Properties of gases • • • Ideal & real gases ! • Applications of kinetic theory Kinetic theory of gases! Maxwell-Boltzmann distribution! Collision Frequency, z • • • Volume swept-out by a single particle ⇡d2 hvrel i t Number of collisions for one particle = volume × number density Collision frequency for one particle = number of collisions in unit time Collision Density, Z • Collision density is the total number of collisions occurring per unit volume Mean free path = v⇥ z Applications of kinetic theory • • • • Effusion! Diffusion! Thermal Conductivity! Viscosity Effusion • Gas escapes into a vacuum through a small hole! • No collisions occur after the particles pass through the hole.! • The rate of effusion will be proportional to the rate at which particles strike the area defining the hole! • The diameter of the hole is smaller than the mean free path in the gas! • No collisions occur as the particles pass through the hole Effusion Collisions with container walls ⇥V ⇤⇤ ⇥V = = m 2 kB T P (vx )dvx = ⇤ ⇤ VP P (v (vxx)dv )dvxx V ⌅ ⇤ ⌅ ⇤ =A A tt = vvxx 2 0 2 0 0 0 m m exp T exp BT kkB ⇥1/2 e 2 mvx 2kB T ⇥ 2⇥ 2 mv mvxx dv x 2k T dvx BT 2kB Collision frequency with wall (per unit area, per unit time) ⇢N = N p = V kB T Effusion Collisions with aperture Rate = dN pa = zwall a = 1/2 dt (2 mkB T ) N kB T V ⇥ = kB T dN V dt p dp d = dt dt t p = p0 exp t/⇥ where ⇥= 2 m kB T ⇥1/2 V a dvx Effusion dN pa = zwall a = 1/2 dt (2 mkB T ) p = p0 exp t/⇥ p Rate = t • Graham’s law of effusion! • Rate of effusion of a gas is inversely proportional to the square root of the mass of its particles Transport Properties Property Transported Quantity Diffusion Matter Thermal Conductivity Energy Viscosity Momentum Kinetic Theory D= 1 3 v⇥ 1 = ⇥ v⇥ CV [X] 3 Units m2s-1 J K-1 m-1 s-1 kg m-1 s-1 where ⇥= Flux Describes the amount of matter, energy, charge... passing through a unit area per unit time For matter flux, Jz d N where ρN is the number density. dz Fick’s first law of diffusion Jz = D= D 1 3 d N dz v⇥ ⇥N ( ) = ⇥N (0) zwall Jleft ⇥v⇤ ⇥N ( = 4 ) ⇥v⇤ = 4 d⇥N dz N = ⇢N hV i = V ⇥N (0) Jz = Jleft For matter flux, Jz ⇥ d⇥N dz ⇥N (+ ) = ⇥N (0) + 0 ✓ kB T 2⇡m ⇥ ⇥ ◆1/2 Jright 0 Jright = 1 2 = ⇢N ⇥ 0 ⇥ 0 hvi 4 v⇥ ⇥N (+ ) v⇥ = = 4 4 d⇥N dz d⇥N dz d⇥N dz ⇥N (0) + ⇥v⇤ d N where ρN is the number density. dz Diffusion in a Gas D= 1 2 v⇥ ⇥ ⇥ 0 ⇥N ( d⇥N dz ) = ⇥N (0) zwall Jleft ⇥v⇤ ⇥N ( = 4 ) ⇥v⇤ = 4 ⇥N (0) ⇥N (+ ) = ⇥N (0) + 0 N = V d⇥N dz Jz = Jleft For matter flux, Jz ⇥ kB T 2 m ⇥ ⇥ ⇥1/2 = Jright 0 Jright = 1 2 v⇥ ⇥N (+ ) v⇥ = = 4 4 d⇥N dz ⇥ Diffusion D= D 1 3 ⇥ 0 N v⇥ V 4 0 d N dz v⇥ d⇥N dz ⇥N (0) + ⇥v⇤ d N where ρN is the number density. dz Jz = d⇥N dz D= D 1 23 v⇥ v⇥ ⇥ ⇥ 0 Diffusion in a Gas Thermal Conductivity in a Gas T T Previously ⇥N ( d⇥N dz ) = ⇥N (0) ⇥ d⇥N dz ⇥N (+ ) = ⇥N (0) + 0 ⇥ 0 Now zwall Jz = Jleft ⇥= N = V kB T 2 m Jright = 1 kB ⌅N ⇤ v⇥ 3 ⇥1/2 = 1 2 N v⇥ V 4 dT dz ⇥ 0 kB ⇤N ⇥ ⇥v⇤ 1 = ⇥ v⇥ CV [X] 3 Transport Properties Property Transported Quantity Diffusion Matter Thermal Conductivity Energy Viscosity Momentum Kinetic Theory D= 1 3 v⇥ 1 = ⇥ v⇥ CV [X] 3 Units m2s-1 J K-1 m-1 s-1 kg m-1 s-1 Transport Properties Property Transported Quantity Diffusion Matter Thermal Conductivity Energy Viscosity Momentum Kinetic Theory D= 1 3 v⇥ 1 = ⇥ v⇥ CV [X] 3 Units m2s-1 J K-1 m-1 s-1 kg m-1 s-1 Classical Mechanics Summary • • Translation! • • • • Newton’s laws! Equations of motion! Linear momentum! Frames of reference & reduced mass! Rotation! • • • Angular momentum! Moments of inertia! • Vibration! • • Simple harmonic motion! The wave equation! Work & Energy! • Elastic & inelastic Collisions! • Centre of Mass Properties of Gases Summary • • Ideal & real gases! • • • pV=nRT! Virial expansion! Compression factor! Kinetic theory of gases! • • • ! • Derivation! Classical equipartition! Predicting Cv! • Maxwell Boltzmann! • • • (Derivation)! Collision frequencies! Mean free path! Applications! • • Effusion! Diffusion
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