui. 19th century me Maxwell, Boltzmann ne kinetic theory develop ki. Remarkably successful - pressure aur temperature ka molecular interpretation diya, gas laws aur Avogadro hypothesis explain ki, specific heats explain ki, viscosity conduction diffusion jaise measurable properties ko molecular sizes masses se relate kiya.

Ek key idea - solids liquids me interatomic forces important hain, short range attraction long range? Actually short range repulsion, few angstrom pe attraction. Gases me average distance 10 times molecule size se zyada, isliye forces neglect kar sakte hain except collision time. Isliye ideal gas simple model hai.

Feynman kehta hai agar saara science destroy ho jaye aur next generation ko ek sentence dena ho, to Atomic Hypothesis: All things are made of atoms - little particles that move around in perpetual motion, attracting when little apart, repelling when squeezed. Yehi sabse important.

Ancient India me Kanada ne Vaiseshika school me paramanu ka idea diya - Bhoomi, Ap, Tejas, Vayu chaar types. Two atoms milke dvyanuka diatomic, three se tryanuka. Greece me Democritus ne atom word diya indivisible. Unke ideas ingenious the par quantitative experiments se test nahi hue, isliye modern science nahi bane tab.

Dalton ne atomic theory ko laws of definite and multiple proportions explain karne ke liye diya. Gay Lussac ka law: gases combine karte hain volumes small integer ratio me. Avogadro hypothesis: Equal volumes of all gases at same T,P have same number of molecules. Dalton atomic + Avogadro se Gay Lussac explain ho gaya, because elements molecules ke form me hote hain, hydrogen oxygen diatomic.

Sizes: Electron microscope se ab atoms dekh sakte hain. Atom size ~1 angstrom 1e-10 m. Solids me spacing few angstrom 2 Å. Liquids me bhi similar. Gases me average distance tens of angstroms. Mean free path - bina collision travel average distance - gases me thousands of angstroms, thousands times size. Isliye gases free. Density estimate: water density 1000 kg/m3, 1 mole 18g =0.018kg contains 6.02e23 molecules, so mass of one molecule ~3e-26 kg, volume ~3e-29 m3, radius ~2e-10 m =2 Å. Water vapour at 100C 1 atm density 0.6 kg/m3, so molecular volume fraction ~6e-4, vapour me volume 1671 times liquid, distance increases 10 times ~20 Å average separation 40 Å.

Gases ke properties solids liquids se easy hain kyunki molecules far apart, interactions negligible except collision. Real gases low pressures high temperatures pe ideal relation follow karte hain PV=KT where T absolute. K = N kB where kB Boltzmann constant 1.38e-23 J/K. So PV/NT = kB constant. If P,V,T same, N same for all gases - Avogadro hypothesis. 22.4 litres any gas at STP has 6.02e23 molecules - Avogadro number, mole definition. Mass of 22.4 L equals molecular weight in grams.

Perfect gas equation: PV=mu R T where mu moles, R=NA kB =8.314 J/molK. Other form P = kB n T where n number density molecules per volume, or P = rho R T / M0 where rho mass density, M0 molar mass.

Ideal gas defined as one satisfying PV=mu RT exactly at all P,T. No real gas truly ideal, but Fig real gases approach ideal at low P high T. At low P or high T molecules far apart, interactions negligible.

Fix mu and T: PV=constant Boyle's law. P vs V experimental curves match dotted Boyle curves at high T low P. Fix P: V proportional T Charles law. Mixture of nonreactive ideal gases: PV=(mu1+mu2+...)RT, partial pressure P1=mu1 RT/V, total P=P1+P2+... Dalton's law of partial pressures. Example neon and oxygen ratio partial pressures 3:2 given, so mu1/mu2=3/2, N1/N2=3/2, density ratio rho1/rho2 = (mu1 M1/V)/(mu2 M2/V) = (3*20.2)/(2*32)=0.947.

