Thermodynamics

Front 1

zeroth law?

Back 1

If A in thermal equilibrium with B and B is in thermal equilibrium with C, then C is is thermal equilibrium with A

Front 2

1st law? and for infinitesimal changes?

Back 2

∆U = q + w (for a closed system ∆n = 0)

dU = δq + δw

Front 3

What does internal energy depend on?

Back 3

V and T i.e. U(V,T)

Front 4

Define enthalpy?

Back 4

H(p,T) = U + pV

Front 5

Link the two heat capacities?

Back 5

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

Front 6

Define entropy?

Back 6

dS = dqrev/T

Front 7

2nd Law?

Back 7

∆Stot ≥ 0

tot = system and surroundings if not isolated

tot = system if isolated

Front 8

Clausius inequality?

Back 8

dS ≥δq/T

for an irreversible process dS>δq/T

for a reversible process dS = dqrev/T

Front 9

Helmholtz free energy?

Back 9

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

Front 10

helmholtz spontaneous change?

Back 10

at dT = 0, dV = 0

then dA ≤0 for a spontaneous change

Front 11

Gibbs free energy?

Back 11

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

Front 12

Gibbs spontaneous change?

Back 12

When dT = 0, dp = 0

then requirements for spontaneous change => dG ≤ 0

Front 13

Spontaneity for different systems?

Back 13

All Systems:
∆Stot ≥ 0

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

Front 14

Third law?

Back 14

The entropy of all perfect crystalline substances is 0 at T = 0

Front 15

Link U and S + discuss probrem?

Back 15

dqrev = TdS, dwrev = -pdV, dU = dqrev + dwrev

=> 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

Front 16

derive dG(T,p) (closed system)

Back 16

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)

Front 17

dA(T,V) for a closed system? when is it used?

Back 17

dA(T,V) = -pdV -SdT

dG(T,p) is usually used, but dA(T,V) is used in statistical thermodynamics

Front 18

dG(T,p) as partial?

Back 18

dG(T,p) = Vdp-SdT

=> dG(T,p) = (∂G/∂p)T dp + (∂G/∂T)p dT

(∂G/∂p)T = V, (∂G/∂T)p = -S

Front 19

Gibbs for an open system?

Back 19

dG(T,p,n) = (∂G/∂p)T,n dp + (∂G/∂T)p,n dT + (∂G/∂n)T,p dn

where (∂G/∂n)t,p = µ = chemical potential

Front 20

Chemical potential for a one-component system?

Back 20

dG(p,T,n) = Vdp -SdT + µdn

dp = 0, dT = 0

=>∫dG = ∫µdn

=> G = nµ

=> Gm = µ

where Gm is the molar Gibbs free energy

Front 21

Chemical potential of a perfect gas?

Back 21

[∂/∂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

Front 22

Chemical Potential Variation with p andT (one component)?

Back 22

(∂G/∂p)T = V, (∂G/∂T)p = -S

one component system µ = Gm

=> (∂µ/∂p)T = Vm

=> (∂µ/∂T)p = -Sm

Front 23

Derive Clapeyron equation?

Back 23

dµ(p,T)=dGm(p,T) = Vmdp-SmdT

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

Front 24

Derive Claussius-Clapeyron equation?

Back 24

dp/dT = ∆H/T∆V

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

Front 25

How can the phase boundaries be approximated when one of the phases isa vapour?

Back 25

Clausius-Clapeyron equation

Front 26

dG for open system containing two components?

Back 26

dG(T,p,nA,nB) = (∂G/∂p)T,nA,nB dp + (∂G/∂T)p,nA,nB dT + (∂G/∂nA)T,p,nB dnA + (∂G/∂nB)T,p,nA dnB

=> 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)

Front 27

Define partial molar volume? partial molar Gibbs free energy?∫dG?∫dV?

Back 27

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)

Front 28

Gibbs-Duhem eqn?

