Liquids and Solutions Flashcards
What is an extensive/intensive variable?
Extensive - depends on size of system
Intensive - doesn’t depend on size
What are some examples of intensive and extensive variables?
Extensive: A, V, n (therefore dA, dV, dn)
Intensive: T, P, μ
What is the condition for chemical eqm?
μiα = μiβ
where μ is derivative of G wrt n
What is an ideal solution?
Equivalent to ideal gas
All interactions are the same - molecule 1 doesn’t care if surrounded by 1 or 2
Ideal entory of mixing
What is the ΔmH (of mixing) in an ideal solution?
ΔmH = 0
As all interactions are the same
What is the model used for ideal ΔmS?
Lattice model of a binary mixture
N1</sub molecules of type 1
N2 molcules of type 2
N lattice sites
Random mixing
What is Boltzmann’s eqn of entropy?
S = kb lnΩ
Ω = number of ways you can realise a given conformation
Ω = N!/[N1!N2!]
What is the ideal ΔmS?
ΔmS = -R[n1lnx1 + n2lnx2]
where ΔmS = Smix - Sunmix
What is x and sterling’s approx?
x is mole fraction
For small values of x:
lnx! = xlnx - x
What is the ΔmG of an ideal solution?
ΔmG = ΔmH - TΔmS = 0 + RT[n1lnx1 + n2lnx2]
Totally entropic
What is the μ of an ideal solution?
μ = (δG/δn)
G = Gunmix +RT[n1lnx1 + n2lnx2]
μ = μi + RTlnxi
Vapour pressure linearly related to molar pressure
What is Raoult’s law?
For an ideal solution on a boundary with ideal gas
Pi = xiPi*
How can you derive Raoult’s law?
μisolution = μivapor
Gas and solution is ideal
1) μi* + RTlnxi = μθ + RTln(Pi/Pθ)
2 (when @xi)) μi* = μθ + RTln(Pi/Pθ)
1-2 so:
RTlnxi = RTln(Pi/Pθ) - RTln(Pi*/Pθ)
gives:
xi = Pi/Pi*
How does a solid in an ideal solution change temeprature?
Solid is solvent
ΔT = T - T* = - X2 (RT*2/ΔfusH)
where T* is the freezing point of water, and X2 is mole fraction of ideal solution
What is a non-ideal solution?
Interactions not identical
Non-ideal entropy of mixing
Combination of both
What is a regular solution?
Same as ideal soltuion but interactions not identical
so Δm =! 0
How is a polymer solution different to an ideal one?
Non-ideal (so different) entropy of mixing
What is the potential of a non-ideal solution?
μi = μi* + RTlnki + RTlnfi
where the activity coefficient is fi
What is Henry’s law?
Pi = xiKH
Not derived
When component is the minority - so linear vapour pressure when low or high x
When is Raoult’s and Henry’s law followed?
Roult’s - for a substance when pure (xi~1)
Henry’s law - for a substance when in minority (xi~0)
What is ΔmS in a regular solution?
Same as ideal
ΔmS = -R[n1lnx1 + n2lnx2]
How do you derive the sum of interactions, w, in a regular solution?
ΔH = Hmix - Hunmix
Hunmix = (1/2)zN1w11 + (1/2)zNw2w22 = (z/2)(N1+N2)(w11x1 + w22x2)
Hmix = (z/2)(N1+N2)(w11x12 + w22x22 + 2w12x1x2
Then work out difference and eventually gives
ΔmH = z(n1+n2)x1x2 [w12 - (w11+w22)/2]
What is sum of interactions in a regular solution?
w = NAz[w12 - (w11+w22)/2]
What model is used for the enthalpy of mixing in a regular solution?
Lattice model of binary mix
What are the assumptions in enthalpy of regular solution?
Random mixing
Only nearest neighbour interactions, w
Volume constant, and as H = U + PV, so ΔH = ΔU
What is the coordination number, z, in nearest neighbours?
Number of neighbours at a lattice site
z=3, triangle
z=4, square
z=6, hexagonal
What is the ΔmH of a regular solution?
ΔmH = (n1 + n2) Wx1x2
where W = NAz[w12 - (w11+w22)/2]
What is ΔmG of a regular solution?
ΔmG = ΔmH - TΔmS
ΔmG = (n1+n2)Wx1x2 + RT[n1lnx1 + n2lnx2]
What is the chemical potential of a regular solution?
μ1 = μ1* + RTlnx1 + wx22
can relate to f1 = exp[wx22/RT]
What is the ideal limit of activity coefficient?
f1 = exp[wx22/RT]
ideal limit when w=0 so f1 = 1
What is nearest neighbour interactions in an ideal solution?
w11 = w12 = w21 = w22
How do you relate the vapour pressure of regular solutions?
