Download Some ideas from thermodynamics

Physics of the Human Body
Chapter 6
Some ideas from thermodynamics
53
Some ideas from thermodynamics
The branch of physics called thermodynamics
deals with the laws governing the behavior of
physical systems at finite temperature. These laws
are phenomenological, in the sense that they represent empirical relations between various observed
quantities. These quantities include pressure, temperature, density and other measurable parameters
that define the state of a given system. Thus, in
discussing magnetic systems, e.g., the variables
might include temperature, magnetization and external magnetic field; in discussing elastic systems
we might want to use temperature, stress and strain
as defining variables. Some useful references for
the student who wants to pursue the subject further are
1. F.W. Sears, Thermodynamics, 2nd ed. (AddisonWesley Publishing Co., Reading, MA, 1963).
2. K. Huang, Statistical Mechanics (John Wiley and
Sons, Inc., New York, 1963).
3. R.P. Feynman, Statistical Mechanics (W.A. Benjamin, Inc., Reading, MA, 1972).
1.
The First Law of Thermodynamics
We imagine a system such as a gas, where temperature, pressure and density are the appropriate system-defining variables. Then conservation of
energy states that the following differential relationship must hold:
dU = dQ − dW
(1)
where dQ represents (a small amount of) heat
flowing into the system from some heat source;
dW the mechanical work done by the system, and
dU the change in internal energy.
The quantity U depends, by assumption, only on
the present state of the system and not on the
history of how the system got into that state. In
that case, dU must be, in the mathematical sense,
an exact differential. To understand what this
means, consider the work done by sliding a heavy
block up an inclined plane. There is a force acting
downhill (gravity) as well as friction. Obviously the
work done traversing the wiggly path up the hill is
greater than that along the straight path, since
friction always acts opposite to the direction of
motion. That is, the part of the work arising from
overcoming gravity is independent of path (because gravity is a conservative force) but that used
to overcome friction is path-dependent (because
friction is a dissipative force).
In other words, the work output depends on how
the job is done, so dW is not an exact differential.
Similarly, the heat input depends on the details of
how the heat enters the system—for example, it
can depend on the rate at which heat is supplied—
so dQ is also not an exact differential. But experience indicates that the internal energy of a system
does not depend on how it got there, hence we
postulate that dU is exact.
Now suppose this postulate were wrong: then conclusions we can deduce from it would at some point
disagree with experience. So far no such cases have
arisen; moreover, the microscopic theory of thermodynamics (that is, statistical mechanics) suggests strongly that dU is exact.
54
2.
Physics of the Human Body
Equation of state
Equation of state
We can use different sets of variables to define the
state of a thermodynamic system. For example we
can use temperature and pressure, pressure and
volume, or temperature and volume. The reason
we have only two independent variables is that the
three variables p, V, T are related by an equation of
state which takes the form
f(p, V, T) = 0 .
relationships. That is, imagine it depends on pressure and volume:
U ≡ U(p, V)
so that
 ∂U 
 ∂U 
dU =   dp +   dV .
 ∂p V
 ∂V p
(5)
(2)
For example one typical equation of state is the
perfect gas law,
pV = νRT
(3)
where R is the perfect gas constant (numerical
value ≈ 8.3 J ⁄ oK ⁄ gm−mol) and ν is the number of
gram-moles present.
Another well-known equation of state is the Van
der Waals equation,
p + an2 1 − nV0 = n kBT
(4)



where n is the number-density of the molecules, a
is a constant representing an average (attractive)
inter-molecular interaction energy, and V0 is proportional to the volume of a molecule (that is, if
there are N molecules the available volume is
V − N V0 rather than V). Also kB = R ⁄ NA is
Boltzmann’s constant.
The Van der Waals equation of state represents an
attempt to take into account two physical effects
that cause actual gases to deviate from the behavior of an ideal gas: intermolecular forces, and the
finite size of molecules. Although not a precise
representation of any gas (except possibly over a
limited range of the variables) Eq. (4) is useful in
understanding qualitatively the behavior of gases
in phase transitions.
Now imagine that we go from internal energy
U1 = U(p1, V1) to U2 = U(p2, V2) via two different paths as shown above. Since the result is
independent of path, it must be true that
∫
Γ
dU =
∫
Γ
f(p, V) dp + g(p, V) dV = 0 ,


