Chapter 4
Math Tools: Series and Approximations
1. Taylor Series.
Derive equation: (1 + x)p 1 + px + p (p 1)/(2!) x 2 , to second order using the
Taylor series method, around x 0.
Begin with the Taylor series expansion about x = 0, Equation (4.21),
f (x)
Chapter 5
Math Tools: Multivariate Calculus
1. Applying the Euler test.
How much kinetic energy does a 1700 kg car have, if it travels 100 km h1 ? Which of
the following are exact dierentials?
(a) 6x 5 dx + dy
(b) x 2 y 2 dx + 3x 2 y 3 dy
(c) (1/y)dx (x/y
Chapter 25
Phase Transitions
1. Protein aggregation.
You have a solution with mole fraction x of proteins. The proteins can aggregate and
thus change their local concentration. The entropy of mixing is
S = k[x ln x + (1 x) ln(1 x)],
and the internal energ
Chapter 16
Solvation and Transfer of Molecules
1. Mechanism of anesthetic drugs.
Anesthetic drug action is thought to involve the solubility of the anesthetic in the hydrocarbon region of the lipid bilayer of biological membranes. According to the classic
Chapter 2
Extremum Principles Predict Equilibria
1. A lattice gas.
How many arrangements are there of fteen indistinguishable lattice gas particles distributed on:
(a) V = 20 sites?
(b) V = 16 sites?
(c) V = 15 sites?
(a)
W (N = 15, V = 20) =
20 19 18 17
Chapter 9
Maxwells Relations and Mixtures
1. How do thermodynamic properties depend on surface area?
The surface tension of water is observed to decrease linearly with temperature (in
experiments at constant p and a): (T ) = b cT , where T = temperature C
Chapter 7
Thermodynamic Driving Forces
1. The work of compression.
One mole of a van der Waals gas is compressed quasi-statically and isothermally from
volume V1 to V2 . For a van der Waals gas, the pressure is
p=
a
RT
2,
V b V
where a and b are material
Chapter 15
Solutions and Mixtures
1. Ternary lattice mixtures.
Consider a lattice model liquid mixture of three species of spherical particles, A, B , and
C . As with binary mixtures, assume that all N = nA + nB + nC sites are lled.
(a) Write an expressio
Chapter 13
Chemical Equilibria
1. Iodine dissociation
Compute the dissociation constant Kp for iodine at T = 300 K.
Follow Example 13.3. But now
kT =
0
e RT
35,600
= e (1.987)(300)
qrI2 =
qvI2 =
2
qtI
qtI2
so
300
1.363 1025 m3 atm
1000
=
=
300
9310
1000
1
Chapter 3
Heat, Work and Energy
1. The time dependence of a mass on a spring.
(a) For the harmonic motion of a mass on a spring, the kinetic energy is K =
(1/2)mv 2 , and the potential energy is V = (1/2)ks x 2 , where ks is the spring
constant. Using the
580.321 HW1
assigned 8/28/2015
due Fri 9/4/2015
On separate sheets of paper (this sheet is optional) label your: Name, Section, Assignment, Due Date.
List collaborators and staple.
1. You have applied to only three Med Schools: Harvard (H), Johns Hopkins(
580.321 HW2 assigned (9/2/2015) due (9/9/2015)
On separate sheets of paper (this sheet is optional) label your: Name, Section, Assignment, Due Date.
List collaborators and staple.
1. How many arrangements are there of ten indistinguishable lattice gas par
HW6 posted due Wed 10/14
If a system is isolated, so that its U, V, and N are constant, the release of any internal
constraint will maximize the systems entropy:
S(U,V,N)
These variables are system variables Ssys, Usys, Vsys, Nsys.
The systems energy U(S,
4. Consider a protein with three energy levels, 30 = 0 , 1 2:1 ; 2 = 4.
a) Write down the Lagrange multiplier equations which maximize the entropy for each
17, given the constraints that p, is a distribution function and that the average energy
(a )of the
/
Section: 2.
BME 580.321
Statistical Mechanics and Thermodynamics
Exam I
September 30,2011
You have 50 minutes for this exam: 11:00am-11 :50am
1 page of notes allowed
Show all work. Points will be deducted if your explanations are not sufficiently clear.
Chapter 14
Phase Equilibria
1. Applying the ClausiusClapeyron equation.
(a) The vapor pressure of water is 23 mm Hg at T = 300 K, and 760 mm Hg at T =
373 K. Calculate the enthalpy of vaporization, hvap .
(b) Assuming that each water has z = 4 nearest nei
Chapter 12
Temperature, Heat Capacity
1. Heat capacity peaks from phase transitions.
The peak heat capacity in the gure below shows that helium gas adsorbed on graphite
undergoes a transition from an ordered state at low temperature to a disordered state
BME 580.321
Exam I Practice Problems
1. Briefly:
a) Why do systems tend to maximize entropy?
b) What is the 1st Law of Thermodynamics?
c) What is the 2nd Law of Thermodynamics?
2. A protein has 5 binding sites, A, B, C, D, and E, which can interact with 2
BME 580.321
Exam I Practice Problems
1. Briefly:
a) Why do systems tend to maximize entropy?
b) What is the 1st Law of Thermodynamics?
c) What is the 2nd Law of Thermodynamics?
2. A protein has 5 binding sites, A, B, C, D, and E, which can interact with 2
Exam I Practice Problems
BME 580.321
1. A 90 amino acid (aa) protein has one trypsin cleavage site (which cuts at lysine or
arginine), at residue 30. You incubate trypsin with your protein for 20 minutes, after which
the cleavage site has a probability p
Exam I Practice Problems
BME 580.321
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c) What is the 2 nd Law of Thermodynamics?
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BME 580.321
Exam I Practice Problems
1. Briefly:
a) Why do systems tend to maximize entropy?
b) What is the 1st Law of Thermodynamics?
c) What is the 2nd Law of Thermodynamics?
2. A protein has 5 binding sites, A, B, C, D, and E, which can interact with 2
Ch 25. Phase Transitions
Mixing two liquids (oil & water)
Fmix
= AB x(1 x) + x ln x + (1 x) ln(1 x)
NkT
U mix
S mix > 0
AB =
(TS mix < 0)
x(1 x)
x ln x + (1 x) ln(1 x)
z
w + wBB
wAB AA
kT
2
Fmix
= AB x(1 x) + x ln x + (1 x) ln(1 x)
NkT
increasing T