P1 = x1P1* exp[wx22/RT) = x1P1* f1
Remember: for ideal gas and regular solution chemical eqm
How do you derive eqn for vapour pressure of regular solution?
Boundary of ideal gas and regular solution in eqm,
μsolution = μvapour
μ1* + RTlnx1 + Wx22 = μ1θ + RTln(P1/Pθ)
when x1 = 1 and x2 = 0,
μ1* = μ1θ + RTln(P1/Pθ)
then 1st - 2nd equation
RTlnx1 + Wx22 = RTln(P1/P1*)
rearrange to
P1 = x1P1* exp[Wx22/RT]
What is seen when two phases separate?
Starts with mix
Finishes with two phases, neither are pure but just richer in one
What factors determine if phase separation occurs in a regular solution?
g = wx1x2 + RT[x1lnx1 + x2lnx2]
when high w then encourages demixing, when low then mixing
when high T encourages mixing, when low T is demixing
Wants to min Gibbs free energy
How does W change if mixing/demixing occur?
W = NARz[w12 - (w11+w22)/2]
if W +ve then w11 and w22 more -ve so more attractive than w12
What is Tc?
Critical temperature - T at beyond which there is a one-phase region
What is seen in a molar gibbs plot for mixing?
Double well needed for phase separation
As T increases then one phase more likely
What is the chemical potential of two separate phases?
μ1 = μ1* - X2(dg/dX2) + g
μ2 = μ2* - (1 - X2)(dg/dX2) + g
What is teh common tangent construction?
Minima gives coesting densities after separation
(dg/dx2a) = [g(x2a) - g(x2b)]/(x2a-x2b)
What is the coexistence condition for two phases?
(dg/dx2)a = (dg/dx2)b = 0
This is required for 2 phases to be present
What is the condition (derivative) for the critical temp of two phases?
(dg/dx2)Tc = (d2g/dx22)Tc = 0
When this satisfied then at the Tc
What is a binodal or spinodal region of a gibbs graph?
Binodal - line on graph when (dg/dx2)a = (dg/dx2)b = 0 , gives 2 phases, dominates at lower T
Spinodal - line on graph when (d2g/dx22)Tc = 0 , gives mix, dominates at higher T
What is the approach for understanding colloids?
Treat as huge (slow and see-able) atoms
Same stat thermo and similar phase behaviour
What is the random flight model for a polymer?
Polymer is a random walk of N statistical segments of length l
Stat segments can include multiple momomers
What is the Flory-Huggins theory?
Volume fraction formula for entropy of mixing of polymer solutions
What is the lattice model for a polymer solution?
N1 molecules of type 1 (solvent)
N2 molecules of type 2 (polymer, with r stat segments per molecule N2
Number of lattice sites, N = N1 + rN2
What is volume fraction?
Equivalent to mole fractions but for space taken by solvent (1) or polymer (2)
φ1 = N1/(N1 + rN2)
φ2 = rN2/(N1 + rN2)
What is Ωunmix for a polymer solution?
Ωunmix = [rN2!/N2)(z/rN2)](r-1)N2
What is Ωmix for a polymer solution?
Expected without corrections:
Ωmix = N!/(N1! rN2!)
Must implement corrections:
1) random mix doesnt apply, so z/n for each segment other than first one (where z is coord)
2) Cannot switch chem segments unlike separate molecules, so multiply by rN2!/N2!
Actual result
Ωmix = N!/(N1! N2!) (z/N)(r-1)N2
What is ΔmS for a polymer solution?
Find by taking logs of Ω and using stirling’s approx as for ideal
ΔmS = kb[ln(N!/(N1! N2!) + (r-1)N2 ln(rN2/N)] = -R[n1lnφ1 + n2lnφ2]
What is Hunmix for a polymer solution?
Hunmix = (z/2)N1w11 + (z/2)rN2w22 = (1/2)(N1+rN2)[w11N1/N + w2rN2/N]
Hunmix = z/2 (N1+rN2)(w11φ1 + w2φ2)
What is Hmix for a polymer solution?
Start with Hmix = (z/2)N1w11ϕ1 + (z/2)N1w12ϕ2 +(z/2)N1w11ϕ1 + (z/2)N1w12ϕ2 + (z/2)rN2w22ϕ2 + (z/2)rN2w21ϕ1
Simplifies down to:
Hmix = (z/2)(N1 + rN2)[w11ϕ12 + w22ϕ22 + 2w12ϕ1ϕ2]
What is ΔmH for a polymer solution?
ΔmH = (n1 + rn2)wφ1φ2 = RT(n1 + rn2)χφ1φ2
What is the flory-huggins parameter?
χ = w/RT
What is ΔmG for a polymer solution?