where the integral is taken around the closed path
Γ. Now, by noting that if the result is true for a large
path it must also be true for the small square path
shown below, we find
−g(p + dp)dV
−f(V)dp
f(V+dV)dp
g(p)dV
3.
Thermodynamic relations
Because the internal energy differential dU is (by
hypothesis) perfect, we immediately derive certain
Physics of the Human Body
Chapter 6
Some ideas from thermodynamics
 ∆Q 
 ∂H 
cp =   ≡  
 ∆T p
 ∂T V
g(p,V) − g(p+dp, V) dV +


or
+ f(p, V+dV) − f(p, V) dp = 0


55
(7)
where the enthalpy H is defined as
H = U + pV .
If a thermally isolated ideal gas is allowed to expand
slowly into a vacuum, as shown below,
∂ g
∂ f
 ∂V − ∂p  dp dV = 0 ,


or finally
∂  ∂U 
∂  ∂U 
=
.


∂V  ∂p 
∂p  ∂V p
V
In other words, for dU to be an exact differential
the order of its second derivatives with respect to
p and V is immaterial. (This result is just Stokes’s
Theorem in two dimensions.)
Since the the work output can be written
dW = pdV ,
we can express the heat input into a system in the
alternate forms
  ∂U 
 ∂U 
dQ =   dp + p +  
∂V
 ∂p V
  p

 dV (5a)

Joule’s free-expansion experiment
  ∂V 
 ∂U  
+ p   +    dp
∂p
 ∂p  
  T
T
it is found that the temperature does not change.
On the other hand the volume may change substantially. Since the system is thermally isolated
the change in heat must be zero. On the other
hand, no work is done in the expansion since no
force is exerted on anything. Therefore we deduce
that the internal energy of an ideal gas must be
unchanged in the experiment; that is, the internal
energy of an ideal gas depends neither on volume
nor on pressure, but solely on temperature.
 ∂U 
  ∂U  
dQ =   dT + p +    dV . (5c)
∂V
 ∂T V
  T 
From the above experimental result we further
deduce that the internal energy of the ideal gas has
the form
  ∂V 
 ∂U  
dQ = p   +    dT +
∂T
 ∂T  
  p
p
(5b)
The specific heat at constant volume is defined by
 ∆Q 
 ∂U 
cV =   ≡   ,
 ∆T V
 ∂T V
(6)
where the last relation follows from letting dV=0
above. The specific heat at constant pressure is
defined as
U = cV T + constant .
(8)
Since the origin of the energy scale is arbitrary, we
may set the constant to zero (that is, it is unmeasureable).
56
Physics of the Human Body
Heat engines and Carnot efficiency
For an ideal gas, since pV = νRT, we see the
enthalpy is also a function of temperature only,
H = U + pV ≡ cV + R T ;


