Kinetic Theory: Gas Behavior and Atomic Theory

• Kinetic theory elucidates gas behavior by positing that gases comprise rapidly moving atoms or molecules
• Inter-atomic forces are negligible for gases, contrasting with their significance in solids and liquids

Key Figures and Development

• Boyle, Newton, and others initially proposed gases consist of tiny atomic particles
• The actual atomic theory was established over 150 years later
• Key developers include Maxwell and Boltzmann, developing it in the 19th century

Successes of Kinetic Theory

• Provides a molecular interpretation of gas pressure and temperature
• Consistent with gas laws and Avogadro's hypothesis
• Accurately explains specific heat capacities of many gases
• Relates measurable gas properties to molecular parameters, estimating molecular sizes and masses

Molecular Nature

• Richard Feynman emphasizes matter being composed of atoms
• Atomic Hypothesis states that all things are made of atoms

Atomic Concepts

• Atoms are in perpetual motion, attracting when slightly apart and repelling when squeezed together
• Speculation about non-continuous matter existed across cultures, including Kanada in India and Democritus in Greece

Ancient Atomic Theories

• Kanada (6th century B.C.) in India and Democritus in Greece conjectured about atoms long before modern science
• Kanada's Vaiseshika school detailed eternal, indivisible atoms as ultimate matter parts
• Four atom types postulated: Bhoomi (Earth), Ap (water), Tejas (fire), and Vayu (air) with mass and attributes
• Akasa (space) was considered structure less, continuous, and inert.
• Atoms combine into molecules, like diatomic dvyanuka or triatomic tryanuka, with properties depending on the nature and ratio of constituent atoms.
• Atom size was also estimated
• Estimates in Lalitavistara (2nd century B.C.) are close to modern estimates of atomic size ~10^-10 m
• Democritus believed atoms differed physically in shape, size, and properties, resulting in different substance properties

Dalton's Atomic Theory

• Proposed to explain laws of definite and multiple proportions
• Fixed proportion by mass of constituents in a given compound
• Multiple compounds formed by two elements have masses in small integer ratios for a fixed mass element.
• Atoms of one element are identical but differ from other elements
• Small numbers of atoms from each element combine to form a molecule

Gas behavior

• Gay Lussac's law: Combining gases yield another gas in small integer volume ratios
• Avogadro's law: Equal gas volumes at equal temperature and pressure have the same number of molecules
• Avogadro's law combined with Dalton's theory explains Gay Lussac's law
• Dalton's atomic theory can also be referred to as molecular theory
• Now well accepted, although some 19th-century scientists did not believe in atomic theory

Modern confirmation

• Molecules (one or more atoms) constitute matter
• Electron and scanning tunnelling microscopes enable visualization of atoms and molecules
• Atom size: ~1 angstrom (10^-10 m)
• Solids: Atoms tightly packed, spaced a few angstroms (2 Å) apart
• Liquids: Atoms have similar separation but are not rigidly fixed, enabling flow

Gases

• Interatomic distances in gases: tens of angstroms
• Mean free path: average molecule distance without collision
• Mean free path in gases: thousands of angstroms
• Atoms are much freer in gases and travel long distances without colliding

Intermolecular Forces

• Closeness in solids and liquids makes interatomic force important, with long-range attraction and short-range repulsion
• Atoms attract at a few angstroms but repel when closer

Gas Dynamics

• Gas appearance misleading, full of dynamic activity
• In dynamic equilibrium, molecules collide and change speeds
• Only average properties are constant
• Atoms consist of a nucleus and electrons
• Nuclei comprise protons and neutrons, which also made of quarks

Gases vs. Liquids

• Easier to understand gas properties compared to liquids and solids
• Molecules are far from each other, which leads to their interactions being negligible
• At low pressures and high temperatures gases satisfy a simple relation between pressure, temperature and volume

Ideal Gases

• PV = KT, where T is temperature in Kelvin and K is constant for a sample
• K is proportional to the number of molecules N, we can write K = Nk,Observation tells us that k is same for all gases called Boltzmann constant denoted by KB