Back 28

G = ∑niµi (at constant T and p)

=> 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

Front 29

Gibbs-Duhem eqn for binary system?

Back 29

dµA = -(nB/nA)dµB

Front 30

Gibbs-Duhem applied to molar volume?

Back 30

dVA = -(nB/nA) dVB

Front 31

Chemical potential of pure liquids?meaning?

Back 31

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

Front 32

Chemical potential of liquid mixture (or solvents in solutions)?

Back 32

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)

Front 33

Raoult's law?

Back 33

empirical observation that for solvent A:

pA = p*A xA

where xA is the mole fraction of A in the solution

The law holds increasing well as xA →1

Front 34

Define ideal solution?

Back 34

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

Front 35

Perfect gas A is gas mixture?

Back 35

µA(g) = µ⁰A(g) + RTln(pA/p⁰)

Front 36

Gibbs energy of mixing in an ideal solution?

Back 36

unmixed: G = nAµ*A + nBµ*B

mixed: G = nAµA + nBµB

= nA(µ*A + RTlnxA) + nB(µ*B + RTlnxB)

=>∆G = nARTlnxA + nBRTlnxB

= nRT(xAlnxA +xBlnxB)

also ∆H = 0, ∴ ∆G = -T∆S

Front 37

When is Henry's law obeyed?

Back 37

when a solution is dilute (mole fraction <<0.1) it is found that the pressure of solute B often obey's Henry's Law.

Front 38

Ideal dilute solutions?

Back 38

solutions that obey Henry's Law

Front 39

Henry's Law?

Back 39

pB = KBxB

where KB does not necessarily equal p*B

Front 40

Chemical potential of an ideal dilute solution?

Back 40

µB(l) = µ⁰B(l) + RTln(mB/m⁰)

-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

Front 41

Why Solute has effect on solvents Tm and Tb?

Back 41

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

Front 42

What effect solute has on solvents Tm and Tb?

Back 42

Adding a solute lowers the melting point and raises the boiling point of the ideal solution relative to the pure liquid

∆Tb < ∆Tm

Front 43

∆Tm and ∆Tb?

Back 43

∆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

Front 44

osmotic pressure?

Back 44

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)

Front 45

Van't Hoff equation for osmotic pressures?

Back 45

small xB, ln(xA) ≈ -xB, xB = nB/(nB+nA)≈ nB/nA

=> ΠV = nBRT

Front 46

Ideal solubility of a pure solid?

Back 46

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)

Front 47

eutectic point (e)?

Back 47

The minimum melting point of the mixture

-isopleth at e correspondns to the eutectic composition

Front 48

lever rule?

Back 48

nαlα = nβlβ

Where nα is the number of mols of phase α

Front 49

What is p assuming Raoults' law holds over entire concentration range?

Back 49

p = xAp*A + xBp*B

Front 50

Discuss mol fraction of vapour phase in a binary liquid-vapour system?

Back 50

-yA = mol fraction in vapour

-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)

Front 51

T-x phase diagrams and p-x diagrams?

Back 51

Front 52

Gibbs phase rule?

Back 52

F = C - P + 2

F = variance (degrees of freedom)

C = number of components

P = numberof phases in equilibrium

Front 53

Biological standard state?

Back 53

1 molal concentration for all reagnts except H⁺ and OH⁻ which are 10⁻⁷ M (pH = 7)

Labelled as µ⁰' or µ^⊕

Front 54

∆rG⁰?

Back 54

Difference in free energy of the products (pure, unmixed) in tehir standard states and the reactants (pure, unmixed) in their standard states

∆rG⁰ = -RTlnQ = -RTlnK

at Equilibrium ∆rG = 0, Q = K

Front 55

∆rG?

