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• Equation of state

  A relation between state variables in physics and thermodynamics, describing the state of matter under a given set of physical conditions
• Equation of state


  • It is a constitutive equation which provides a mathematical relationship between two or more state functions associated with the matter, such as its temperature, pressure, volume
• Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and even the interior of stars
• Boyle's law


  The gas volume varies inversely with the pressure, mathematically: PV = Constant
• Charles's law


  Indicating a linear relationship between volume and temperature

  
• Dalton's law of partial pressures

  The pressure of a mixture of gases is equal to the sum of the pressures of all of the constituent gases alone, mathematically: P = Σ Pi
• Ideal gas law


  PVm = R(TC + 273.15), where P is pressure, V is volume, n is amount of substance, R is the gas constant, and T is absolute temperature
• The ideal gas law is a good approximation to the behavior of many gases under many conditions, although it has several limitations
• The ideal gas law was first stated in 1834 as a combination of Boyle's law, Charles' law and Avogadro's Law
• The ideal gas law is often written as: PV = nRT
• 1 atm = 1 bar = 1 torr = 760 mmHg = 105 Pa
• The ideal gas law has been presented as an empirical relation, but it doesn't work perfectly for all gases under all conditions because it is based on imperfect assumptions
• Molar volume


  The volume of 1 mole of gas or liquid
• Specific gas constant


  The ratio of the universal gas constant to the molar mass of the gas
• Three forms of equation of state of ideal gas

  Using initial and final state

  1. Using gas constant for a specific gas
  2. Using universal gas constant
• Real gas


  Non-hypothetical gases whose molecules occupy space and have interactions, adhering to gas laws
• Real gases


  • Need to account for: compressibility effects, variable specific heat capacity, van der Waals forces, non-equilibrium thermodynamic effects, issues with molecular dissociation and elementary reactions with variable composition
• Compressibility factor (Z)


  Measures the deviation of a gas from its ideal state, equal to PV/nRT
• As pressure increases


  Compressibility factor (Z) increases to a number larger than one, distorting the ideality
• As temperature decreases


  Compressibility factor (Z) rises above 1 as the temperature approaches a smaller number
• Van der Waals equation of state

  An equation of state that takes into account molecular stickiness and molecular size
• The conditions assumed for an ideal gas are: 1) Molecules are perfectly elastic, 2) Molecules are point masses, 3) Molecules move at random
• The first two assumptions for an ideal gas are clearly wrong for all gases, because at low temperatures all gases condense or form a liquid phase due to molecular stickiness
• Correcting the ideal gas equation for molecular stickiness

  PIdeal = PReal + a (n^2/V^2)

  (P + a n^2/V^2) V = nRT
• Correcting the ideal gas equation for finite molecular volume

  VIdeal = VReal - nb, where b is the volume of a mole of gas molecules at rest
• Probability of the second molecule being in the same place

  Same as the number density (n/V)
• Reduction in pressure due to stickiness

  Proportional to (n/V)^2
• Free volume

  Real (container) volume minus the volume taken up by the gas molecules
• van der Waals constants a and b

  Different for different gases, generally increase with increase in mass and complexity of gas molecule
• van der Waals constants for different substances

  • He: a=0.0341, b=0.0237
  • H2: a=0.244, b=0.0266
  • O2: a=1.36, b=0.0318
  • H2O: a=5.46, b=0.0305
  • CCl4: a=20.4, b=0.1383
• Avogadro's law

  Under same temperature and pressure, equal volumes of different gases contain equal number of molecules (6.022140857 × 10^23 molecules per mole)