### Ch.3 - Interactions and Implications

```Chapter 3
Interactions and Implications
Entropy
Entropy
Let’s show that the derivative of entropy with respect to energy
is temperature for the Einstein solid.
Let’s show that the derivative of entropy with respect to energy
is temperature for the monatomic ideal gas.
Let’s prove the 0th law of thermodynamics.
An example with the Einstein Solid
Heat Capacity, Entropy, Third Law
•
•
•
•
•
Calculate W
Calculate S = kB ln(W)
Calculate dS/dU = 1/T
Solve for U(T)
Easy
Cv = dU/dT
Difficult to impossible
Easy
Easy
Easy – we’ll see a
better way in Ch . 6 w/o
needing W
Heat capacity of aluminum
Let’s calculate the entropy changes in our heat capacity experiment.
Heat Capacity, Entropy, Third Law
What were the entropy changes in the water and aluminum?
DS = Sf – Si = C ln(Tf/Ti)
Heat Capacity, Entropy, Third Law
As a system approaches absolute zero temperature, all processes within the
system cease, and the entropy approaches a minimum.
The Third Law
It doesn’t get that cold.
limS  0
T 0
limCV   0
T 0
As a system approaches absolute zero temperature, all processes within the
system cease, and the entropy approaches a minimum.
Stars and Black Holes modeled as orbiting particles
m1
Show the potential energy is equal to
negative 2 times the kinetic energy.
r
r
m2
Stars and Black Holes modeled as orbiting particles
m1
Show the potential energy is equal to
negative 2 times the kinetic energy.
r
r
m2
Stars and Black Holes modeled as orbiting particles
m1
What happens when energy is added? If
modeled as an ideal gas what is the total
energy and heat capacity in terms of T?
r
r
m2
Stars and Black Holes modeled as orbiting particles
m1
Use dimensional analysis to argue potential
energy should be of order -GM2/R. Estimate
the number of particles and temperature of
our sun.
r
r
m2
Stars and Black Holes modeled as orbiting particles
m1
What is the entropy of our sun?
r
r
m2
Black Holes
What is the entropy a solar mass black hole?
Black Holes
What are the entropy and temperature a solar
mass black hole?
S
U
Mechanical Equilibrium
Mechanical Equilibrium
Mechanical Equilibrium
Diffusive Equilibrium
Diffusive Equilibrium
Chemical potential describes how particles move.
The Thermodynamic Identity
Diffusive Equilibrium
Chemical potential describes how particles move.
Diffusive Equilibrium
Chemical potential describes how particles move.
Diffusive Equilibrium
Chemical potential describes how particles move.
Diffusive Equilibrium
Chemical potential describes how particles move.
Entropy
The Thermodynamic Identity
The Thermodynamic Identity
Paramagnet
Paramagnet
U
+mB Down, antiparallel
0
-mB
Up, parallel
Paramagnet
Paramagnet
Paramagnet
• M and U only differ by B
Nuclear Magnetic Resonance
wo = 900 MHz
B = 21.2 T
wo = g B
g = 42.4 (for protons)
Nuclear Magnetic Resonance
B
S
Inversion recovery
Quickly reverse magnetic field
NmB
Low U (negative  stable)
Work on system lowers entropy
but it will absorb any available
energy to try and slide towards max S
U
M
B
NmB
NmB
High U (positive  unstable)
Work on system lowers entropy
but it will absorb any available
energy to try and slide towards max S
t
Analytical Paramagnet
Analytical Paramagnet
Analytical Paramagnet
Analytical Paramagnet
Paramagnet
Paramagnet Properties
Paramagnet Properties
Paramagnet Heat Capacity
Magnetic Energies
```