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82 Cards in this Set
- Front
- Back
zeroth law?
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If A in thermal equilibrium with B and B is in thermal equilibrium with C, then C is is thermal equilibrium with A |
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1st law? and for infinitesimal changes?
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∆U = q + w (for a closed system ∆n = 0) dU = δq + δw |
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What does internal energy depend on?
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V and T i.e. U(V,T) |
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Define enthalpy?
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H(p,T) = U + pV |
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Link the two heat capacities?
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Cp = (∂H/∂T)p, H = U +pV => Cp = (∂(U+pV)/∂T)p => Cp = (∂U/∂T)p + ∂/∂T(pV)p ∂/∂T(pV)p = ∂/∂T(nRT) = nR (perfect gas) =>Cp = Cv + nR |
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Define entropy?
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dS = dqrev/T |
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2nd Law?
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tot = system and surroundings if not isolated tot = system if isolated |
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Clausius inequality?
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dS ≥δq/T for an irreversible process dS>δq/T for a reversible process dS = dqrev/T |
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Helmholtz free energy?
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Using dU = dq -pdV, when dV = 0, => dU = dqV => dS ≥dU/T => dU - TdS ≤ 0 Define A = U -TS => dA = dU -d(TS) = dU -TdS - SdT (dT = 0) => dA = dU -TdS |
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helmholtz spontaneous change?
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at dT = 0, dV = 0 then dA ≤0 for a spontaneous change |
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Gibbs free energy?
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dH = dqp when dp = 0 using dS≥δq/T =>dS≥dH/T => dH -TdS≤0 Define G = H - TS =>dG = dH -d(TS) = dH - TdS -SdT dT = 0 => dG = dH -TdS |
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Gibbs spontaneous change? |
then requirements for spontaneous change => dG ≤ 0 |
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Spontaneity for different systems?
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All Systems: Isolated System: ∆Ssys ≥0 ∆Ssys = 0 at equilibrium Closed or Open System: dA ≤ 0, (dT = 0, dV = 0) dA = 0 at equilibrium dG ≤ 0, (dT = 0, dp = 0) dG = 0 at equilibrium |
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Third law?
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The entropy of all perfect crystalline substances is 0 at T = 0
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Link U and S + discuss probrem?
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=> dU = TdS - pdV (for reversible change, but applies whether reversible or not due to all being state functions) S can not be varied in a controlled way |
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derive dG(T,p) (closed system)
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H = U +pV => dH = dU +pdV + Vdp dU = TdS - pdV => dH = TdS + Vdp G = H -TS => dG = dH - TdS - SdT => dG(T,p) = Vdp - SdT (Closed system) |
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dA(T,V) for a closed system? when is it used?
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dG(T,p) is usually used, but dA(T,V) is used in statistical thermodynamics |
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dG(T,p) as partial?
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=> dG(T,p) = (∂G/∂p)T dp + (∂G/∂T)p dT (∂G/∂p)T = V, (∂G/∂T)p = -S |
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Gibbs for an open system?
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where (∂G/∂n)t,p = µ = chemical potential |
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Chemical potential for a one-component system? |
dp = 0, dT = 0 =>∫dG = ∫µdn => G = nµ => Gm = µ where Gm is the molar Gibbs free energy |
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Chemical potential of a perfect gas?
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[∂/∂p(∂G/∂n)] = [∂/∂n(∂G/∂p)] => (∂µ/∂p) = (∂V/∂n) V = nRT/p =>(∂V/∂n) = RT/p =>(∂µ/∂p) = RT/p =>∫dµ = RT∫dp/p => µ = µ⁰ + RTlnp/p⁰ |
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Chemical Potential Variation with p andT (one component)?
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one component system µ = Gm => (∂µ/∂p)T = Vm => (∂µ/∂T)p = -Sm |
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Derive Clapeyron equation?
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µ(l) = µ(v) => dµ(l) = dµ(v) =>Vm(l)dp -Sm(l)dT = Vm(v)dp - Sm(v)dT =>dp/dT = ∆Sm/∆Vm ∆G = 0 (because at equilibrium) =>∆H -T∆S = 0 =>dp/dT = ∆H/T∆V |
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Derive Claussius-Clapeyron equation?