that is, the molar specific heats cp and cV are related
by
cp = cV + R .
4.
Heat engines and Carnot efficiency
The first law of thermodynamics is really nothing
more than the law of conservation of energy, taking into account that heat is a form of energy. The
key difference between heat and other forms of
energy, however, becomes manifest in the process
of conversion from one form of energy to another.
Electrical energy—in the form of charge on a
capacitor or a current in a wire—can in principle1
be converted 100% into mechanical energy
(work), chemical energy (charging a battery, or
electrolysis, e.g.), electromagnetic energy (radio,
light), or heat (radiant heater, oven, etc.). For that
reason we say that electricity is “high grade” energy. Similarly chemical energy can be converted
into electrical energy (battery, fuel cell), or into
mechanical work, with high efficiency.
Heat is different. If we want to convert heat into
mechanical work or electricity using engines or
engine-driven generators, the fundamental nature
of heat imposes a limitation on the maximum
efficiency that can be achieved with the most
perfect machine conceivable. This limitation was
discovered by the French engineer Sadi Carnot.
Carnot’s argument was based on a heat engine
using a perfect gas as its working fluid. The first
part of the argument is the assumption that perpetual motion, in the form of an engine that cre-
1.
ates more energy than it consumes, is impossible.
Thus if we consider heat engines, we are entitled
to imagine one that has no heat leaks, friction or
other ways in which energy can be used unproductively.
The most perfect kind of heat engine is one that
begins in a certain state (pressure, volume), takes
heat Q1 from a source at temperature T1, produces
some work
W ≤ Q1 ,
and if there is any leftover heat,
df
Q2 = Q1 − W ,
that has not been converted to work, dumps that
into a heat sink at temperature T2 . Then the
system is restored to its original state and the
engine is in a position to repeat these actions.
The key to getting the most work possible for a
given heat input is to make all parts of the cycle
reversible. That is, when heat is moved from the
source to the engine, it is done with the engine at
the same temperature as the source, and so slowly
that one can reverse the engine and put the heat
back with no loss or change of state.
This is illustrated below. In the first step, a cylinder
of gas at temperature T1 is placed in contact with
the source and slowly expanded, extracting heat
Practical considerations—friction, resistance, etc.—restrict the achievable conversion efficiencies to less
than 100%, of course.
Physics of the Human Body
Chapter 6
Some ideas from thermodynamics
and doing work. The cylinder is then removed
from the source and insulated so no heat can flow
in or out. The piston is further expanded, doing
more work and allowing the temperature to fall to
T2 . The cylinder is then placed in contact with
the sink and the gas compressed so that heat flows
into the sink. (Work has to be put in in order to
compress the gas.) Finally, the cylinder is removed
from the sink and insulated so no heat flows in or
out, and the gas compressed still further until the
temperature has come back to T1 . The gas is now
back in its original state.
How do we know the system returns to its original
state at the end of a cycle? The answer is that we
must show it is possible to construct a cycle that
does precisely this. If the working fluid is a perfect
gas we may find relations for the two types of
expansion (or compression). When the cylinder is
in contact with the heat reservoir, we say a change
of volume is isothermal. When it is isolated, the
change of volume is adiabatic. In mathematical
terms, isothermal means
Carnot cycle
Isothermal
Adiabatic
Pressure
1
2
3
4
Volume
dT = 0
whereas adiabatic means
dQ = 0 .
In an isothermal process (we assume we are working with 1 mole of gas)
pV = RT = constant ,
whereas in an adiabatic process,
57
dU ≡ cV dT = −p dV .
Applying the perfect gas equation of state we have
cV dT = −R dT + V dp
or with
dT =
1
d(pV)
R
we have
(η − 1) V dp + η p dV = 0
(9)
or
cp dV
dp
+
= 0,
cV V
p
and
−cp ⁄ cV
V
p = p0  
V
 0
Since
df
γ =
.
(10)
cp
> 1,
cV
the curve of pressure vs. volume is steeper than
that of an isotherm passing through the same
point. Thus on the diagram to the left we see the
initial isothermal expansion (from point 1 to point
2) lies on a curve that falls more slowly than the
subsequent adiabatic expansion (from point 2 to
point 3). Similarly, when the gas is compressed
isothermally (point 3 to point 4) the pressure rises
less steeply than in the subsequent adiabatic compression (4 to 1) that returns to the initial point.
The possibility of returning to the initial state then
depends on being able to return to the initial
volume and pressure (point 1 on the diagram). The
equation of state guarantees that the temperature
will then be the initial temperature, T1 . Moreover,
since the internal energy of an ideal gas depends
only on temperature, U will also have returned to
its initial value.
A solution that looks like the figure is possible if
the compression ratio λ satisfies
58
Physics of the Human Body
Heat engines and Carnot efficiency
1 ⁄ (γ − 1)
Vmax
 T1 
λ =
>  
Vmin
 T2 
Then it is easy to see that
.
(11)
 T2 
V3 = λVmin  
 T1 
and
η =
1 ⁄ (γ − 1)
Vmin < V3 < Vmax
and
Vmin < V1 < Vmax .
What about the net work done by the engine in
this cycle? It is clearly the area contained within
the intersecting curves,
W =
1
2
∫2 p dV + ∫3 p dV + ∫4 p dV + ∫1 p dV
1 ⁄ (γ − 1)
.
= RT1 − T2 ln λ T2 ⁄ T1


  


It is worth noting that the areas under the adiabatic parts of the curves,
W34 =
4
∫3 p dV
and
W12 =
.