ΔmG = RT[n1lnφ1 + n2lnφ2 + (n1+ rn2)χφ1φ2
What is the chemical potential of a polymer solution?
μ1 = μ1* + RT[lnφ1 + (1-1/r)φ2 + χφ22]
How can you find the vapour pressure & phase separation of polymer solutions?
Follow same process as with regular, just replace values
What is the source osmotic pressure of polymer solutions?
Semi-permeable membrane which polymer cannot pass through, but solvent can
Osmotic pressure is difference in pressure on either side
What is the osmotic pressure of polymer eqn in terms of pressure difference?
Π = Pβ - Pα
Where β contains the polymer
What drives osmosis and how is this shown in equation?
Π = Pβ - Pα = (μ1 - μ1*)/V1*
then sub in: μ1= μ1* + RT[lnφ1 + (1-1/r)φ2 + χφ22
gives:
Π = RT[ (1/M2)c2 + (1/2 - χ)/(ρ22V1*) c22 + ..]
How does osmotic pressure relate to virial expansion for gases?
Π = RT[ (1/M2)c2 + (1/2 - χ)/(ρ22V1*) c22 + ..]
Virial expansion for gases:
P = RT[(1/V) + B(T)(1/Vmean2) + …]
What is the Van’t Hoff law?
(ideal osmotic pressure)
Π = RT[c2/ M2]
This is from the first term of expansion
What is the experimental osmotic pressure?
Π = RT[ (1/M2)c2 + (1/2 - χ)/(ρ22V1*) c22 + ..]
2nd osmotic virial coefficient, A2 = (1/2 - χ)/(ρ22V1*)
Plot c2 on x and Π/c2 on y
y-intercept = RT/M2
gradient = RTA2
What are a good, theta, and poor solvents?
Good - max interactions with solvent, polymer spread out
Theta - ideal osmotic pressure
Poor - max interactions with self, polymer folded in
How does the Flory-Huggins parameter (χ) and A2 show solvent quality?
Good solvent : χ < 1/2 and A2 > 0, T = θ
Theta: χ = 1/2 and A2 = 0, T =θ
Poor solvent: χ > 1/2 and A2 < 0, T =θ
What is end-to-end distance of a polymer?
R = Σ I
<R> = Σ <I> = 0
This is 0 due to the randomness of the polymer
What is mean square distance of a polymer?
Removes effect of -ve contribution to the mean
<R2> = Nl2 = Ml2/m
where M is mol weight and m is segment mol weight
What is the radius of gyration?
Rg is just useful experimental value
< Rg2 > = < R2 > / 6
so Rg proportional to sqrt(M)
What is the composition of an electrolyte?
Av+Bv- -> v+ AZ+ + v- BZ-
What interactions are present in electrolytes?
Long-range electrostatic interactions
Presence of ionic atmosphere of counter-ions
What is an ionic atmosphere?
-ve of ion surrounded by +ve ions and vice cersa
What is the chemical potential of one component of electrolyte solutions?
μ+ = μ+θ + RTlnc+ + RTlny+
ci is molarity, equivalent for mole fraction
yi is activity coefficient equivalent
What is μ of electrolyte solutions?
μ = μθ + vRTlnc+- + vRTlny+-
How do you derive the potential of a charge in solution?
Potential of an isolated charge: ψ(r) = (1/r)(e/4πε0εr)
in solution: ψ(r) = (exp-κr/r) (e/4πε0εr)
then taylor expansion: ψ(r) = (1/r - κ) (e/4πε0εr)
where κ = debyte length
What is κ, debye length?
κ-1 is range of ionic atmosphere
κ-1 = Sqrt[ε0εrkBT/2e2n0]
n0 = N+/-/V
κ-1 α 1/Sqrt[c]
where c is salt conc
What is the potential of an ionic atmosphere?
ψatm = -eκ/4πε0εr
How do you find the work required to bring in an ion to a electrolyte?
wel = ∫ψatm dq = ∫-eκ/4πε0εr dq
where ψatm is potential of ionic atmosphere
wel = -κe2/8πε0εr
What is the relation activity coefficient of an ion in electrolyte?
1:1 electrolyte
RTlny = NAwel = -κe2/8πε0εrkBT
What is relation of activity coefficient and concentration of an electrolyte?
logy+- = - A Sqrt[c]
What is ionic strength (I) in debye-huckel law?
I = 1/2 Σ cizi2
where c is concentration and z is ionic charge
What is the debye-huckel limiting law?
logy+- = -z+z-A Sqrt[I]
where I is an i
What is ionic strength of a 1:1 electrolyte with 1+ charges (AgCl)?