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Assume perfect gas Vm(g)=RT/p Assume Vm(g) >> Vm(l) => ∆Vm = RT/p =>dp/dT = p∆H/RT² =>1/p dp/dT = dlnp/dT = ∆H/RT² |
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How can the phase boundaries be approximated when one of the phases isa vapour?
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Clausius-Clapeyron equation |
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dG for open system containing two components?
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=> dG(p,T,nA,nB) = Vdp - SdT + µAdnA +µBdnB => dG(p,T,ni) = Vdp - SdT + ∑µidni => ∫dG = G = ∑niµi (constant p,T, composition) (State function so equ. always true) |
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Define partial molar volume? partial molar Gibbs free energy?∫dG?∫dV?
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Vi = (∂V/∂ni)p,T,nj≠i Gi = (∂G/∂ni)p,T,nj≠i ∫dG = G = ∑niµi (constant p, T, composition) ∫dV = V = ∑niVi (constant p, T, composition) |
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Gibbs-Duhem eqn?
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=> dG = ∑nidµi + ∑µidni dG(p,T,ni) = Vdp - SdT + ∑µidni => dG = ∑µidni (constant T, p) ∴=> ∑nidµi = 0 (at constant T and p) When you change the composition at T,p = 0, the chemical potentials do not vary independently |
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Gibbs-Duhem eqn for binary system?
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dµA = -(nB/nA)dµB |
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Gibbs-Duhem applied to molar volume?
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dVA = -(nB/nA) dVB |
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Chemical potential of pure liquids?meaning?
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for a liquid in equilibrium with its vapour phase: µ*A(l) = µA(g) =>µ*A(l) = µ⁰A(g) + RTln(p*A/p⁰) - (assum perfect gas for vapour) *indicates pure liquid -we can obtain chemical potential of A in its liquid phase by measuring its vapour pressure |
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Chemical potential of liquid mixture (or solvents in solutions)?
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µ*A(l) = µ⁰A(g) + RTln(p*A/p⁰) (pure liquid) µA(l) = µ⁰A(g) + RTln(pA/p⁰) (mixture or solution) substitute for µ⁰A(g) => µA(l) = µ*A(l) + RTln(pA/p*A) |
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Raoult's law?
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pA = p*A xA where xA is the mole fraction of A in the solution The law holds increasing well as xA →1 |
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Define ideal solution?
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µA(l) = µ*A(l) + RTln(pA/p*A) pA = p*A xA => µA(l) = µ*A(l) + RTln(xA) obtain chemical potential of a solvent in an ideal solution from just how much solvent there is |
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Perfect gas A is gas mixture?
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µA(g) = µ⁰A(g) + RTln(pA/p⁰) |
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Gibbs energy of mixing in an ideal solution?
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mixed: G = nAµA + nBµB = nA(µ*A + RTlnxA) + nB(µ*B + RTlnxB) =>∆G = nARTlnxA + nBRTlnxB = nRT(xAlnxA +xBlnxB) also ∆H = 0, ∴ ∆G = -T∆S |
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When is Henry's law obeyed?
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when a solution is dilute (mole fraction <<0.1) it is found that the pressure of solute B often obey's Henry's Law. |
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Ideal dilute solutions?
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solutions that obey Henry's Law |
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Henry's Law?
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where KB does not necessarily equal p*B |
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Chemical potential of an ideal dilute solution? |
-where m is molarity (moles per Kg of solvent) -m⁰ is 1 molal -µ⁰B(l) = limit (µB(l) - RTln(mB/m⁰)) as xB -> 0 µ⁰B(l) does not physically exst as it is a 1 molal solution in which the molecular interecations are the same as in an infinitely dilute solution |
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Why Solute has effect on solvents Tm and Tb?
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µA(l) = µ*A(l) + RTln(xA) (ideal solution) =>µA = µ*A + RTln(xA)= µ*A + RTln(1-xB) for small xB => ln(1-xB) ≈ -xB => µA ≈ µ*A -xBRT Solutes do not dissolve in solid solvent (some exceptions) and gas interactions are much weaker so µA(s) and µA(g) are not affected by B |
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What effect solute has on solvents Tm and Tb?
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Adding a solute lowers the melting point and raises the boiling point of the ideal solution relative to the pure liquid ∆Tb < ∆Tm |
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∆Tm and ∆Tb?
