BSCI 207 – Write down the version of Henry’s Law that gives the solubility

BSCI
207 Fall 2015
Homework
#3: Gas Exchange & Bulk Flow

Please
type your answers into this document and upload to ELMS by
11:59
P.M., Wednesday, 14 October

You may work on this homework on your own or discuss it with
others. If you work through questions
with others, be sure to turn in your own work, at the level at which you
understand it. Do not simply copy
answers from or for someone else.
Questions such as these will show up on exams, so be sure to work to
understand how to arrive at the answers.

(1) In class, we have made the point that the solubility of oxygen in
water (or blood plasma) is relatively low and that hemoglobin helps increase
the amount of oxygen that can be carried by blood so that cells in large
animals with circulatory systems can get enough oxygen to support their
metabolism. This problem asks you to quantify
the help that hemoglobin provides.

A. Write down the version of Henry’s Law that
gives the solubility constant (including units) in terms of partial pressure (which
is measured in atmospheres of pressure, or atm) and concentration (which is
measured in mol/L, or M).

B.
The partial pressure of oxygen in the lung alveoli is a bit less than in
air, being about 100 mm of mercury, or 0.13 Atm (it is lower than the partial
pressure in air mainly because oxygen is continually taken up by the alveolar
capillaries and carbon dioxide is continually released into the alveoli). In a saline/glucose solution (formulated to
match the composition of blood plasma and lacking hemoglobin, so we can think
of this as approximating cell-free blood plasma) at 37° C, the
concentration of oxygen will equilibrate at about 3 mg/L (or 0.093 mM). In oxygen-saturated blood (with hemoglobin),
however, the O2 concentration is about 0.008 M (8 mM).

What
is the Henry’s Law constant, K, for each of these two situations (with hemoglobin
vs. without hemoglobin)? Show your work.

C. By
what factor does the presence of hemoglobin enhance oxygen solubility in
blood? Show your work.

D. Given the above, imagine that you are an
emergency room doctor treating a patient who lost half of his blood in an
accident. A paramedic replaced this lost blood with cell-free plasma to keep
his blood pressure up (the plasma contains no hemoglobin since it contains no
red blood cells). The patient is short of breath and not doing well. If you had
a choice between administering pure (100%) oxygen or giving a blood infusion to
restore the red blood cell count, which would be more helpful? Why? In
comparing the two options, be as quantitatively explicit as possible.

(2) As mentioned in class, some air-breathing aquatic beetles carry a
bubble of air underwater with them and can stay submerged for long periods of
time using the bubble as a sort of “gill” to extract oxygen from the water. In
this problem, you will explore how this works.

Consider the situation shown below. Assume
a uniform temperature of 20 C and a body of pure water that is at equilibrium
with the atmosphere. At 20 C, K for oxygen in water is 1.4 mmol/Latm (= 1.4 X
10-3 mol/Latm). The molar concentration of oxygen in air is around
8.6 mmol/L (simply derived using Ideal Gas Law, for those of you who have
learned about that in chemistry).

Answer
the questions below (show your work, where appropriate) and fill in the missing
numbers in the two diagrams below (if you have trouble figuring out how to get
the numbers to show up in the right place on the figures, you can type the
answers as a clear list above the figures).

What
is the partial pressure of oxygen in the atmosphere?

What
is the partial pressure (in the physiologist’s sense) of oxygen in the water?

What
is the molar concentration of oxygen in the water?

What
is the initial partial pressure of oxygen in the beetle’s air bubble?

What
is the initial molar concentration of oxygen in the beetle’s air bubble?

Is
the initial equilibrium concentration of oxygen in the air bubble higher or
lower than the equilibrium concentration in the surrounding water?

Why
are the concentrations of oxygen in the air bubble and in the surrounding water
not equal at equilibrium?

After
the beetle has consumed half the oxygen in the air bubble, what is the partial
pressure of oxygen in the bubble?

After
the beetle has consumed half the oxygen in the air bubble, what is the
concentration of oxygen in the bubble?

After
the beetle has consumed half the oxygen in the air bubble, is the concentration
of oxygen in the air bubble higher or lower than the concentration in the
surrounding water?

After
the beetle has consumed half the oxygen in the air bubble, are the gas phase in
the bubble and the surrounding water still at equilibrium with respect to
oxygen? If not, what will happen to return the system to equilibrium? Will
oxygen move from a higher concentration to a lower concentration? Why or why
not?

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INITIAL AFTER CONSUMED 50% OF BUBBLE O2

(3) As we discussed in class, if we accept some assumptions (not
strictly met for blood flow through blood vessels, but close enough for our
purposes), the flow of a fluid through a tube is proportional to the 4th
power of the radius of the tube—or, equivalently, to the square of the
cross-sectional area. Thus, if cross-sectional area is reduced by 10%, the new flow
will be (0.9)2 = 81% of unobstructed flow. In humans, occlusion
(blockage) of the coronary arteries (which feed oxygenated blood to the heart
tissue itself) is usually symptomless even with a 50% reduction in
cross-sectional area. Significant symptoms are typically not experienced until
the reduction in cross-sectional area reaches about 70% (above 75-80% serious
problems often develop).

Consider whether the non-linear
relationship between flow and degree of reduction of cross-sectional area can explain
why a 50% blockage typically does not produce symptoms whereas a 70% blockage
does:

What is the predicted new flow rate given a 50%
reduction in cross-sectional area? Show your work.

What is the predicted new flow rate given a 70%
reduction in cross-sectional area? Show your work.

Given these results, does it seem plausible that no
symptoms would be felt until 70% occlusion? Explain.

Why might
the analysis above overestimate the effect of a given reduction in
effective artery cross section on blood flow through an artery? The artery
itself provides resistance to flow, but other blood vessels with which the
artery in question is in series contribute resistance to flow as well. Not
until occlusion of the coronary arteries exceeds around 75% do these
vessels become the dominant elements of resistance to blood delivery to
the heart muscle.

Suppose the artery in question is just one of ten
blood vessels contributing equally to the total resistance to flow. If
resistance doubles in one of these blood vessels, by what percentage does total
resistance increase?