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9 Bioenergetics
9.3 If your brain is currently using 20 W of work energy per unit time to make new
neuronal structures and populate higher molecular energy states, what is the rate of
entropy production in the air surrounding you due to this activity?
9.4 Suppose the muscle attached to your Achilles tendon exerts a force of 705
newtons lifting your weight a distance of 2.6 cm as you rise up onto your toes, and
then the muscle relaxes again, returning you to your original height. In the process
of raising you up, 12 calories of heat is released from the muscle. What was the
efficiency of this muscle? If the environment was at 22 ◦ C, how much entropy was
produced? Ideally, what was the change in the internal energy of the muscle for the
whole process? What was the change in the entropy of the muscle for the whole
process?
9.5 A certain thermodynamic system is described by the equilibrium energy
relation
U (S, V , N) =
KN 5/3
(V − Nb)
2/3
exp
2S
3N
−
aN 2
V
where K, a, and b are constants. Find the internal pressure p as a function of
temperature T and volume V for this system.
9.6 Suppose at some moment the concentration of sodium ions on the inside of a
cell membrane of thickness 5 nm is 304 mM and outside the membrane it is 302 mM
(1 nm is one nanometer, 1 mM is one millimolar), with ions passing through at the
rate of 4.3 × 10 14 s/cm 2 . What is the drift velocity of an ion through the membrane?
What is the diffusion constant for sodium through this wall?
9.7 The rate of diffusion of small particles in the vitreous humor of the eye is of
physiological and medical interest. Estimate the diffusion time for a concentration
of protein molecules to spread from an injection point in the center of the eye to
a RMS distance of 10 mm across the vitreous if the diffusion constant for these
molecules is measured to be D = 0.47 mm 2 / h.
9.8 If a set of macromolecules each of diameter 812 nm are observed to spread
in water a root-mean-square distance of 10.51 μm after 30 s, estimate Avogadro’s
number. The viscosity of water at room temperature (27 ◦ C) is 0.852 centipoise.
9.9 A small vacuole containing a concentrated solution of salt, NaCl, is opened
on the interior of one side of a cell of diameter 12 μm. The diffusion constant for
salt in water is about 1.0 × 10 3 (μm) 2 /s. Estimate the time for this salt to reach the
opposite side of the cell by diffusion.
9.10 Two membranes, #1 and #2, permeable to a small organic molecule, are found
at either end of an impermeable cylinder of length L within a cell. The cell maintains
a concentration C 1 on the outside of membrane #1, and C 2 outside membrane #2.
The diffusion constant for the molecules in the fluid contained in the cylinder is D.
What is the steady-state concentration of the small organic molecules at a distance
x from the membrane #1?
9 Bioenergetics
9.3 If your brain is currently using 20 W of work energy per unit time to make new
neuronal structures and populate higher molecular energy states, what is the rate of
entropy production in the air surrounding you due to this activity?
9.4 Suppose the muscle attached to your Achilles tendon exerts a force of 705
newtons lifting your weight a distance of 2.6 cm as you rise up onto your toes, and
then the muscle relaxes again, returning you to your original height. In the process
of raising you up, 12 calories of heat is released from the muscle. What was the
efficiency of this muscle? If the environment was at 22 ◦ C, how much entropy was
produced? Ideally, what was the change in the internal energy of the muscle for the
whole process? What was the change in the entropy of the muscle for the whole
process?
9.5 A certain thermodynamic system is described by the equilibrium energy
relation
U (S, V , N) =
KN 5/3
(V − Nb)
2/3
exp
2S
3N
−
aN 2
V
where K, a, and b are constants. Find the internal pressure p as a function of
temperature T and volume V for this system.
9.6 Suppose at some moment the concentration of sodium ions on the inside of a
cell membrane of thickness 5 nm is 304 mM and outside the membrane it is 302 mM
(1 nm is one nanometer, 1 mM is one millimolar), with ions passing through at the
rate of 4.3 × 10 14 s/cm 2 . What is the drift velocity of an ion through the membrane?
What is the diffusion constant for sodium through this wall?
9.7 The rate of diffusion of small particles in the vitreous humor of the eye is of
physiological and medical interest. Estimate the diffusion time for a concentration
of protein molecules to spread from an injection point in the center of the eye to
a RMS distance of 10 mm across the vitreous if the diffusion constant for these
molecules is measured to be D = 0.47 mm 2 / h.
9.8 If a set of macromolecules each of diameter 812 nm are observed to spread
in water a root-mean-square distance of 10.51 μm after 30 s, estimate Avogadro’s
number. The viscosity of water at room temperature (27 ◦ C) is 0.852 centipoise.
9.9 A small vacuole containing a concentrated solution of salt, NaCl, is opened
on the interior of one side of a cell of diameter 12 μm. The diffusion constant for
salt in water is about 1.0 × 10 3 (μm) 2 /s. Estimate the time for this salt to reach the
opposite side of the cell by diffusion.
9.10 Two membranes, #1 and #2, permeable to a small organic molecule, are found
at either end of an impermeable cylinder of length L within a cell. The cell maintains
a concentration C 1 on the outside of membrane #1, and C 2 outside membrane #2.
The diffusion constant for the molecules in the fluid contained in the cylinder is D.
What is the steady-state concentration of the small organic molecules at a distance
x from the membrane #1?