BIOLPHYS 438

1
BIOL/PHYS 438

• Logistics
• Corrections from last week
• Review of Mechanics
• Introduction to Entropy and Temperature
• Back to Ch. 2 Energy Management

2
Logistics

• Assignment 1
• Login and Update your Profile!
• Please Email us about yourself!

3
Notation a few symbols
X Any abstract quantity dX An
infinitesimal change in x ?X A finite change in
x t Time s, x,y,z,d,r,R Distances
m (usually) M Mass kg U Energy J
(usually potential energy) K Kinetic energy
J W Mechanical Work J P dW/dt
Power W H Enthalpy J (usually stored
chemical energy) Q Heat energy J h
Mechanical efficiency of a heat engine G
Metabolic rate W
4
Physics Model of an Animal
?Min
?Hin
Mass is Conserved!
Energy is Conserved!

• In steady-state,
• ?Mstored 0 and ?Hstored 0
• Mechanical efficiency
• h ?Wout /?Hin
• Resting Metabolic Rate G0
• minimum dHin/dt to stay alive.

?Wout
?Mstored
?Hstored
?Hout
?Qout
?Mout
5
The Emergence of Mechanics(a mathematical
fantasy)

• Newton's Second Law F m a dp/dt p
• Dot Notation for Time Derivatives
• Time Integral ?F(t) dt ?p
• Impulse changes Momentum
• Dot Product with r Path Integral ?F(r) dr
  ?(½ mv 2)
• Work changes Kinetic Energy
• Cross Product with r r x F G r x p L
• Torque changes Angular Momentum

6
Poll Within the context of Classical Newtonian
Mechanics, assuming your weight is 600 N,
approximately what net force do you exert on the
Earth ? a) 600 N upward b)
0 N c) 600 N downward d) Other
7
Newton and the Free Body Diagram
Newton's Second Law ? F m a
Doh! du jour
Not as simple as it sounds! What forces? Mass
and acceleration of what? In the above picture,
the Free Body is the (nearly massless) sock
that the dogs are pulling on. Ergo Fa Fb
almost exactly, or else the sock would have a
huge acceleration!
8
Newton and the Free Body Diagram
Newton's Second Law ? F m a
Fb
Fa
Fa
Fb
A correct FBD from which we can calculate the
common acceleration of the entire system (both
dogs plus the sock) involves the forces Fa and
Fb exerted on the dogs' feet by the ground, in
reaction to the forces the dogs exert with their
feet. (Newton's Third Law)
9
Statistical Mechanics
An Abstract Introduction
Total energy
U
The System is composed of many irreducible
components, each of which can contain a share dUi
of the total energy U. There are many ways U
can be distributed among all the components of
the system. How many? Let's call the number O
(U), since it will be a function of U. For any
macroscopic system, O will be a big number, so
let's take its natural logarithm. . . .
dU1
dU5
dU3
dU2
dU4
Isolated Closed System
10
Statistical Mechanics
An Abstract Introduction
Total energy
U
The System is composed of many irreducible
components, each of which can contain a share dUi
of the total energy U. There are many ways U
can be distributed among all the components of
the system. How many? Let's call the number O
(U), since it will be a function of U. For any
macroscopic system, O will be a big number, so
let's take its natural logarithm. . . .
dU1
dU5
dU3
dU2
dU4
Isolated Closed System
11
Entropy s ln O
Remember, O (U) is the number of ways a given
total energy U can be distributed among all the
components of the system. Usually this goes up
as U increases. So does the entropy s. How
fast? Define ß ds/dU. Hold that thought.
Total energy
U
dU1
dU5
dU3
dU2
dU4
Isolated Closed System
12
Thermal Contact
The number O1 of ways U1 can be distributed
within system 1 is independent of the number O2
of ways U2 can be distributed within system 2,
so there are O O1 O2 ways that the total
energy
U U1 U2 can be distributed within the
combined system. Thus the total entropy is s
s1 s2 . Since we assume these redistributions
occur at random, the most probable configuration
is one in which there are the most possibilities
? the one with the highest total entropy.
Recall ß ds/dU.
dU2 - dU1
dU1
ds2 ß2 dU2
ds1 ß1 dU1
2
1
Two Systems can exchange U.
13
Thermal Equilibrium
Any exchange of energy (heat) that increases the
net entropy produces a macrostate that is more
probable than before, and so will tend to occur
spontaneously through utterly random processes.
How can we predict whether heat will flow?
ds ds1 ds2 ß1 dU1 ß2 dU2 ,
but dU2 - dU1, so ds (ß1 ß2) dU1 .
When ß1 ß2 a transfer of energy will have no
effect on the total energy. This is called
thermal equilibrium. It nicely corresponds to
our notion of two systems having the same
termperature. Is ß the temperature, then?
dU2 - dU1
dU1
ds2 ß2 dU2
ds1 ß1 dU1
2
1
Recall ß ds/dU.
14
Cold Hot
Any exchange of energy (heat) that increases the
net entropy produces a macrostate that is more
probable than before, and so will tend to occur
spontaneously through utterly random processes.
How can we predict which way the heat will flow?
ds ds1 ds2 ß1 dU1 ß2 dU2 ,
but dU2 - dU1, so ds (ß1 ß2) dU1 . If
ß1 gt ß2 then transferring energy from 2 to 1
increases s and will therefore happen
spontaneously. This is what we expect to happen
when 2 is hotter than 1 ? implying that a cold
system has a larger ß than a hot system, opposite
to our idea of temperature. The solution is
trivial . . . .
dU2 - dU1
dU1
ds2 ß2 dU2
ds1 ß1 dU1
2
1
Recall ß ds/dU.
15
Temperature t kBT
The definition ß ds/dU 1/t restores our
common sense notion of temperature a system
with high t is hot and will spontaneously give
up heat to a cold (low t) system. However, we
must still deal with units. Since s is a pure
number, t has units of energy (J). What
happened to degrees? The answer is, Degrees
are bogus! but we must live with bogosity, so
Boltzmann invented a conversion constant kB
1.38066 x 1023 J/K (where K means degrees
Kelvin which are the same size as oC but start
273.15o lower). Likewise the conventional form
of entropy S kBs