Hence the Carnot efficiency is
 T1 
V1 = Vmin  
 T2 
so that
4
1 ⁄ (γ − 1)
df
Q1 = W23 = RT 1 ln λ T2 ⁄ T1

 
1 ⁄ (γ − 1)
3
Now to get the efficiency we need to evaluate the
energy extracted from the heat source. This is just
the area
2
∫1 p dV
cancel exactly. This should not be surprising since
on these segments of the cycle, ∆Q = 0, and therefore
W34 = cV T1 − T2


and
W12 = cV T2 − T1 ≡ −W34 .


T1 − T2
∆W
=
.
T1
Q1
Carnot’s proof that this is the maximum possible
efficiency of any heat engine is based on the assumption that a machine that creates work from
nothing (that is, a perpetuum mobile of the first
kind) is impossible. Suppose some other kind of
engine could do better than a Carnot cycle engine
that uses an ideal gas. Then we could use it to
produce energy, and run the Carnot engine backward to dump heat back into the heat source. That
is, if engine X can produce work
∆WX > ∆WCarnot
from the same heat input Q1, then running the
Carnot engine backwards uses work ∆WCarnot to
put the heat Q1 back, leaving the source and the
sink in their initial conditions, but with a net
output of work,
∆WX − ∆WCarnot > 0 ,
having been produced.
This argument of Carnot’s is so general and clever
that it has been applied in many other circumstances. For example, Einstein used it to show that
all forms of energy content must contribute to the
weight of an object according to
g ∆m = g
∆E
c2
—for if one form of energy did not contribute to
weight, one could in principle use pulleys, ropes
and an energy converter to make a perpetual motion machine.
Physics of the Human Body
Chapter 6
Some ideas from thermodynamics
5.
The Second Law of Thermodynamics
Carnot’s discussion of the efficiency of heat engines led naturally to the concept of entropy, although it was not until the end of the 19th Century
that Boltzmann was able to give a physical interpretation of entropy in terms of degree of order and
probability. Now in the Carnot engine, we note
that the heat taken out of the reservoir at T1 is
Q1 = W23
1 ⁄ (γ − 1)
= RT 1 ln λ T2 ⁄ T1

 



and that dumped in the sink at temperature T2 is
1 ⁄ (γ − 1)
Q2 = W41 = RT 2 ln λ T2 ⁄ T1

 
;