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∆Tm = xBRT*²/∆fusH ∆Tb = xBRT*²/∆vapH Where T* is the phase transition temperature for the pure solvent ∆T depends only on the mole fraction of the solute not on its identity |
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osmotic pressure?
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Pure liquid has potential µ*A(l) = µ⁰A(g) + RTln(p*A/p⁰) The solute side has potential µA(l) = µ*A(l) + RTln(xA) + ΠVm where ∆p = Π = ρgh (density x gravity x height) µ*A(l) =µA(l) at equilibrium => ΠVm = -RTln(xA) |
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Van't Hoff equation for osmotic pressures?
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small xB, ln(xA) ≈ -xB, xB = nB/(nB+nA)≈ nB/nA => ΠV = nBRT |
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Ideal solubility of a pure solid? |
μ*B(s) = μ*B(sol) = μ*B(l) + RTlnxB =>lnxB = (μ*B(s) - μ*B(l))/RT =-(∆fusGm)/RT =>dlnxB = -1/R (d∆fusGm/T) (∂(∆G/T)/∂T)p = -∆H/T² =>dlnxB = 1/R (∆fusH/T² dT) => lnxB = -∆fusH/R (1/T - 1/Tm) |
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eutectic point (e)?
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The minimum melting point of the mixture -isopleth at e correspondns to the eutectic composition |
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lever rule?
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Where nα is the number of mols of phase α |
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What is p assuming Raoults' law holds over entire concentration range?
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p = xAp*A + xBp*B |
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Discuss mol fraction of vapour phase in a binary liquid-vapour system?
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-xA = mol fraction in liquid phase -yA ≠ xA because vapour will be enriched in the more volatile component usually assigned A -yA = pA/p = xAp*A/(xAp*A + (1-xA)p*B) |
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T-x phase diagrams and p-x diagrams?
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Gibbs phase rule?
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F = variance (degrees of freedom) C = number of components P = numberof phases in equilibrium |
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Biological standard state?
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Labelled as µ⁰' or µ^⊕ |
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∆rG⁰?
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∆rG⁰ = -RTlnQ = -RTlnK at Equilibrium ∆rG = 0, Q = K |
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∆rG?
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= ∑niµi - ∑njµj (i = products, j = reactants) ∆rG = ∑xµi - ∑xµj (∆rG in Jmol⁻¹ )x=stoichoimetry using µA = µA⁰ + RTln(mA/m⁰) => ∆rG = ∆rG⁰ +RTlnQ |
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Q?
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= (Πprod(mj/m⁰)^vj)/(Πreact(mi/m⁰)^vi) where vx is the stoichiometry |
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extent of reaction?
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dξ = 1/vj dnj |
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∆rG(ξ)? |
∆rG=dG/dξ |
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Values of K?
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Liquids: K in mole fractions, x; standard state = pure liquid Solutions: K in m/m⁰, standard state = 1 molal(with the same interactions as at infinite dilution) |
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Van't hoff equation?
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=> lnK = -∆rG⁰/RT => (∂lnK/∂T)p = -(1/R) (∂/∂T) (∆rG⁰/T)p ∂/∂T (G/T)p = -H/T² =>(∂lnK/∂T)p = ∆rH⁰/RT² |
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Temperature dependence on equilibrium constant?
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(Note for gases, K is defined at 1 bar pressure) => dlnKp/dT = ∆rH⁰/RT² |
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Pressure dependence of the equilibrium constant?
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lnK = -∆rG⁰/RT ∴ K is independent of pressure 2)For solutions we can define the standard state at any pressure (∂G/∂p)T = V => (∂lnK/∂p)T=-∆rV⁰/RT -weak dependence |
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Pressure dependence of the composition at equilibrium?
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Worked example A(g) -> B(g) + C(g), K = 0.0501, T = 900K, A initial = 1 mol increasing pressure increases A, you're going to have to look at notes for reasoning |
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non-ideal systems?
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Activities for different states?
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Ideal solution; a = γx Ideal dilute solution: a = γm/m⁰ γ = activity coefficient, measures deviation from ideality, γ=1 for an ideal system (called fugacity and fugacity coefficient for gases) |
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Equilibrium constant in activities?
