Hi Rahul, this is the last assignment in the matlab series. If you require any further documentation please let me know. I have included the last 6 lectures some may be relevant some maybe not.. you...

Matlab_thermo_11062020/__MACOSX/._basic_crystallisation_model_assignment3.m
Matlab_thermo_11062020/__MACOSX/._basic_ideal_gas_model_assignment_2(1).m
Matlab_thermo_11062020/__MACOSX/._task.png
Matlab_thermo_11062020/__MACOSX/Lectures/._Lecture 11-12 R(slides).pdf
Matlab_thermo_11062020/__MACOSX/Lectures/._Lecture 1-2 (R).pdf
Matlab_thermo_11062020/__MACOSX/Lectures/._lecture 3-4-r.pdf
Matlab_thermo_11062020/__MACOSX/Lectures/._lecture 5-6(1).pdf
Matlab_thermo_11062020/__MACOSX/Lectures/._lecture 7-8 (slides).pdf
Matlab_thermo_11062020/__MACOSX/Lectures/._Lecture 9-10 (Slides).pdf
Matlab_thermo_11062020/~$SIGNMENT 3_Q1.docx
Matlab_thermo_11062020/~WRL2355.tmp
ASSIGNMENT 3
a Matlab Code for answe
clc;
clear all;
%Basic simulation of trajectories of particles in a two
%dimensional interacting gas inlcuding energy dissipation leading to
%cooling
%Saves postions and temperature of particles as a function of time, and a
%movie showing the trajectories as they crystallise
%Written by Warwick Bowen, April 2020. Can be used as a basis for the
%computational question in Assignment 3 of PHYS2020 in 2020.,

clear all

Ti=300; %Temperature in Kelvin
k=1.38e-23; %Boltzmann constant
m=1e-26; %Mass in kg

v_rms=sqrt(k*Ti/m); %Mean particle velocity expected
N=100; %number of particles
L=1; %length of cube side in meters

gamma=0.99; %factor of velocity not decayed after one time step (1-gamma) is the amount the veclocity reduces).

C_repulsive=-1e-4; %Coefficient defining magntitude of repulsive force (needs to be negative)
d=L/30; %separation of two particles that gives minimum potential energy (i.e. molecular bond length)
C_attractive=-C_repulsive*d^2; %Coefficent defining magnitude of attractive force, calculated do that the potential minima occurs at d. I calcuated this for a potential of the form V = a
^2 +
^4.

tres=d/v_rms/30; %time step: chosen to be small compared to time for average velocity to make two particles in ground state lattice overlap to avoid large e
oneous jumps in potential energy
N_time=16000; %number of time steps to calculate
tmax=tres*N_time; %duration of simulation

%particle position: randomly locating particles
x_i=L*rand(N,1);
y_i=L*rand(N,1);

%particle velocity: randomly selecting their vwlocity components from a Gaussian
%distribution
vx_i=v_rms*randn(N,1);
vy_i=v_rms*randn(N,1);

vx=vx_i; %Initialising velocities in loop
vy=vy_i;

x=zeros(N,N_time); %Initialising positions in loop
x(:,1)=x_i;
y=zeros(N,N_time);
y(:,1)=y_i;

T=[Ti]; %Initialising temperature of ensemble

figure(1) %plotting the initial positions of the particles
plot(x_i, y_i,'.')
axis([0 1 0 1])
f = getframe;
[im,map] = rgb2ind(f.cdata,256,'nodither');
im(1,1,1,20) = 0;
for ring=2:N_time
x(:,ring)=x(:,ring-1)+vx*tres;
y(:,ring)=y(:,ring-1)+vy*tres;

xtemp=x(:,ring);
ytemp=y(:,ring);

ind=find(xtemp>L);
vx(ind)=-abs(vx(ind));
xtemp(ind)=L;

ind=find(xtemp<0);
vx(ind)=abs(vx(ind));
xtemp(ind)=0;
x(:,ring)=xtemp;

ind=find(ytemp>L);
vy(ind)=-abs(vy(ind));
ytemp(ind)=L;

ind=find(ytemp<0);
vy(ind)=abs(vy(ind));
ytemp(ind)=0;
y(:,ring)=ytemp;