that is,
Q2
Q1
=
.
T1
T2
(12)
Like many results in physics, Eq. 12 is more general
than the arguments used to derive it. If we imagine
a lot of heat reservoirs (at different temperatures)
we could add up all the heats extracted from them,
divided by their temperatures, to obtain the integral
df
S =
∫
dQ
.
T
2.
3.
dQ
T
(14)
is sometimes called the second law of thermodynamics. One of its chief consequences is that in a closed
system the entropy always increases or remains the
same.
6. Statistical Mechanics
Thermodynamics is somewhat of a misnomer—it
deals with equilibrium relations rather than timedependent ones !2 .Thetheory consistsoftwolaws:
the First Law, which is just conservation of energy;
and the Second Law, which posits a new thermodynamic function, entropy, that (in a closed system) always increases or remains the same. The
remainder of thermodynamics consists of empirical
relations (equation of state, specific heat) and
relationships that can be derived from them.
Statistical mechanics is an attempt to derive the
laws of thermodynamics from a microscopic description at the atomic level. That is, historically
statistical mechanics accepted the reality of atoms
and molecules, even though its early practitioners
did not know how exactly big atoms were, and had
no means by which to observe them directly!3 .
(13)
It turns out that S is independent of the path the
system takes in getting from state 1 to state 2,
hence it is another thermodynamic function like
the internal energy, U, that depends only on the
state of the system. This new thermodynamic function is called entropy.
The differential relation
dS =
59
Boyle’s Law
Daniel Bernoulli seems to have been the first to
use the statistical approach to obtain a practical
result, Boyle’s Law, which states that at constant
temperature the product of pressure and volume
of a gas is constant:
pV = constant .
(15)
…that is, it should really be called thermostatics.
Of course technological progress has somewhat altered this—we can now isolate and work with individual
atoms, as well as visualize the atomic granularity of the surfaces of solids.
60
Physics of the Human Body
Statistical Mechanics
N
N
i=1
i=1
m
m
1
Fx =
v2x (i) ≡
N ⋅ ∑ v2x (i)
∑
L
L
N
Nm 2
〈vx 〉 ,
L
=
(19)
where 〈v2x 〉 is the mean-square velocity component
in the x-direction. But there is nothing special
about one direction or another, so we may write
〈v2x 〉 ≡
His (vastly oversimplified) picture of a gas consisted of a box of point masses whizzing around
randomly, and colliding elastically with the walls
of the box. The particles do not interact with each
other. We shall imagine a box in the form of a
rectangular parallelepiped of length L and crosssectional area A, as shown below!4 .
A single molecule collides with the face at the right
end of the box (say, lying in the y-z plane) and
reverses the x component of its velocity. The momentum transferred to the wall is thus
∆px = 2mvx .
(16)
The time between such collisions is
2L
∆t =
vx
(17)
1 2
1
〈vx + v2y + v2z 〉 = 〈v2〉 .
3
3
Now, finally, the force per unit area (that is, the
pressure) acting on the rightmost face is
df
p =
Fx
N 2 m 2
N2
=
〈v 〉 ≡
〈ε〉 . (20)
LA 3 2
V 3
A
Not only does Eq. 20 express Boyle’s Law, it even
tells us (once we extend Boyle’s Law to the ideal
gas law) how the average kinetic energy of a molecule is related to the absolute temperature:
RT =
N2
〈ε〉 .
ν 3
(21)
If we take one mole of molecules, then ν = 1 and
N = NA so we have
〈ε〉 =
df
3
3 R
T = kBT .
2
2 NA
(22)
so the average force exerted by this molecule is
mv2x
=
.
fx =
L
∆T
∆px
(18)
If there are N such molecules whizzing about, the
total force they exert on the rightmost face of the
box is
4.
Maxwell-Boltzmann distribution
When we study matter from the atomic viewpoint
we can ask more detailed questions than those
permitted by thermodynamics. For example we
might ask what fraction of molecules of a perfect
gas, at temperature T, have velocities between v
and v + dv .
The shape of the container is immaterial. A derivation for a spherical container may be found in G.B.
Benedek and F.M.H. Villars, Physics with Illustrative Examples from Medicine and Biology: Statistical Physics
(Springer Verlag, New York, 2000) pp. 250-252.
Physics of the Human Body
Chapter 6
Some ideas from thermodynamics
James Clerk Maxwell first derived the probability
distribution of molecular velocities in a dilute gas
in thermal equilibrium. He argued that if the temperature is constant, then the probability that a
molecule has x-component of velocity in the range
[vx , vx + dvx] must be the same function as the
probability that it has y-component in the range
[vy , vy + dvy]. That is, the motions in orthogonal
directions (and their corresponding probabilities)
are statistically independent. The total probability
function therefore has the form5
→
pv d3v = f(vx) f(vy) f(vz) dvx dvy dvz . (23)
 