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K = [Πprod(ai^vi)/Πreact(aj^vj)]eq where ax = activity of x vx = stoichiometry of x NB activities still depend on the standard state, its just no longer explicit in the equation for K |
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Discuss gas-liquid critical point? |
T>Tc: there is no gas-liquid phase transitions T gas phase transition will take place upon decompression inflexion appera in pV curve => dp/dVm = 0, d²p/dVm² = 0 |
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Van der Waals loops?
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How well does Van der Waals eqn model the system?
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d²p/dVm² = 2RT/(Vm-b)³ - 6a/Vm⁴ = 0 => pc = a/27b², Vm,c = 3b, Tc = 8a/27Rb Zc = pcVm,c/RTc = 3/8 |
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Virial equation? |
Z=pV/nRT = (1+B(n/V) + C(n/V)².....) (virial eqn) =(1+B'p+C'p²...) (∂µ/∂p) = (∂V/∂n) = ∂/∂n [(nRT/p)(1 + B'p + C'p²...)] =>µ = RT(lnp + B'p +½C'p²...) + constant p->0, µ = RTlnp + constant p->0, gases behave as perfect gases =>µ = µ⁰ +RTln(p/p⁰) =>constant = µ⁰ -RTln(p⁰) => µ = µ⁰ + RTln(p/p⁰) + RT(B'p+½C'p²....) µ = µ⁰ + RTlnf =µ⁰ + RTln(p/p⁰) + RTlnγ => lnγ = B'p +½C'p²....
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Where does non-ideal behaviour arise from? and how is it accounted for? |
-probability of collision is proportional to the number density of the gas (n/V) -using Virial equation where: B: accounts for pairwise collisions C: acconts for 3-body collisions etc |
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Boyle Temperature?
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Z = 1 + B'p + C'p²... dZ/dp = B' + 2C'p + ... ≈ B' (at close to p=0) At Boyle Temperature (TB): B' = 0 pVm ≈ RTB (at p≈0) |
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B' values?
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Positive B': Repulsive forces dominate (high temperature) |
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Fugacity?
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-fugacity coefficients can be obtained from experimental data -as p-> 0, µreal = µideal |
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activities for non-ideal solutions? |
µA(l) = µA(g) (at equilibrium) ideal: aA = xA = pA/p*A real: aA = γA(l)xA = γA(g)pA/p*A assume γA(g) = 1 => γA(l) = pA/xAp*A (γ is the ration of actual pressure to Raoult's Law pressure) |
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∆mixH for regular solutions?
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Before Hint = ½zNAεAA+½zNBεBB After Hint = ½z(NAxAεAA+NAxBεAB + ....) NA = xAN, NB = xBN, ∆mixH = ½zN(xA²εAA + xAxBεAB + xB²εBB....) write xA²= xA(1-xB), xB² = xB(1-xA) ∆mixH = ½zN(-εAA + 2εAB - εBB)xAxB ∆mixH = NβxAxB = NβxA(1-xA) β = ½z(2εAB-εAA-εBB) |
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Favourable and unfavourable interactions between A and B in a regular solution?
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β = ½z(2εAB-εAA-εBB) β<0; εAB more negative than ½(εAA + εBB) - favourable interactions between A and B - ∆mixH <0 β>0: εAB is less negative than ½(εAA + εBB) -unfavourable interactions between A and B - ∆mixH >0 |
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Free energy of ∆mixG =
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∆mixG= ∆mixH-T∆mixS ∆mixG = nβxAxB + nRT(xAlnxA +xBlnxB) |
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Chemical potential of a real solution?
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=> G = nAµA* + nBµB* +βnAnB/(nA+nB) + RT(nAln(nA/(nA+nB))+nBln(nB/(nA+nB))) =>dG/dnx =>µA = µA* + RTlnxA + βxB² =>µB = µB* + RTlnxB+ βxA² |
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Temperature dependence of mixing?
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∆mixG= ∆mixH-T∆mixS Enthalpic term is independent of temperature in the regular solution model entropic term has more weighting at higher temperatures, and so mixing is more favoured at higher T. |
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spinodal?
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line separating the unstable and meta stable regions (the line that passes through the inflection points |
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binodal?
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line separating the themodynamically stable one and two phase regions (line that passed through the minima)
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