Gas ko large number molecules ka collection mano order Avogadro number, incessant random motion me. Ordinary P,T pe average distance 10 times molecule size 2 Å, so interaction negligible, free straight lines Newton first law. Occasionally close aate hain forces experience karte hain velocities change - collisions. Molecules incessantly collide each other or walls, velocities change, collisions elastic total KE conserved momentum conserved.

Assumptions: (1) All molecules identical, (2) Large number, random motion, (3) Size negligible compared to container, (4) No intermolecular forces except collisions, (5) Collisions elastic, (6) Time between collisions much larger than collision time.

Socho cube side l, axes sides ke parallel. Molecule velocity vx,vy,vz. Wall yz plane parallel area A=l^2 hit kare. Elastic collision, y,z unchanged, x reverses to -vx, vy,vz same. Momentum change of molecule = -mvx - (mvx)= -2m vx, so wall ko momentum 2m vx milta.

Time dt me wall hit karne wale molecules: x direction me vx dt distance ke andar wale hi pahunch sakte hain, volume A vx dt. Isme n number density hai to molecules count n A vx dt, but half average moving towards wall half away, so hitting count = (1/2) n A vx dt? Actually derivation me half factor later include karte hain careful: total number with velocity vx hitting = n A vx dt * (1/2). Momentum transferred in dt = 2m vx * (n A vx dt /2) = n m vx^2 A dt. Force = momentum per time = n m vx^2 A, pressure = Force/Area = n m vx^2.

Par sab molecules same vx nahi, distribution hai. Pressure due to group n. Total pressure sum over groups: P = n m average vx^2. Isotropic gas no preferred direction, so average vx^2 = vy^2 = vz^2 = (1/3)(vx^2+vy^2+vz^2)=1/3 average v^2. So P = (1/3) n m average v^2. Same as (2/3) * (N * 1/2 m average v^2)/V = (2/3) E/V where E total translational KE.

Remarks: Cube shape immaterial, any small planar area choose kar sakte, A and dt final result me cancel, Pascal law pressure anywhere same. Collisions ignored in derivation? Actually collisions random but steady state maintain, number molecules hitting wall calculation unaffected if collisions not too frequent and time in collision negligible vs between collisions.

Equation P=(1/3) n m ko PV = (1/3) N m = (2/3) N * (1/2) m likho. Quantity in bracket average translational KE per molecule. Total internal energy E of ideal gas purely kinetic (monoatomic) E=N*1/2 m . So PV=(2/3)E. Combine with ideal gas PV=N kB T, we get E=(3/2) N kB T. So average KE per molecule = (1/2) m = (3/2) kB T. Proportional to absolute temperature, independent of P,V,nature. Fundamental result connecting macroscopic T to molecular KE via kB.

Isse internal energy of ideal gas depends only on T, not P,V. Kinetic theory fully consistent with gas laws.

For mixture, total pressure P=(1/3)[n1 m1 + n2 m2 +...]. In equilibrium average KE of molecules of different gases equal, i.e. (1/2) m1 = (1/2) m2 = (3/2) kB T. So P=(n1+n2+...)kB T Dalton law recovered.

Typical speed: T=300K nitrogen M=28g/mol=28e-3 kg/mol, m=M/NA≈4.65e-26 kg. =3kB T/m = (3*1.38e-23*300)/4.65e-26≈2.66e5, sqrt =516 m/s order of speed of sound. Lighter molecules greater rms speed at same T. Argon chlorine flask mass ratio 2:1, atomic mass Ar 39.9, Cl2 70.9, average KE per molecule same at same T ratio 1:1, rms speed ratio sqrt(M2/M1)=sqrt(70.9/39.9)=1.33, Ar faster. Mass composition irrelevant to KE ratio, only T matters.

