NBE-E4100 Molecular biophysics
Exercise 2 January 27 2016
LH2-1 Rubber bands are entropic springs.
Experiments show that the retractive force f of polymeric elastomers as a function of
temperature T and expansion L is approximately given by f (T,L) = aT (L − L0), where a
and L0 are constants.
(a) Use Maxwell relations to determine the entropy and enthalpy, S(L) and H(L), at
constant T and p.
(b) If you adiabatically stretch a rubber band by a small amount, its temperature increases,
but its volume does not change. Derive an expression for its temperature T as a function
of L, L0, a, and its heat capacity C = (∂U/∂T).
LH2-2 Pressures in Surfactant Monolayers.
A Langmuir trough is shown below. N surfactant molecules float on water on the left side
of a bar. When you push the bar to the left with pressure π, the N surfactant molecules
occupy an equilibrium area A, where
Consider a two-dimensional van der Waals model in which the entropy is given by
S = k ln (A - A0 ) ,
where A0 is the minimum area the molecules can occupy when they crowd together. The
enthalpy is given by
b
H = .
a
(a) You can express the pressure in terms of enthalpic p H and entropic pS
components:
p = p H + pS
Write a Maxwell relation for
æ¶ S ö
÷
.
pS = T çç ÷
÷
çè¶ A ÷
øT , p, N
(b) Express p H in terms of measurable quantities T and A.
(c) Write the equation of state of the model, giving the quantity p (T , A, N ) .
(d) Write an expression for the free energy D G (T , A, N ) at constant p.
LH2-3 Torque on DNA.
One end of a DNA molecule is fixed to a surface. Using single-molecule methods, a torque
(twisting force) is applied to the other end, twisting the molecule as shown below:
When the molecule is twisted through an angle θ, the DNA gives a restoring torque τ that
depends on temperature T:
t = (k 0 + k1T )q
where k0 and k1 are measured constants.
(a) Write an expression for the free energy G(θ) at fixed T and p.
(b) Use a Maxwell relation to express (∂S/∂θ)T,p in terms of experimental observables,
where S is entropy and p is pressure.
(c) Derive an expression for S(θ), at constant T and p.
(d) Derive an expression for the enthalpy H(θ) at constant T and p.
(e) Under what conditions, i.e., for what values of k0, k1, and T, will this torsional
retractive force be entropy-driven?
LH2-4 Thermodynamics of a single molecule.
One end of a molecule is fixed to a surface. Another end is pressed down by the tip of an
atomic force microscope, as shown below.
Using the tip of an atomic force microscope as a probe, we measure the force of molecule
retraction. The force depends on the temperature at which the experiment is performed:
f = (aT + b)x
where a and b are constants.
(a) Write a fundamental equation describing the energy of the molecule. Write an
expression for dG at fixed T and P. Derive an expression showing how G depends on x.
(b) Use a Maxwell relation to express (¶ S / ¶ x )
T ,p
.
(c) Derive an expression for S(x), at fixed T and P.
(d) Derive an expression for enthalpy H(x) at fixed T and P.
(e) What values of a and b will cause the retraction force to be driven by a change in
Entropy
LH2-5 Surface tension components.
The following figure shows the surface tension of water as a function of temperature:
(Redrawn from data in CRC Handbook of Chemistry and Physics, 47th edition, RC Weast, ed.,
The Chemical Rubber Company, Cleveland, OH, 1962.)
(a) From the figure, determine the numerical value of
(¶ g /
¶T
)p, N , A
where T =
temperature, p = pressure, and a = surface area.
(b) Find the Maxwell relation for the partial derivative equal to (¶ g / ¶ T
)p, N , A .
(c) Write an expression for the enthalpic and entropic components of the surface tension γ.
(d) Combining the results from above, compute the numerical values of the enthalpic and
entropic parts of γ at T = 300 K, and comment on which component dominates the surface
tension