Temperature, Ideal Gases and Related Topics

Temperature, Ideal Gases and Related Topics

1. Temperature

   • Temperature: The temperature of a substance is a measure of the mean translational kinetic energy associated with the disordered microscopic motion of its constituent atoms or molecules.
   • A thermodynamic temperature scale is one that does not depend on properties of substances that are used to measure temperature. e.g. kelvin

2. Equations of State

   • An equation of state for a thermodynamic system is a mathematical relationship between state variables
     • Isotherm - plot p vs V const T. Isobar - plot V ts T. const p. Isochors - plot p vs T. const V
   • EQUATION OF STATE FOR AN IDEAL GAS
     • p V = n R T
   • VAN DER WAALS EQUATION OF STATE
     • ( P + (a(n^2))/(v^2) )●(V-nb) = n R T
       • nb = molecular volume . so the volume for molecules around it = V - nb.
       • Intermolecular attraction = (a(n^2))/(v^2)
   • VIRIAL EQUATION OF STATE
     • (p V) / (n R T) = 1 + B_2(n/v) + B_3(n/v)^2 + B_4(n/v)^3 + ....
     • Valid for any isotropic substance if enough terms are used
   • EQUATION OF STATE FOR SIMPLE SOLIDS
     • V = V_0 [ 1 + β (T - T0) - K_t (P-P0) ]
       • β = ISOBARIC VOL EXPANSIVITY = ( + ΔV / V_0 ) / ΔT
       • K_t = ISOTHERMAL COMPRESSIBILITY = ( - ΔV / V_0 ) / ΔP

3. HEAT

   • Heat: a measure of the energy transferred between two systems as a result of a temperature difference
     • Heat Transfer Mechanisms = radiation, conduction, convection
       • STEFAN BOLTZMANN LAW FOR POWER RADIATED
         • P = Ɛ σ A (T^4)
     • HEAT TRANSFER RATE = Q dot = dQ/dT
       • Q dot = ( κ A / L ) ( T_1 - T_2 ) = - κ A (dT/dx)
         • THERMAL RESISTANCE = R_TH = L / κ A
   • SPECIFIC HEAT CAPACITY
     • Δ Q = c M Δ T
     • Specific Heat Cap = c Heat Capacity = C
     • Specific Heat Capacity depends on Temperature so you use derivatives to define it.
       • c_p (T) = (1/M) (δ Q / d T) _ p
       • c_V (T) = (1/M) (δ Q / d T) _ V

4. Kinetic Theory of Gases

   • Assumptions
     • Molecular radius small compared with avg distance between molecules. Constant rapid motion. Obey Newtons Laws. No force acting between - all collisions perfectly elastic. Container walls are perfectly rigid and infinitely massive. Gas in equilibrium.
     • Isotropic = same in all directions
       • < (V_x) ^2 > = 1/3 < V^2>
   • p V = 1/3 m N < V^2 > = 1/3 m N V^2 _rms
     • Comapring this to pressure eqn ( p V = N k_b T ) gives k_b T = 1/3 m < V ^2 > = 2/3 E_TR
       • E_TR = MEAN TRANSLATIONAL KINETIC ENERGY / MOLECULE
         • E_TR = 1/2 m <V ^2> = 3/2 k_b T
   • INTERNAL ENERGY ASSUME: no intermolecular forces, no rotational or vibrational KE