Fixed mass gas me sab molecules same velocity nahi, distribution hai. Collisions direction speed change karte hain, but equilibrium pe speed distribution constant. Jaise city me logon ki age groups - children up to 20, adults 20-60, old above 60, smaller intervals pe number proportional dx, dN(x)=nx dx. Similarly molecular speed distribution dN(v)=4πN a^3 exp(-b v^2) v^2 dv = n_v dv. Plot n_v vs v shows fraction molecules with speeds v to v+dv area of strip. Average of any quantity = (1/N) ∫ v^2 dN(v). More elementary considerations se =3kBT/m agrees with earlier. Smaller mass faster at same T, peak shifts.

Uranium isotopes masses 235 and 238 units, UF6 gas mass 349 and 352 units, speed ratio v349/v352 = sqrt(352/349)=1.0044 difference 0.44% lighter faster leaks more through porous wall enrichment method. Rate of diffusion inversely proportional sqrt mass Graham law.

Single molecule KE = 1/2 m vx^2 +1/2 m vy^2 +1/2 m vz^2. Equilibrium average <ε_t> = 3/2 kBT, and due to symmetry no preferred direction, each term average 1/2 kBT. So (1/2) m =1/2 kBT etc.

Molecule free to move in space needs 3 coordinates - 3 degrees of freedom for translation. Constrained to plane 2, line 1. Each translational degree contributes term square of velocity variable e.g. 1/2 m vx^2. In thermal equilibrium average of each such term = 1/2 kBT.

Monoatomic gas like argon only translational 3 DOF. Diatomic O2 or N2 has 3 translational + 2 rotational about axes normal to bond (rotation about bond axis moment inertia very small quantum reasons not counted). So 2 rotational DOF each contributes term containing square of angular speed ω. Energy total = 1/2 m vx^2+... +1/2 I1 ω1^2+1/2 I2 ω2^2 = 5 terms. If O2 rigid rotator no vibration.

CO even at moderate T has vibration mode atoms oscillate along interatomic axis like 1D oscillator contributes vibrational energy ε_v = 1/2 m (dy/dt)^2 +1/2 k y^2 - two squared terms y and velocity. So while each translational rotational contributes one squared term, one vibrational mode contributes two squared terms KE+PE.

Each quadratic term in energy expression is mode of absorption energy by molecule. In equilibrium total energy equally distributed in all possible energy modes, each mode average 1/2 kBT. This is law of equipartition: each translational and rotational degree contributes 1/2 kBT, each vibrational frequency contributes 2*1/2 kBT = kBT (KE+PE). First proved by Maxwell.

Monoatomic gas only 3 translational DOF, average energy per molecule (3/2)kBT, total internal energy per mole U=(3/2)RT NA? Actually U= (3/2) NA kB T = (3/2) RT. Molar specific heat at constant volume Cv = dU/dT = (3/2)R. Cp-Cv=R so Cp=5/2 R. Ratio gamma=Cp/Cv=5/3=1.67.

Diatomic rigid 5 DOF 3 trans+2 rot, total energy U=5/2 RT, Cv=5/2 R, Cp=7/2 R, gamma=7/5=1.40.

If diatomic not rigid but has vibration mode, U=5/2 RT + kB T per vibrational? Actually one vibrational mode adds kBT per molecule average = RT per mole, so U=5/2 RT + RT =7/2 RT, Cv=7/2 R, Cp=9/2 R, gamma=9/7≈1.29. Usually vibrational frozen at moderate T for O2, active at high T.

Polyatomic general 3 translational, 3 rotational, f vibrational modes, U=(3+3+f?) Wait: Actually (3 translational *1/2 +3 rotational *1/2 + f vibrational *1) kBT per molecule? The formula in book: U= (3kBT +kBT + f kBT) NA? Let's correct: Translational 3*1/2=3/2, rotational for non-linear 3*1/2=3/2, so 3kBT plus f*kBT for f vibrational modes (each kBT). So U=(3+f)RT? Actually for polyatomic with f vibrational modes: U=(3+ f) RT? No, need factor. Simplified: General Cv=(3+f)R/2? Wait check Table 13.1: Triatomic predicted Cv=3R, Cp=4R, gamma=4/3=1.33 which corresponds to 6 DOF (3 trans+3 rot). That is f=0. So U=3RT Cv=3R. Yes matches. So general formula: For polyatomic with f vibrational modes, Cv=(3+3/2+f?) Let's keep simple as book: U=(3/2+3/2+f)kBT NA = (3+f)RT, Cv=(3+f)R, Cp=(4+f)R. But for non-linear without vibration f=0 gives Cv=3R as observed. For linear triatomic like CO2 might be different. Use table values.

