The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
Interactions and Implications
Temperature
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Definitions of temperature
So far, we have defined temperature in three ways (one operational
and two theoretical):
Temperature is what you me
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
The Second Law
Very Large Numbers
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Numbers in statistical mechanics
Three kinds of numbers are of interest in statistical mechanics:
Ordinary numbers
Large numbers
Very large numbers
S
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
The Second Law
Large Systems
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Large Einstein solid
Consider the multiplicity of an Einstein solid:
How does this function behave in a large system?
If both N and q are large numbers,
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
The Second Law
Large Interacting Systems
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Motivation
We are now in a position to derive the most fundamental result in
statistical mechanics
Namely, in a large interacting system, the
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
The Second Law
Interacting Systems
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Interacting solids
Consider now a system of two Einstein solids, labeled A and B
The solids are allowed to interact and exchange energy:
N oscillat
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
The Second Law
Entropy
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
The second law
We have now seen many examples in support of the following
statement:
Any large system in equilibrium will be found in the macrostate
with the gre
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
The Second Law
Interacting Ideal Gases
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Monatomic ideal gas
We have seen that the multiplicity of a monatomic ideal gas is
When the and dependence is of primary interest, we can also
wr
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
The Second Law
Entropy of an Ideal Gas
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Entropy of an ideal gas
As another example, consider a monatomic ideal gas:
The corresponding entropy is called the SackurTetrode equation:
Ent
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
The Second Law
Multiplicity of a Monatomic Ideal Gas
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Many particles
The next step is to consider a system of N noninteracting particles
in 3 dimensions
The energy in this case is
Thi
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
The Second Law
Geometry in Very High Dimensions
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Volume of a hypersphere
Our next objective is to calculate the multiplicity of an ideal gas
To do this, we need to know the following f
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
PHYS 4050 Assignment 1
Energy in Thermal Physics
Due 18 September 2015
1. Estimate the average temperature of the air inside a spherical hotair
balloon of diameter 15 m. Assume that the total mass of the unfilled
balloon and payload is 500 kg. What is th
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
PHYS 4050 Assignment 4
Interactions and Implications
Due 9 October 2015
1. The figure below shows graphs of entropy versus energy for two objects,
A and B. Both graphs are on the same scale. The energies of these
two objects initially have the values indi
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
PHYS 4050 Assignment 7
Phase Transformations
Due 30 October 2015
1. Calcium carbonate, CaCO3 , has two common crystalline forms, calcite
and aragonite. Thermodynamic data for these phases can be found at
the back of the textbook.
(a) Which is stable at th
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
PHYS 4050 Assignment 5
Mechanical and Diffusive Equilibrium and Heat Engines
Due 16 October 2015
1. Sketch a qualitatively accurate graph of the entropy of a substance
(perhaps H2 O) as a function of temperature, at fixed pressure. Indicate
where the subs
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
PHYS 4050 Assignment 3
The Second Law
Due 2 October 2015
1. Consider an ideal monatomic gas that lives in a twodimensional universe (flatland), occupying an area A instead of a volume V .
(a) Derive an expression for the multiplicity of this gas as a fun
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
PHYS 4050 Assignment 2
Microstates, Macrostates, and Multiplicity
Due 25 September 2015
1. Suppose you flip 50 fair coins.
(a) How many possible outcomes (microstates) are there?
(b) How many ways are there of getting exactly 25 heads and 25 tails?
(c) Wh
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
Microstates,
Macrostates, and
Multiplicity
The Einstein Model of a Solid
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Einstein model of a solid
Previously we modeled the atoms in a
solid as classical harmonic oscillators
Lets no
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
Energy in Thermal
Physics
Heat Capacities
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Definition of heat capacity
The heat capacity of an object tells us how much heat is needed
to raise its temperature by one degree:
Since thi
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
Interactions and Implications
Mechanical Equilibrium and Pressure
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Thermal and mechanical interaction
Consider two systems separated by a movable partition
They can therefore interact
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
Free Energy and Chemical
Thermodynamics
Free Energy as Available Work
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Enthalpy
+
Previously we introduced the enthalpy function
The purpose of adding was to account for the work nee
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
The Second Law
Geometry in Very High Dimensions
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Volume of a hypersphere
Our next objective is to calculate the multiplicity of an ideal gas
To do this, we need to know the following f
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
Energy in Thermal Physics
Thermal Equilibrium
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Motivation
Thermodynamics deals with the properties of very large systems
(typically containing 1023 or more particles)
Ultimately, we wi
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
Microstates, Macrostates,
and Multiplicity
TwoState Systems
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Motivation
Why does heat always flow from hot objects to cold objects?
The ultimate reason is that this is the most probab
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
Boltzmann Statistics
The Boltzmann Probability Distribution
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
New statistical methods
The fundamental postulate tells us how to calculate probabilities in
an isolated system
We shall no
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
Free Energy and Chemical
Thermodynamics
Phase Transformations of Pure Substances
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Phase transformations
The most common phases of matter are solid, liquid, and gas
In a phase transform
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
Boltzmann Statistics
The Maxwell Speed Distribution
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Speed of molecules
Previously, we used the equipartition theorem to find the rootmeansquare speed of the molecules in an ideal gas:
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
Quantum Statistics
Indistinguishable Particles in Quantum Mechanics
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Classical indistinguishable particles
Consider a system of N indistinguishable particles
Our classical correction f
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
Quantum Statistics
Degenerate Fermi Gases
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Fermi gas
Lets now use the density of states to calculate the number of
particles in a gas of noninteracting fermions with spin 1/2:
= FD
F
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
Quantum Statistics
Massive Bosons at Low Temperatures
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Massive bosons
Bosons with mass are very different from bosons with zero mass
The number of zeromass bosons (e.g., photons) is n
The Hong Kong University of Science and Technology
thermodynamics
PHYS 4050

Fall 2015
Energy in Thermal
Physics
Heat and Work
PHYS 4050: Thermodynamics and Statistical Physics
Prof. B. A. Foreman
Temperature and energy
We have seen that temperature is related to energy averages by
the equipartition theorem:
However, this is not the defin