T=[T, m*mean(vx.^2+vy.^2)/k/2];

for ring2=1:N %applying the interparticle forces. here, I'm taking a potential of the form V = a
^2 +
^4
vxtemp=vx+C_repulsive./((x(:,ring)-x(ring2,ring)).^2+(y(:,ring)-y(ring2,ring)).^2).^(3/2).*(x(:,ring)-x(ring2,ring))+C_attractive./((x(:,ring)-x(ring2,ring)).^2+(y(:,ring)-y(ring2,ring)).^2).^(5/2).*(x(:,ring)-x(ring2,ring)); %This calculates all x-component velocity shifts from particle "ring2" on other particles simultaneously including a spurious (infinite!) shift due to that particle interacting with itself (which obviously doesn;t actually happen). I fix this self-interaction issue below.
vytemp=vy+C_repulsive./((x(:,ring)-x(ring2,ring)).^2+(y(:,ring)-y(ring2,ring)).^2).^(3/2).*(y(:,ring)-y(ring2,ring))+C_attractive./((x(:,ring)-x(ring2,ring)).^2+(y(:,ring)-y(ring2,ring)).^2).^(5/2).*(y(:,ring)-y(ring2,ring));
ind=isfinite(vxtemp); %finding the infinity due to the spurious interaction of the particle with itself ("ind" is a set of indices of the non-infinite values).
vx(ind)=vxtemp(ind); %only updating the non-infinite valued velocities (i.e. all those except the self-interaction)
ind=isfinite(vytemp);
vy(ind)=vytemp(ind);

end

vx=gamma*vx; %damping
vy=gamma*vy;

if round(ring/50)==ring/50 %updating the particle positions only every 50th cycle of the loop
plot(x(:,ring), y(:,ring),'.');
axis([0 1 0 1])
f = getframe;
im(:,:,1,ring/50) = rgb2ind(f.cdata,map,'nodither'); %saving the time steps of the plot in a movie
end

ing/N_time; %counting up to one so we know how far through the loop is
end
imwrite(im,map,'test2.gif','DelayTime',0,'LoopCount',inf) %saving the movie as a gif

time=[1:N_time]*tres; %creating time axis for figure 2

figure(2)
plot(time,T)
ylim([-100,300])
ylabel('Temperature')
xlabel('Time')
Output

The temperature vs time graph is best representation to see the phase change. The given MATLAB code is mainly used to show the cooling process and impact of the cooling on the motion of the particales. Due to cooling the velocity of the particales is also decreased and due to this there is also decreased in the temperature. One of the best way to see the phase changes by using the behaviour of the temperature with respect to time. We can see the during initial time temperature was start decreasing rapidly but after some time there is not much changes in the temperature with respect to time. In the last temperature is almost constant with respect to time. This is best way to see the phase change during cooling. When temperature remains constant during cooling then there is meaning that phase is changing. During the cooling there is only one time when temperature decreases and then remain constant. During the decreasing of the temperature there is only one phase, If we start with gas, then during the decreasing of the temperature there is only one phase exist. But when temperature become constant. Then 2 phase exist during that interval.

