Temperature, Ideal Gases and Related Topics

Temperature
1. 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.
2. A thermodynamic temperature scale is one that does not depend on properties of substances that are used to measure temperature. e.g. kelvin
Equations of State
1. An equation of state for a thermodynamic system is a mathematical relationship between state variables
  1. Isotherm - plot p vs V const T. Isobar - plot V ts T. const p. Isochors - plot p vs T. const V
2. EQUATION OF STATE FOR AN IDEAL GAS
  1. p V = n R T
3. VAN DER WAALS EQUATION OF STATE
  1. ( P + (a(n^2))/(v^2) )●(V-nb) = n R T
    1. nb = molecular volume . so the volume for molecules around it = V - nb.
    2. Intermolecular attraction = (a(n^2))/(v^2)
4. VIRIAL EQUATION OF STATE
  1. (p V) / (n R T) = 1 + B_2(n/v) + B_3(n/v)^2 + B_4(n/v)^3 + ....
  2. Valid for any isotropic substance if enough terms are used
5. EQUATION OF STATE FOR SIMPLE SOLIDS
  1. V = V_0 [ 1 + β (T - T0) - K_t (P-P0) ]
    1. β = ISOBARIC VOL EXPANSIVITY = ( + ΔV / V_0 ) / ΔT
    2. K_t = ISOTHERMAL COMPRESSIBILITY = ( - ΔV / V_0 ) / ΔP
HEAT
1. Heat: a measure of the energy transferred between two systems as a result of a temperature difference
  1. Heat Transfer Mechanisms = radiation, conduction, convection
    1. STEFAN BOLTZMANN LAW FOR POWER RADIATED
      1. P = Ɛ σ A (T^4)
  2. HEAT TRANSFER RATE = Q dot = dQ/dT
    1. Q dot = ( κ A / L ) ( T_1 - T_2 ) = - κ A (dT/dx)
      1. THERMAL RESISTANCE = R_TH = L / κ A
2. SPECIFIC HEAT CAPACITY
  1. Δ Q = c M Δ T
  2. Specific Heat Cap = c Heat Capacity = C
  3. Specific Heat Capacity depends on Temperature so you use derivatives to define it.
    1. c_p (T) = (1/M) (δ Q / d T) _ p
    2. c_V (T) = (1/M) (δ Q / d T) _ V
Kinetic Theory of Gases
1. Assumptions
  1. 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.
  2. Isotropic = same in all directions
    1. < (V_x) ^2 > = 1/3 < V^2>
2. p V = 1/3 m N < V^2 > = 1/3 m N V^2 _rms
  1. Comapring this to pressure eqn ( p V = N k_b T ) gives k_b T = 1/3 m < V ^2 > = 2/3 E_TR
    1. E_TR = MEAN TRANSLATIONAL KINETIC ENERGY / MOLECULE
      1. E_TR = 1/2 m <V ^2> = 3/2 k_b T
3. INTERNAL ENERGY ASSUME: no intermolecular forces, no rotational or vibrational KE

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Temperature, Ideal Gases and Related Topics

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