Back 55

∆rG=Gprod - Greact (∆rG in J)

= ∑niµi - ∑njµj (i = products, j = reactants)

∆rG = ∑xµi - ∑xµj (∆rG in Jmol⁻¹ )x=stoichoimetry

using µA = µA⁰ + RTln(mA/m⁰)

=> ∆rG = ∆rG⁰ +RTlnQ

Front 56

Q?

Back 56

reaction quotient

= (Πprod(mj/m⁰)^vj)/(Πreact(mi/m⁰)^vi)

where vx is the stoichiometry

Front 57

extent of reaction?

Back 57

ξ = 0 at start, 1 at end

dξ = 1/vj dnj

Front 58

∆rG(ξ)?

Back 58

∆rG=dG/dξ

Front 59

Values of K?

Back 59

Gases: K in p/p⁰, standard state = 1bar

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)

Front 60

Van't hoff equation?

Back 60

∆rG⁰ = -RTlnK

=> lnK = -∆rG⁰/RT

=> (∂lnK/∂T)p = -(1/R) (∂/∂T) (∆rG⁰/T)p

∂/∂T (G/T)p = -H/T²

=>(∂lnK/∂T)p = ∆rH⁰/RT²

Front 61

Temperature dependence on equilibrium constant?

Back 61

we get Van't Hoff equation

(Note for gases, K is defined at 1 bar pressure)

=> dlnKp/dT = ∆rH⁰/RT²

Front 62

Pressure dependence of the equilibrium constant?

Back 62

1)For gases ∆rG⁰ is defined at 1 bar

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

Front 63

Pressure dependence of the composition at equilibrium?

Back 63

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

Front 64

non-ideal systems?

Back 64

write the chemical potential in terms of activities, (a)

Front 65

Activities for different states?

Back 65

gas: a = γ p/p⁰

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)

Front 66

Equilibrium constant in activities?

Back 66

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

Front 67

Discuss gas-liquid critical point?

Back 67

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

Front 68

Van der Waals loops?

Back 68

cannot exist, what we really get is coexistence of a liquid and a gas which then have different molar volumes,

Front 69

How well does Van der Waals eqn model the system?

Back 69

dp/dVm =-RT/(Vm-b)² +2a/Vm³ =0

d²p/dVm² = 2RT/(Vm-b)³ - 6a/Vm⁴ = 0

=> pc = a/27b², Vm,c = 3b, Tc = 8a/27Rb

Zc = pcVm,c/RTc = 3/8

Front 70

Virial equation?

Back 70

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....

Front 71

Where does non-ideal behaviour arise from? and how is it accounted for?

Back 71

-collisions between molecules

-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

Front 72

Boyle Temperature?

Back 72

pVm = RT(1+B'p+C'p²....) = RTZ

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)

Front 73

B' values?

Back 73

Negative B': Attractive forces dominate (low temperature)

Positive B': Repulsive forces dominate (high temperature)

Front 74

Fugacity?

Back 74

-γp = effective pressure or fugacity (f)

-fugacity coefficients can be obtained from experimental data

-as p-> 0, µreal = µideal

Front 75

activities for non-ideal solutions?

Back 75

µA(l) = µ*A(l) + RTln(pA/p*A) = µ*A(l) + RTlnaA

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)

Front 76

∆mixH for regular solutions?

Back 76

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)

Front 77

Favourable and unfavourable interactions between A and B in a regular solution?

Back 77

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

Front 78

Free energy of ∆mixG =

Back 78

∆mixG= ∆mixH-T∆mixS

∆mixG = nβxAxB + nRT(xAlnxA +xBlnxB)

Front 79

Chemical potential of a real solution?

Back 79

G = nAµA*+nBµA* +nβxAxB + nRT(xAlnxA + xBlnxB)

=> 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

Front 80

Temperature dependence of mixing?

Back 80

∆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.

Front 81

spinodal?

Back 81

line separating the unstable and meta stable regions (the line that passes through the inflection points

Front 82

binodal?

Back 82

line separating the themodynamically stable one and two phase regions (line that passed through the minima)