I = 1/2 Σ cizi2
I = (1/2) (c) (1*12 + 1*12) = c
How does conc of electrolyte relate to K of solubility?
log c = (1/2)logKs + A Sqrt[I]
Sol increases if inert electrolyte conc increases
How does increasing salt change solubility?
Adding more inert salt decreases work to bring in more ions, so more soluble
As decreases debye length
What is surface tension?
Force per unit length which resists an external force
Minmises surface area
What is the notation for surface tension and interface?
γ = surface tension
σ = interface
What is required to increase surface area?
Work against surface tension which wants to min surface area
F = 2γl, where l is length and 2 as bottom and top surface of film
dw = Fdx = 2γldx = γdσ
What effects dominate at different size of systems?
Bulk dominates for large systems (length3)
Surface dominates for smaller systems (length2)
What is the origin of surface tension?
Molecules attracted to surrounded in bulk
Costs energy to move a molecule from bulk to surface as loses attraction so it costs work to create a surface
This is surface tension
What happens when you have a small and large bubble connected?
Small bubble goes into large
This is because pressure in smaller bubble is larger
What is Laplace’s law?
ΔP = 2γ/R
where R is radius of droplet
when isothermal enlargement
What is required for soap bubbles?
Pinside > Poutside
so inside can act against force preventing it from forming
How do you derive Laplace’s law?
outside force (expansion work then expansion work against outside P0) = inside force (work to increase bubble size)
(P0+ΔP)dV - P0dV = γdσ
σ = 4πR2
dσ = 8πRdR
ΔP (4πR2dr) = γ8πRdR
ΔP = 2γ/R
What is the Kelvin equation relating to?
Vapour pressure of a liquid droplet
What is Kelvin’s equation?
Pr = P0 exp[2γvI/rRT]
where Pr is vap pressure of a liquid droplet with radius r, with respect to P0
What is the vapour pressure of a small droplet?
Small droplet has large lapace pressure so large (2γ/r)
Pr»_space; P0
What is the vapour pressure of a large droplet?
Large droplet has small lapace pressure so small (2γ/r)
Pr ~ P0
How do you derive Kelvin’s equation?
Eqm between liquid droplet and gas
1) μliq(P0)) = μgas (P0)
2) μliq(Pr + 2γ/r) = μgas(Pr)
then do 2-1 and then integrate
use molar volume liquid (vl) = molar volume gas (vg) = RT/P
Then RTln(Pr/P0)»_space; vl(Pr-P0) so approx to give
(2γvl)/r = RTln(Pr/P0)
rearrange to give
Pr = P0 exp[2γvl/rRT)
What is the critical droplet size?
Smallest droplet size possible in a pressure P
r* = 2γvl/RTln(P/P</sub>0</sub>)
How does vapour pressure realte to nucleation?
When smaller vap pressure (Pr) than required due to bigger size of a droplet then is more likely to nucleate
What is the relation of gibbs adsorption equation?
Relation between adsorption and surface tension
What is a concentration profile?
What is the Gibbs dividing surface?
Approx to conc profile
Gibbs dividing surface has no colume
No excess or depletion of solvent molecules
What is the volume of gibbs dividing surface?
Dividing surface volume, Vσ = 0
Vtotal = Vα + Vβ
How does gibbs dividing surface change the excess/depletion of solvent?
No excess/depletion of solcent molecules (no molecules in interface)
n1σ
n1,total = n1α + n1β
What is the surface-excess amount?
niσ = ni,total - (n1α + n1β)
XS of this amount of this component actually present in system over that present in a reference system
What is the amount of phase α & β in a conc profile?
n1α = Vαciα
n1β = Vβciβ
These come from integral beneath conc profile
How is the Gibbs dividing surface nσ=0 ?
Over and underestimates the curve by the same amount so interface is 0
What are the different types of adsoprtion?
Positive - surfactants wants to be at interface more than bulk
Negative - surfactant does not want to be at the surface
What are the different types of adsoprtion?
Positive - surfactants wants to be at interface more than bulk
Negative - surfactant does not want to be at the surface
What is the surface-excess conc, Γ?
Γi1 = niσ/σ
this is for solvent - component 1
What is the surface xs conc of positive and negative adsorption?
What is dA of a surface?
dAσ = -SσdT + Σμidniσ + γdσ
What is dγ, the surface analogue of Gibbs-Duhem?
dγ = -(Sσ/σ)dT - Σ (niσ/σ) dμ
Where the sum is from i to c
SσdT + Σμidniσ + γdσ
What is the gibbs adsorption equation?
dγ = - Σ Γi1 dμi
What is the Gibbs adsorption equation for ideal binary solutions?
Sub in μ2 = μ2θ + RTlnc2
Into following equation: dγ = - Σ Γi1 dμi
Gives dγ = - RTΓ21 dlnc2
Surface tension therefore a function of conc c2