→
Moreover, p( v ) can depend only on the squared
→
→ →
magnitude, |v|2 ≡ v ⋅ v , of the velocity, since it
must be6 invariant under rotations and reflections
of the coordinates. Thus we have
p(v2x + v2y + v2z ) = f(v2x ) f(v2y ) f(v2z ) .
Now suppose we rotate the coordinates so that
v2y = v2z = 0 ;
2
2
p(v ) = A f(v ) ,
or
ln f(v2x + v2y + v2z ) + 2 ln A
= ln f(v2x ) + ln f(v2y ) + ln f(v2z ) .
It is easy to show that the only function for which
this is true is
f(u) = −α u + Γ
where α is an undetermined constant and
Γ =
1
2
ln A2
is a normalization constant that we determine from
the condition that the total probability that a
particle has some velocity is 1.
Therefore the velocity distribution function is
2
p(v2) = A3 e−α v .
To determine the constant α we need to calculate
the average kinetic energy, since the kinetic theory
of gases, as we just saw, relates the average kinetic
energy of a molecule to temperature via
1
2
〈 mv2 〉 =
3
k T;
2 B
therefore, since
df
〈v2〉 =
2
∫ d3v v2 e−αv
∫ d3v e−αv
6.
=
3
,
2α
we find
1
2
α = m ⁄ kBT ,
p(v2) = A3 e−ε ⁄ kBT ,
(24)
where ε is the kinetic energy.
Ludwig Boltzmann noticed that Maxwell’s velocity
distribution has the form (24), and saw that the
law for the variation of gas density with altitude in
a uniform gravitational field,
ρ(z) = ρ(0) e−mgz ⁄ kBT
(25)
was the same law, with kinetic energy replaced
with potential energy. He therefore proposed a
probability distribution law governing the energies
of a complicated system:
p(E) = g e−E ⁄ kT ,
5.
2
or in other words,
then it becomes clear that (let A = f(0) )
2
61
(26)
The sparseness of the Latin alphabet leads us to use the symbol p to mean both probability, as in this section; and pressure, as elsewhere.
…that is, by symmetry. This is an illustration of the enormous power of symmetry in theoretical physics.
62
Physics of the Human Body
Applications of Maxwell-Boltzmann distributions
of which Maxwell’s result was a special case.
7.
Applications of Maxwell-Boltzmann
distributions
We now consider three applications of the distribution law, Eq. 26, each with biological implications.
Liquid-vapor equilibrium
The first concerns the equilibrium of a liquid and
a gas in contact, say water and water vapor. We
note the well known fact that energy is required to
convert a molecule of water from the liquid to the
gaseous phase. This is basically the binding energy
of water molecules to the body of liquid. In thermodynamics it is called the chemical potential. In
general the calculation of chemical potentials from
first principles—that is, from a knowledge of the
microscopic forces acting between water molecules, and using the laws of quantum mechanics—
is a difficult task that has only been performed for
some simple systems. So we shall assume it is a
given (slowly varying) function of temperature and
pressure. We call this energy µ.
Reaction rates
A second application of Boltzmann’s Law concerns
activation and reaction rates. Suppose that before
two molecules can undergo a chemical reaction
(that ultimately releases energy, to be sure!) they
have to approach each other within a certain
distance. If there is a repulsive potential between
Then according to Boltzmann, the probability for
a molecule to be in a state of energy µ relative to
the liquid state—that is to be gas rathor than
liquid—is proportional to e−µ ⁄ kBT and we may write
the partial pressure of vapor molecules as
pvap = p0 e−µ ⁄ kBT
where p0 is a constant that, like µ, could in principle be calculated from fundamental considerations, but has not been because of the difficulty of
the calculation. That is, we regard µ and p0 as
empirical fitting parameters.
If we take the logarithm of the vapor pressure and
plot it against 1 ⁄ T we should obtain a straight line
of negative slope, −µ ⁄ kBT ; as the figure at the
above right shows, this is indeed what we find.
them, as shown below, their motion must surmount the barrier before they can get close enough
to react. The height of the barrier is the “activation
energy” ∆E; not surprisingly, the fraction of molecules with energies equal to or greater than this
energy varies like
f ≈ e − ∆E ⁄ kBT.
Physics of the Human Body
Chapter 6
Some ideas from thermodynamics
Thus we expect the reaction rate to increase very
rapidly with temperature, when kBT << ∆E , but
only slowly thereafter.
The dependence of reaction rate on temperature
explains why life as we know it can exist only
within a narrow range of temperatures. The effects
are especially severe on muscles: in order to generate mechanical power most efficiently, muscles
must be maintained at a definite temperature. It is
interesting to note that certain large fishes—marlin, great white sharks, and so on—actually maintain their cruising muscles well above the ambient
temperature of the water7. They are then able to
use oxygen more efficiently, meaning they need
less muscle for the same propulsive power, can
thereby be more streamlined, and thus obtain a
competitive edge in the evolutionary struggle.
Dissolved gases