Avogadro's Hypothesis

• If P, V, and T are the same, N (number of molecules) is the same for all gases
• Number of molecules in 22.4 litres of any gas is 6.02 × 10^23 (Avogadro number, NA)
• The mass of 22.4 litres of any gas at STP equals its molecular weight in grams
• S.T.P means standard temperature 273 K and pressure 1 atm
• Amount of substance is called a mole

Perfect Gas Equation

• Can be written as PV = μRT, where μ is the number of moles and R is a universal constant
• With Kelvin scale, R = 8.314 J mol-1 K-1
• Another equation is: μ = M/M0 = N/NA, with M as the gas mass, M0 as the molar mass, and NA as Avogadro's number
• PV = N kBT or P = n kBT where n is the molecules per unit volume

Boltzmann Constant

• kB value in SI units is 1.38 × 10^-23 J K-1
• P = (ρ/M0) RT, with ρ as gas density

Ideal vs. Real Gases

• A gas that satisfies Eq. (12.3) at all pressures and temperatures is an ideal gas
• No real gas is truly ideal and at low pressures and high temperatures it behaves like an ideal one
• Low pressures or high temperatures means means molecules are far apart and molecular interactions are negligible

Boyle's Law

• Fix μ and T in Eq. (12.3), then PV = constant i.e., keeping temperature constant, pressure of a given mass of gas varies inversely with volume
• Fig. 12.2 shows comparison between experimental P-V curves and the theoretical curves predicted by Boyle's law
• Good agreement at high temperatures and low pressures

Charles' Law

• Fix P, Eq. (12.1) shows that V ∝ T (volume of a gas is proportional to its absolute temperature)

Dalton's Law of Partial Pressures

• For non-interacting ideal gases, μ1 moles of gas 1, μ2 moles of gas 2, etc., in volume V at temperature T
• PV = ( μ1 + μ2 + ... ) RT
• P = P1 + P2 + ...
• Partial Pressure= μ1 RT/V is the pressure that gas 1 would exert at the same conditions if no other gases were present
• Total Pressure of a mixture of ideal gases is the sum of partial pressures

Molecular Volume

• Density of matter is less if volume is large
• Given mass of water molecules volume of vapour is 1000/0.6 = 1/(6 × 10^-4) times larger.
• If densities of bulk water and water molecules are same, then the fraction of molecular volume to the total volume in liquid state is 1
• Estimated volume of a water molecule is (3 x 10^-26 kg)/ (1000 kg m-³) = 3 x 10^-29 m³ = (4/3) π (Radius)3 which gives a radius of ≈ 2 ×10-10 m = 2Å
• A given mass of water in vapour state has 1.67×103 times the volume of the same mass of water in liquid state

Gas Mixtures

• PV = μ1 RT and P2V = μ2 RT which gives us (P1/P2) = (μ1 / μ2)
• Also μ1 = (N1/NA) and μ2 = (N2/NA) giving us (N1/N2) = (μ1 / μ2)
• Similarly μ1 = (m1/M1) and μ2 = (m2/M2), if ρ1 and ρ2 are mass densities then
• P2 = p₁ = M₁ x µ₁/V = (М₁) µ₁ M2 µ2/V
• So the density ratio ρ1/ρ2 is M₁μ₁ /(M2M2) = M₁ μ₁ /(M2μ2)

Kinetic Theory of Ideal Gas

• Gas is a collection of a large number of molecules in incessant random motion typically of the order of Avogadro's number
• At ordinary pressure and temperature, average distance between molecules is a factor of 10 greater than the typical size of a molecule
• Interaction between molecules is negligible so we assume they move freely in straight lines according to Newton's first law
• They come close to each other occasionally and experience intermolecular forces

Molecular Collisions

• Molecules collide incessantly against each other or with the walls and change their velocities
• Collisions are considered elastic
• Pressure of a gas can be derived based on the kinetic theory

