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5.60 Thermodynamics & Kinetics
Spring 2008
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5.60 Final Exam Review
1. Phase Equilibria- 2 components
a. Drawing P-x,y and T-x,y diagrams
2. Ideal and Non Ideal Solutions
a. Raoult’s Law, Henry’s Law, Dalton’s Law
i. Dalton’s Law: pA = yAp
ii. Raoult’s Law: pA = xApA* and pB = xBpB* = (1-xA)pB*
iii. Henry’s Law: pB = xBKB
b. Chemical potential and Ideal Solutions
c. Entropy of Mixing, Free Energy of mixing
Purely entropic, as in gas mixture.
i. ΔGmix = nRT(xAlnxA + xBlnxB
ii. ΔHmix = ΔGmix + TΔSmix = 0
No enthalpy change, ΔG is due to entropy of
mixing
⎛ ∂ΔG mix ⎞
⎟ =0
⎝ ∂p ⎠T
iii. ΔVmix = ⎜
No volume change, just like ideal gas
d. Non ideal solutions
i. Positive Deviations: Δu > 0 (most common)
Mixing is energetically not favorable in liquid phase
ii. Vapor pressure is higher than expected by Raoult’s Law
iii. Can make same argument for negative deviation.
iv. Don’t forget about azeotropes (exam III)
3. Colligative Properties
a. These are properties of solutions in the dilute limit, where there is a solvent “A” and a
solute “B” where nA>>nB.
pure
b. These properties are a direct result of μ mix
A ( A,T,p) < μ A (A,T,p)
c. Use two measures of concentration:
i. Mole Fraction: xB = n/(nA+nB) ~ nB/nA
ii. Molality: mB = (moles solute)/(kg solvent) = nB/(nAMA)
Where MA is the mass in kg of one mole solvent.
d. There are FOUR Colligative Properties
i. Vapor pressure lowering
ii. Boiling point elevation (You should be able to derive the boiling point elevation
of freezing point depression; look in lecture notes)
iii. Freezing point depression
iv. Osmotic pressure
(Be able to derive)
4. Statistical Mechanics
a. Boltmann probability distribution
For two states i and j with energies εi and εj, the relative probability of being in state i is:
Pi =
e−εi / kT
∑ e−εi / kT
i
b. Partition Functions
i. Molecular partition function:
∑e ε
−
i
≡q
/ kT
i
ii. Canonical partition function:
∑e
− Ei / kT
≡Q
i
∑
iii. qconf =
gi e −ε i / kT
energies
εi
iv. S = −
A U
⎛ ∂ ln Q ⎞
+ = k ln Q + kT ⎜
⎟
T T
⎝ ∂T ⎠ V,N
⎛ ∂A ⎞
⎛ ∂ ln Q ⎞
⎟ = kT ⎜
⎟
⎝ ∂V ⎠T,N
⎝ ∂V ⎠T,N
v. p = − ⎜
⎛ ∂A ⎞
⎛ ∂ ln Q ⎞
⎟ = − kT ⎜
⎟
⎝ ∂N ⎠T,V
⎝ ∂N ⎠T,V
vi. μ = − ⎜
vii. H = U + pV
viii. G = A + pV
ix.
S = − k ∑ pi ln pi
i
c. Energy Levels
d. Model Systems
e. Applications
5. Reaction Kinetics
a. Complex reactions and mechanisms
i. For reaction: aA + bB Æ cC + dD
Rate of reaction: Rate =
1 d [ A ] 1 d[B] 1 d[C] 1 d[D]
=
=
=
a dt
b dt
c dt
d dt
ii. Zero order reactions (rare)
1. Reaction: A Æ products
2. Rate: [A] = -kt + [A]o
3. Half life: t1/2 – [A]o/(2k)
iii. First order reactions
1. Reaction: A Æ products
ln[A] = -kt + ln[A]o
2. Rate: [A] = [A]oe-kt
3. Half life: t1/2 = (ln2) / k=0.693/k
[so k = (0.693)/(t1/2)
iv. Second order reactions
1. Reaction: A Æ products
2. Rate:
1
1
=
+ kt
[A] [A]0
3. Half life: t1/2=1/(k[A]o)
b. Steady State and Equilibrium Approximations
i. Steady state approximation
k1
1.
k2
ARB→C
k -1
2. Assume [B] is small and slowly varying: [B] reaches a steady state
concentration [B]SS and remains there
3.
d[C]
k k [A]
d[A]
= k 2 [B]SS = 1 2
=dt
k -1 +k 2
dt
ii. Equilibrium approximation
k1
1.
k2
ARB→C
k -1
2. Assume k2<<k-1 and k1 (so [BÆC ] is the rate limiting step): Then A and B
quickly come into equilibrium, while C slowly builds up
k1 [B]
≈
k -1 [A]
3.
K eq =
4.
d[C]
kk
=k 2 [B]=k 2 K eq [A]= 1 2 [A]
dt
k -1
c. Chain Reactions
d. Temp Dependence
i. Arrhenius Law: k = Ae − Ea /RT
1. where Ea = Activation Energy
2. where A = Pre-Exponential Factor
ii.
iii.
e. Catalysis
i. Review Enzyme catalysis