The third application of Boltzmann’s Law concerns the solution of gases in liquids, which is
crucial to understanding oxygen transport by
blood. Animals above a certain size cannot rely on
the amount of oxygen that can dissolve to provide
for the demands of sustained metabolism. Therefore they have evolved a transport mechanism of
much greater capacity.
Red blood cells represent about 40% of the blood
volume. The volume of a red blood cell is
1.6×10−16 m3, and it contains 2.8×108 molecules
of hæmoglobin. A single hæmoglobin molecule
binds 4 O2 molecules.
Thus we may ask what is the ratio of O2 dissolved
in plasma to the hæmoglobin-borne O2, in fully
oxygenated blood? To discover the answer to this
question we must understand how a gas dissolves
in a liquid. Assuming the solution is dilute,
7.
63
ngas << n liq ,
we understand that a gas molecule, when it dissolves, releases an energy µ (the binding to the
fluid). It also incurs an energy cost because the
molecules of the fluid are forced to rearrange
themselves to accomodate the intruder. Let us say
the net energy is positive (rearrangement cost
exceeds binding)—then we should expect the
fraction of molecules in the dissolved phase to be
fdis ≈ e − δE ⁄ kT.
At equilibrium we might expect the partial pressure in the fluid to roughly equal the gas pressure
outside, hence
pg δV − δE ⁄ kT df
ngas
∝
e
= pg K(T) .
n liq
kT
The latter is called Henry’s Law, and K(T) is called
the solubility of the gas. For O2 in water at 25 oC,
the solubility is 3.03 × 10−8 mm−1. (That is, the
pressure must be measured in mm of Hg.)
Air pressure is about 760 mm. About 20% of it is
O2 so the dissolved oxygen concentration must be
(at most) 6×10−4 mol/liter of water. With 6 liters
of blood in the body, and 60% of the volume of the
blood being liquid (assumed to be water) we get
0.002 moles of dissolved gas.
The total number of hæmoglobin molecules is
about
NH ≈ 2.8×108 ×
0.4 × 6×10−3
1.6×10−16
≈ 0.44×1022 .
There are 4 oxygen molecules per hæmoglobin
molecule giving about 0.028 mol of oxygen carried
by the hæmoglobin. In other words, the hæmoglo-
C.J. Pennycuick, Newton Rules Biology (Oxford U. Press, New York, 1992) pp. 33-36.
64
Physics of the Human Body
Applications of Maxwell-Boltzmann distributions
bin carries about 14 times as much oxygen as can
dissolve in the blood.
The bends
The physics of solution of gases in liquids also
explains caisson disease, or “the bends”, that afflicted so many laborers working at high pressure
during the 19th Century8, and that remains a
hazard to scuba divers.
Although nitrogen gas (the dominant fraction of
our atmosphere) is not very soluble in blood, as the
external pressure rises, more dissolves (see below).
When a diver brings his own air supply underwater,
it must be supplied at the pressure of the surrounding water or the effort of breathing would soon
exceed the diver’s strength. Thus in a deep dive for
extended periods, a considerable amount of nitrogen dissolves in the blood.
If the external pressure is removed too rapidly—for
example, by the diver rising too swiftly—the nitrogen can no longer remain in solution and forms
bubbles, rather like a bottle of soda that has been
shaken and then opened quickly. These bubbles
can lodge in joints, where they inflict great pain
(causing the victim to bend over in distress); or
they can press on nerves, causing paralysis; or they
can even cause embolisms blocking blood flow to
crucial areas of the body. In extreme cases the
bends can be permanently crippling or fatal.
immediately to high pressure in a decompression
chamber (the pressure is then gradually reduced,
according to an appropriate schedule).
Finally, especially in deep dives, a diver can
breathe a special mixture of oxygen and helium.
(He cannot breathe pure oxygen since that is
poisonous at high pressure!) Using helium as the
inert component of breathing air has two advantages, and one disadvantage. First, helium emerges
from the blood far more readily than nitrogen, and
can pass from the capillaries to the lungs far more
readily. Using it therefore greatly reduces the decompression time. Second, helium is chemically
far more inert than nitrogen, hence does not produce nitrogen narcosis (“rapture of the deep”) at
high pressure. The disadvantage is that the speed
of sound in helium is far greater than in air, causing
divers breathing that mixture to sound like Alvin
the Chipmunk or Donald Duck, with consequent
effects on communication with the surface.
One solution is to return to the surface more
slowly. A diver progresses to the surface in gradual
stages, using empirical “decompression tables” to
schedule how long to remain at each intermediate
depth, based on the maximum depth of the dive
and how long the diver remained there. Alternatively, a diver can return to the surface quickly (for
example, in an emergency) if he can be returned
8.
The workers who sank the footings for the Brooklyn Bridge suffered from this disease in great numbers.
Even John Roebling, the chief engineer, was incapacitated permanently by the bends.