16
Food Chain
Energy unit conversions 1 cal 4.18 J
1 Cal 1 kcal 4.18 kJ
17
Thermal Radiation
Look up Insolation on http//Wikipedia.org
(great resource, but you can't use it as a formal
reference, because it changes). The solar
constant S 1370 W/m2 is out in space near
Earth we get hereabouts a bit less than 1
kW/m2 on a nice day. At night, perfectly
black surfaces at 0oC radiate about 0.3 kW/m2, of
which a large fraction escapes into outer space
on a clear night. Ask any farmer! Stefan-Boltzm
ann Law P sSB A T 4 where sSB
5.67x10-8 W m-2 K-4. (Not an entropy!)
18
Energy Storage
?Hstored
Photosynthesis
19
Food as Energy
?Hstored
Photosynthesis
20
Muscle Work
Specific muscle stress f 2 105 N/m2 Fa f
A
A typical biceps muscle can exert 500N.
?Vg mgh
A
?W Fb h
h
m
lever
Mechanical advantage Fb (a/b) Fa
a
b
21
Thermal Regulation
?Q rad ? t A (sSB T 4 0.3 kW per m2 area)
?W F ? r
?H food
Work force through a distance
?Q in
?Hout
Chemical Waste
22
Thermal Regulation
?Q rad ? t A (sSB T 4 0.3 kW per m2 area)
?Q evap ? m (L v 2.3 MJ/kg)
?W F ? r
?H food
Work force through a distance
?Q in
?Hout
Chemical Waste
23
Thermal Regulation
?Q rad ? t A (sSB T 4 0.3 kW per m2 area)
?Q evap ? m (L v 2.3 MJ/kg)
?W F ? r
?H food
Work force through a distance
Wait! You also get . . . . . . (or
lose heat through) conduction
and convection!
?Q in
?Hout
Chemical Waste
24
Conduction of Heat
HOT side (TH)

Q cond ? A (TH - TC) / l
l
Thermal conductivity W m-1 K-1
COLD side (TC)
For an infinitesimal region in a thermal
gradient, J U ? T
?
25
Assumption Efficient Heat Removal
Conduction across a thin layer can be very
efficient, but the heat must be taken away on the
other side!
This requires cool mass flow past a warm
surface. The formula for Q assumes that the
cold side of the conducting slab is held at TC
by efficient heat removal.

26
Cylindrical Heat Conduction
Good Insulator
A cylindrical vessel is filled with a hot gel.
As the gel cools, a thermal gradient is set up
between the warm centre and the cool outer
surface. To escape, heat must be conducted
through the whole solid mass of the gel.
Hot Gel
Good Insulator
27
Convection Mixing of Hot Fluids
Warm fluid rises, cool fluid sinks, setting up
cells of circulation which mix hot cold and
thus deliver heat to the (thin) container walls
much faster than it would get there via
conduction through a solid.