Physical Chemistry – Kinetic Theory of Gases – Flashcards

Graham's law states that the rate of effusion of a gas from a small hole is inversely proportional to (molar mass)^(1/2) If the wall has hole in it of area A0: *Rate of effusion = Zw.A0 =p.A0/(kT2πm)^(1/2) =p.A0.L/(MRT2π)^(1/2)* {m = M/L, k = R/L} Knudsen applied Graham's law to measure (very low) vapour pressures of solids/liquids by measuring loss of mass, Δm by effusion in a time Δt *[ Δm = Zw.A0.m.Δt ]* {Zw.A0 = rate of loss of molecules, m = molecular mass} Zw = Δm/(A0.m.Δt) = p/(2kTπm)^(1/2) {rearrange} p = (Δm/A0.m.Δt) (2kTπm)^(1/2) *[ p = (Δm/A0.m.Δt) (2kTπ/m)^(1/2) ]* *[ p = (Δm/A0.m.Δt) (2RTπ/M)^(1/2) ]*

Real gases deviate from perfect behaviour Z = p(Vm)/RT {Vm = molar volume, Vm = V/n} For an ideal gas, Z = 1 low pressures - real gases behave like ideal gases i.e Z~1 moderate pressures - *long range attractive forces* draw molecules together, hence *Z1* Z also depends on mean speeds of molecules low T (or high M) slow moving molecules are more readily captured by one another i.e. Z1 At the Boyle temperature, TB (an intermediate temperature), a real gas behaves ideally over a range of p

At large Vm and high T (low p) gas behaves more ideally At small Vm and low T (high p) behaviour is non-ideal So, P(Vm) = RT can be expanded to account for non-ideality: *[ p(Vm) = RT(1 + B'p + C'p^2) ]* *[ p(Vm) = RT(1+B/(Vm) + C/(Vm)^2) ]* There are versions of the virial equations p(Vm) = RT(1+B/(Vm) + C/(Vm)^2) +... {1st virial coefficient = 1, 2nd virial coefficient = B, 3rd virial coefficient = C} Virial coefficients are dependent on T At typical molar volumes B/(Vm) >> C/(Vm)^2) Virial equations predict Z = p(Vm)/RT = 1 when (Vm) -> ∞ (p -> 0) The slope of Z with p (or 1/(Vm)) differs in virial equations *For an ideal gas: Z = 1, dZ/dp = 0 For a real gas Z = 1 + B'p + C'p^2+..., dZ/dp = B' + 2C'P Z= 1+B/(Vm) + C/(Vm)^2) +..., dZ/dp = B + 2C/(Vm) = B as (Vm) -> ∞* The slope of Z with p or 1/(Vm) is 0 when B=0 at the Boyle temperature, TB, B(TB)=0

At low temperatures (below Tc) increasing the p of a gas can lead to liquefaction or even solidification {Tc is the critical T (critical isotherm): Lowest T at which liquid phase can form by compression} above Tc a fluid which fills the entire volume exists, often the fluid is denser than a gas. The critical pressure (pc), volume (vc) and temperature (Tc) are constants for a given gas.

A more realistic model of the gas based on p(Vm) = RT It takes into account: 1) The volume of the gas molecules themselves Volume = Vm - b hence p = RT/(Vm-b) 2) The reduction of the pressure due to: attractive forces between the molecules reducing the *frequency* and *force* of collisions, both α (1/Vm) Hence reduction in p is a/(Vm^2) Thus *[ p = RT/(Vm-b) - a/(Vm^2) ]* or for n moles of gas *[ mRT/(V-nb) - an^2/v^2 ]* as {Vm = V/n} a and b depend only on the nature of the gas, these can be linked to virial coefficients B and C

Look at potential energy of interaction U(R) as a function of molecular separation, R For the 'hard sphere' model, p.e. = 0 until molecules collide. The model molecules are impenetrable spheres, the p.e. of interaction is ∞ at R < 0 where σ = d (in this case)

The simplest improvement is the square well potential. The model still uses hard spheres, but allows the total p.e to be lowered by ε at intermediate values of R. This model accounts for the complex formation , and the existence of condensed phases (liquids and solids)

We must consider the nature of the forces between molecules - Repulsive forces: when molecules are close together the electrons and nuclei of the different molecules repel one another (+ deshielding). Hence, the potential energy goes up with smaller R: * [ Urepulsion(R) = Aexp(-BR) ]* {A and B are constants} - Attractive Forces: At large separation molecules attract one another. Polar molecules do this by Coulombic attraction. Non-polar molecules do so because of: 1) e- motion => instantaneous dipole 2) the molecule polarises a second molecule and there is an attraction 3) induced dipole The attractive interaction between molecules is known as a dispersion interaction or a London interaction. This depends on: 1) The polarisability of molecules (α) 2) The extent of control of the nucleus over the outer electrons (Ionisation energy) (I) 3) And most importantly - on separation (R) Hence, we write: *[ Urepulsion(R) = -C6/R^6 ]* This is the London formula

The overall effect of the attractive and repulsive potentials is obtained as the sum. As an approximation the exponential repulsive part is replaced by R^-12 Hence: *[ Urepulsiion(R) ~ C12/R^12 ]* Thus: *[ Utotal(R) = C12/R^12 - C6/R^6 ]* {C6 and C12 are constants}

A more realistic potential energy to interactions of two molecules: *[ U(R) - 4ε(σ^12/R^12 - σ^6-R^6) ]* {ε = well depth - minimum p.e. of interaction σ = separation when U(R) = 0 i.e. V(σ) = 0} σ0 = zero potential separation, separation at minimum potential ε can be calculated, because at the minimum: dU(R)/dR = 0 The force between molecules, F = -dU(R)/dR hence, at ε, F=0 general expression for force: *[ F = (24ε/σ)[2(σ/R)^(13) - (σ/R)^(7)] ]* {from differentiations} The attractive force maximises at R = (26/7)^(1/6)σ The Lennard-Jones parameters allow calculation of virial coefficients, hence non-ideality can be predicted This is often called the Lennard Jones 12-6 potential