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Gas

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state of matter which neither definite shape nor definite volume

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Equation of State

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thermodynamic equation relating state variables which describes the state of matter under a given set of physical conditions

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when was first description of equation of states seen

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300 years ago

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Boyle's Law

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1662, Constant T, P proportional to 1/V

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When is Boyle's Law followed?

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Low pressure

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use of boyles law

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with constant T, predict changing P or V with P1V1 = P2V2

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Boyle's law limit equation

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lim of p approaching 0 (PV) = A

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Charles Law

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1787, constant P, V proportional to T, V=kt

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when is charles law followed

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low Pressure

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charles law limit equation

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lim as p approaches 0 (V/T) = k

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use of charles law

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using constant P, predict changing changing V or T using V1T1 = V2T2

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Avogadro's Principle

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at same T and P, equal V has same number of molecules

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Ideal Gas Law

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PV = nRT

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limiting law of ideal gas law

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lim as p approaches 0 of PV = nRT

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Ideal gas

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obeys ideal gas law at all T and P and V

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Constant T Graph

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Isotherm (P vs. V)

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Constant P Graph

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Isobar (T vs. V), ideal gas shows straight line

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Constant V Graph

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Isochore (T vs. P) ideal gas shows straight line

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Thermal expansivity

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slope of an isobar; (alpha) = (1/V) (partial, dV/dT)p

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Isothermal compressibility

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slope of an isotherm; (beta) = (-1/V) (Partial, dV/dP)t

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why negative sign on isothermal compressibility

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to make beta positive since V decrease when P increases

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Total differential of V

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dV = (partial, dV/dV)p dT + (partial, dV/dP)t dP

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Why do gases behave ideally at low P

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molecules far apart, low attractive forces, low repulsive forces

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long range microscopic forces

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coulombic r^-2

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short range microscopic forces

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r^-6

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repulsive vs attractive force strength

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repulse - short range (high P) attract - mid-range (moderate P)

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3 potential interactions

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dipole-dipole, dipole-induced dipole, and london dispersion forces (induced dipole- induced dipole)

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dipole-dipole equation

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Vd = (-2/3)(mu^4/kT)(1/r^6)

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dipole-induced dipole equation

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Vind = -2alpha(mu^2)(1/r^6)

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London dispersion equation

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Vl = (-3/4)hv(alpha^2)(1/r^6)

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Collective interactions

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Van Der Waals Forces

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repulsive force equation

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Vr = B/(r^n)

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Lennard-Jones

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said n=12 is acceptable typical repulsive force

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Lennard-Jones 6-12 Potential

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Vlj = (-A/r^6)+(B/r^12)

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Hard sphere model

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first approx. of intermolecular potential r = intermolecular separation rm = molecular radius v=0, r > 2 rm v=infinity, r <or equal to 2rm

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Hard Sphere Model graph

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straight vertical line to 0

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Square Well Model graph

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negative graph shaped like rectangle

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L-J Model Graph

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starts high, dips to negative to minimum, comes back up to curve out at 0

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Compressibility Factor

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Z, used to measure how non-ideally a gas is behaving

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Compressibility Factor equation

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Z = (PV/nRT) = (PVm/RT)

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Virial Equation of state

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general, power series of Z in P or inverse V Z (V,T) = PVm/RT = 1 + B'T/Vm + C'T/(Vm)^2 Z (P,T) = PVm/RT = 1 + BTP + CTP^2 + DTP^3

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Virial coefficients

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constants which depend on temp in virial equation (Not Density)

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calculating B' for virial equation

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B' = 2piN (int(0 to inf) (1-e^-V(r)/kT) r^2 dr

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Estimate for B (virial)

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B approx= B'/RT

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Boyle Temperature Tb

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temperature at which virial coefficient vanishes

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Van der Waals Equation

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(P + (a/Vm^2))(Vm - b) = RT

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what does a stand for in VDW equation

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a is the cohesion or internal pressure term involving a accounts for attractive forces

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what does b stand for in VDW equation

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co-volume or excluded volume term involving b accounts for repulsive forces

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Berthelot's equation of state

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(P + (a/Vm^2T))(Vm-b) = RT

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difference between VDW and Berthelot's

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a and b are same values, not that much different

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similarity of all equation of states

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must reduce to ideal gas law as P approaches 0

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Critical Temperature

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temperature above which liquefaction cannot be induced at any pressure

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Critical Point means

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gas and liquid are indistinguishable

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Principle of Continuity of State

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one cannot distinguish between liquid and gas unless one observes the discontinuities that arise on passing into and/or out of the two-phase region

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Law of Corresponding States

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equation doesn't contain any constants specific to an individual gas, so it can describe all gases

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reduced variable

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alternative statement of law of corresponding states; every gas will act like every other gas

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Dalton's Law of Partial Pressure

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total pressure of a mixture of gases is equal to the sum of the partial pressures

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partial pressure equation

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Pi = ni(RT/V)

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Curvy Xi

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mole fraction of gas i

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amagat's Law

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similar to partial pressure but in regards to volume

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B in Virial extension

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measure of interactions between pairs of molecules

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