Predicted values ignoring vibrational agree well with experimental values for mono and di rigid gases at room T. For Cl2, C2H6 polyatomic experimental Cv greater than predicted suggesting vibrational modes contribute. Law of equipartition verified experimentally at ordinary temperatures.

Specific heat of solids: solid of N atoms each vibrating about mean position, 3D oscillator average energy per atom 3kBT (2*1/2 kBT per dimension? Actually per dimension KE+PE = kBT, 3 dimensions 3kBT). Per mole U=3RT, Cv=3R Dulong-Petit law. Table 13.3 shows prediction generally agrees at ordinary temperature except carbon exception.

Specific heat of water: Treat like solid, each atom average 3kBT, water molecule has 3 atoms 2H+1O so U=3*3kBT NA=9RT, C=9R=74.8 J/molK ~75 observed very good. In calorie gram degree units water defined unit specific heat 1 cal/gK =4.18J, per mole 18g =>75 J/molK ~9R. More complex molecules alcohol acetone more complicated.

Important aspect: Predicted specific heats independent of temperature classically, but as T→0 specific heats approach zero, degrees freedom get frozen, quantum effect. Einstein first explained low T departure.

Molecules in gas have large speeds order sound, yet gas leaking from cylinder in kitchen takes considerable time to diffuse to other corners, smoke cloud holds together hours. Why? Because molecules have finite though small size, undergo collisions, cannot go straight long, random walk.

Mean free path λ average distance travelled between collisions. Derivation: Collision cross-section σ=π d^2 where d molecular diameter. In time dt molecule sweeps volume σ * average relative speed? Simplified result λ = 1/(√2 π d^2 n) where n number density. Using ideal gas n=P/kBT? Actually n=N/V=P/kBT, so λ = kBT/(√2 π d^2 P). At STP order 1000 Å =1e-7 m. So thousands times molecular size.

If pressure decreases or temperature increases, n decreases, λ increases. Transport properties viscosity conduction diffusion depend on λ and average speed.

1827 Robert Brown observed pollen grains ~1e-5 m suspended in water under microscope continuously moved zigzag random. Kinetic theory explains: Any object suspended in water continuously bombarded from all sides by water molecules random. For ordinary size object number hits from opposite directions about same, small difference negligible vs total hits, no movement noticed.

When object sufficiently small but visible under microscope, difference in molecular hits from different directions not altogether negligible, net impulse torque this or that direction, object moves zigzag tumbles randomly. This Brownian motion visible proof of molecular activity. Last 50 years molecules seen by scanning tunneling microscopes. 1987 Zewail observed molecules and interactions with laser flashes.

When gases diffuse rate of diffusion inversely proportional sqrt mass Graham law rate ∝ 1/√M. Lighter diffuses faster. Uranium has two isotopes masses 235 and 238 units, as UF6 gas masses 349 (235+6*19) and 352 (238+6*19). Smaller mass faster speed average energy same at fixed T, v349/v352=√(352/349)=1.0044 difference 0.44% faster for lighter. Porous cylinder thick narrow molecule wanders individually colliding walls, faster molecule leaks out more enrichment outside. Method not very efficient repeated several times for sufficient enrichment.

Example: Flask contains argon and chlorine ratio 2:1 by mass, temperature 27C, ratio average KE per molecule 1:1 same T, ratio vrms =√(M2/M1)=√(70.9/39.9)=1.33.