Molecular Motion

• Molecule with velocity (vx, vy, vz) hits the planar wall parallel to the yz-plane of area A (= l²)
• After an elastic collision (vx, vy, vz) becomes (-vx, vy, vz)
• Momentum imported to the wall becomes 2mvx
• Number of molecules with velocity (vx, vy, vz) hitting the wall in time Δt is 1 2 nA v Δt
• Total momentum transferred by those molecules becomes Q = (2mvx) (1/2 n A vxΔt),
• Force is the rate of momentum transfer. F = Q/Δt, so the Pressure becomes P = (F/A) = nm
• Molecules in a gas do not have the same velocity there is a distribution in velocity we determine this by groups

Gas Isotropic Properties

• Total pressure is obtained by summing the contribution due to all groups

• P = n m

• Gas is isotropic, so there is no preferred direction in the vessel. Therefore can deduce following symmetry:

• x2= y2= z2=(1/3)[ x2+ y2+ z2]=(1/3) v2So resulting Pressure Equation becomes:

  • P=(1/3)nm(Vx)

• Though we choose the container to be a cube, the shape is immaterial

• Collisions are random; any velocity change due to collision is balanced by others keeping the distribution steady

Temperature Interpretation

• PV = (1/3) nV m v2
• PV = (2/3) N x 1/2 m v2
• The internal energy E of an ideal gas is purely kinetic* so: E = N (1/2)m v2 leading to Pressure equation PV= (2/3)E
• E = (3/2) N kBT this is due to the Ideal Gas equation with the above Pressure equation so (E/N) =3/2 KBT

Boltzmann Constant Effect

• The average kinetic energy of a molecule is proportional to the absolute temperature; it is independent of pressure, volume, or gas nature

• Internal energy depends on temperature, not pressure or volume

• For Non-reactive ideal gases, total pressure gets contribution from each gas

• P = (1/3) [n1m1v12 + n2m2v2 +... ]In equilibrium, the average kinetic energy of molecules of different gases will be equal which means 1/2 m1 v12 = 1/2 m2 v2 = (3/2) KBTso that resulting Pressure can be equal to as:

• P= (n1 + n2 +...) / 2 KBT This is Dalton's law of partial pressures

Speed and Mass

• M= 28/ 6.02x10^23=4.65 x10^-26 , so <v^2> is:

  • 3kBT

  1/2 (485)2
  

  Speed to Sound RMS:

  3kBT /m And the The important speed is known and is of the order of Molecules have larger 516 ms . , molecules S

  Speed with larger

• If the ratio is 2:1 by mass. Given the same temperatures the ratio is always one

• Average kinetic energy of monoatomic like argon; diatomic chlorine and polyatomic is always equal as it is independent of gas

• When gases diffuse, the rate of diffusion is inversely proportional to square root of the masses is fixed

• If the bat is not massive the rebound speed will be less

Energy & Molecules

• Kinetic Energy of a single molecule is known as the following equation Where for a gas at room temperate <&> is: Eι= 1/2mvx2+ 1/2mvy2+ 1/2mvz2=& >= 1/2 mvx^2+ 1/2 mvy^2 + 1/2 mvz^2 =3/2 KBTTherefore no preferred direction exists

• The direction implies the following: the square of (The same for all directions) equals; *KBTTherefor:

  • Mv 2= 1/2 KBT

Molecule Freedom Space

• Space needs 3 coordinates to specify location: plane needs 2 coordinates: line needs 1
• A molecule has 3 transitional degrees of freedom: 2 for a plane and 1 on a line
• Each transitional degree of freedom adds a term to an equation *MV 2 similar for all 3 terms given: 12.24 In thermal we see that:

Each / 2 KBT equilibrium, the average term =

Rotational Axis

• O molecule HAS 3 degrees of freedom but it can can rotate on a centre too. Figure

• The independent axis can use w angular speed or *I - the moment of inertia:

Each energy the Energy contains a squared rotational