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 us about yourself!

3. Notation: a few symbols
X = Any abstract quantity
dX = An infinitesimal change in x
ΔX = 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. = minimum dHin/dt to stay alive.
Physics Model of an Animal
ΔMin
ΔHin
Mass is Conserved!
Energy is Conserved!
In steady-state,
ΔMstored = 0 and ΔHstored = 0
Mechanical efficiency
h = ΔWout /ΔHin
Resting Metabolic Rate G0
= minimum dHin/dt to stay alive.
ΔWout
ΔMstored
ΔHstored
ΔHout
ΔMout
ΔQout

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 ≡ Γ = 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) N upward
b) N
c) 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 Fb Fa
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 Ω (U), since it will be a function of U. For any macroscopic system, Ω 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 Ω (U), since it will be a function of U. For any macroscopic system, Ω will be a big number, so let's take its natural logarithm
dU1
dU5
dU3
dU2
dU4
Isolated Closed System

11. Entropy: σ = ln Ω U Isolated Closed System Total energy
Remember, Ω (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 σ. How fast?
Define β = dσ/dU.
Hold that thought. U
dU1
dU5
dU3
dU2
dU4
Isolated Closed System

12. Thermal Contact 2 1 dU2 = − dU1 dU1 dσ1 = β1 dU1 dσ2 = β2 dU2
The number Ω1 of ways U1 can be distributed within system 1 is independent of the number Ω2 of ways U2 can be distributed within system 2, so there are Ω = Ω1 • Ω2 ways that the total energy
U = U1 + U2 can be distributed within the combined system. Thus the total entropy is σ = σ1 + σ2 .
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 β = dσ/dU.
dU2 = − dU1
dU1
dσ1 = β1 dU1
dσ2 = β2 dU2 2 1
Two Systems can exchange U.

13. Thermal Equilibrium 2 1 dσ = (β1 – β2) dU1 . dU2 = − dU1 dU1
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? dσ = dσ1 + dσ2 = β1 dU1 + β2 dU2 ,
but dU2 = − dU1, so
dσ = (β1 – β2) dU1 .
When β1 = β2 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 β the temperature, then?
dU2 = − dU1
dU1
dσ1 = β1 dU1
dσ2 = β2 dU2 2 1
Recall β = dσ/dU.

14. Cold & Hot 2 1 dσ = (β1 – β2) dU1 . dU2 = − dU1 dU1 dσ1 = β1 dU1
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? dσ = dσ1 + dσ2 = β1 dU1 + β2 dU2 ,
but dU2 = − dU1, so
dσ = (β1 – β2) dU1 .
If β1 > β2 then transferring energy from 2 to 1 increases σ 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 β than a hot system, opposite to our idea of “temperature”. The solution is trivial
dU2 = − dU1
dU1
dσ1 = β1 dU1
dσ2 = β2 dU2 2 1
Recall β = dσ/dU.

15. Temperature τ = kBT
The definition β = dσ/dU = 1/τ restores our “common sense” notion of temperature: a system with high τ is hot and will spontaneously give up heat to a cold (low τ) system. However, we must still deal with units. Since σ is a pure number, τ 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 = x 10–23 J/K (where K means “degrees Kelvin” which are the same size as oC but start o lower).
Likewise the conventional form of entropy: S = kBσ

16. Energy unit conversions:
“Food” Chain
Energy unit conversions:
1 cal = 4.18 J Cal = 1 kcal = 4.18 kJ

17. Thermal Radiation
Look up “Insolation” on (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-Boltzmann Law: P = σSB A T 4
where σSB = 5.67x10−8 W m−2 K−4. (Not an entropy!)

18. Energy Storage
ΔHstored
Photosynthesis

19. Food as Energy
ΔHstored
Photosynthesis

20. Muscle Work f = 2 • 105 N/m2 Specific muscle stress Fa = f • A A
A typical biceps muscle can exert ~ 500N. A
ΔVg = mgh
ΔW = Fb h m h
lever a
Mechanical “advantage”:
Fb = (a/b) Fa b

21. Thermal Regulation ΔQ rad = Δ t • A • (σSB T 4 ≈ 0.3 kW per m2 area)
ΔW = F • Δ r
ΔH food
Work = force through a ║distance
ΔQ in
ΔHout
Chemical Waste

22. Thermal Regulation ΔQ rad = Δ t • A • (σSB T 4 ≈ 0.3 kW per m2 area)
ΔQ evap = Δ m • (L v ≈ 2.3 MJ/kg)
ΔW = F • Δ r
ΔH food
Work = force through a ║distance
ΔQ in
ΔHout
Chemical Waste

23. Thermal Regulation ΔQ rad = Δ t • A • (σSB T 4 ≈ 0.3 kW per m2 area)
ΔQ evap = Δ m • (L v ≈ 2.3 MJ/kg)
ΔW = F • Δ r
ΔH food
Work = force through a ║distance
Wait! You also get . . .
. . . (or lose heat through)
conduction and
convection!
ΔQ in
ΔHout
Chemical Waste

24. Conduction of Heat J U =  • T ∆ ℓ 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.