Density is the order of parameter which we can use for the distinguish between two phases. there is in a certain volume (space). For this liquid-gas transition, density plays the role of the order parameter - basically, some measurement which is different in the two phases and so can be used to tell them apart. One is denser than the other. – Molecules in a liquid are spaced closer together than in a gas, so the density of a liquid is greater than that of a gas. There are two processes at work here: one is the natural attraction that molecules feel toward one another, and the random thermal motion of the molecules (fluid molecules bounce around - the amount of bouncing is what temperature is about!). When thermal motion is more important (at higher temperatures), molecules bounce all over the place rather than sticking together, and so they take up more space. At lower temperatures, the attraction between molecules is more important, and so molecules stay close together.
c
Matlab_thermo_11062020/archive-5co5yn1p/__MACOSX/._basic_crystallisation_model_assignment3.m
Matlab_thermo_11062020/archive-5co5yn1p/__MACOSX/._basic_ideal_gas_model_assignment_2(1).m
Matlab_thermo_11062020/archive-5co5yn1p/__MACOSX/._task.png
Matlab_thermo_11062020/archive-5co5yn1p/__MACOSX/Lectures/._Lecture 11-12 R(slides).pdf
Matlab_thermo_11062020/archive-5co5yn1p/__MACOSX/Lectures/._Lecture 1-2 (R).pdf
Matlab_thermo_11062020/archive-5co5yn1p/__MACOSX/Lectures/._lecture 3-4-r.pdf
Matlab_thermo_11062020/archive-5co5yn1p/__MACOSX/Lectures/._lecture 5-6(1).pdf
Matlab_thermo_11062020/archive-5co5yn1p/__MACOSX/Lectures/._lecture 7-8 (slides).pdf
Matlab_thermo_11062020/archive-5co5yn1p/__MACOSX/Lectures/._Lecture 9-10 (Slides).pdf
Matlab_thermo_11062020/archive-5co5yn1p
asic_crystallisation_model_assignment3.m
clc;
clear all;
%Basic simulation of trajectories of particles in a two
%dimensional interacting gas inlcuding energy dissipation leading to
%cooling
%Saves postions and temperature of particles as a function of time, and a
%movie showing the trajectories as they crystallise
%Written by Warwick Bowen, April 2020. Can be used as a basis for the
%computational question in Assignment 3 of PHYS2020 in 2020.,
clear all;
% V = 1;
Ti=300; %Temperature in Kelvin
k=1.38e-23; %Boltzmann constant
m=1e-26; %Mass in kg
v_rms=sqrt(k*Ti/m); %Mean particle velocity expected
N=100; %number of particles
L=1; %length of cube side in meters
gamma=0.99; %factor of velocity not decayed after one time step (1-gamma) is the amount the veclocity reduces).
C_repulsive=-1e-4; %Coefficient defining magntitude of repulsive force (needs to be negative)
d=L/30; %separation of two particles that gives minimum potential energy (i.e. molecular bond length)
C_attractive=-C_repulsive*d^2; %Coefficent defining magnitude of attractive force, calculated do that the potential minima occurs at d. I calcuated this for a potential of the form V = a
^2 +
^4.
tres=d/v_rms/30; %time step: chosen to be small compared to time for average velocity to make two particles in ground state lattice overlap to avoid large e
oneous jumps in potential energy
N_time=16000; %number of time steps to calculate
tmax=tres*N_time; %duration of simulation
%particle position: randomly locating particles
x_i=L*rand(N,1);
y_i=L*rand(N,1);
%particle velocity: randomly selecting their vwlocity components from a Gaussian
%distribution
vx_i=v_rms*randn(N,1);
vy_i=v_rms*randn(N,1);
% Order paremetr density is used as order paremete
% When gas changes from gasous state to liquid it becomes dense
vx=vx_i; %Initialising velocities in loop
vy=vy_i;
x=zeros(N,N_time); %Initialising positions in loop
x(:,1)=x_i;
y=zeros(N,N_time);
y(:,1)=y_i;
% a = 0.0141;
T=[Ti]; %Initialising temperature of ensemble
% P = (N*k*T/1);
% rho = P/T;
figure(1) %plotting the initial positions of the particles
plot(x_i, y_i,'.')
axis([0 1 0 1])
f = getframe;
[im,map] = rgb2ind(f.cdata,256,'nodither');
im(1,1,1,20) = 0;
for ring=2:N_time
x(:,ring)=x(:,ring-1)+vx*tres;
y(:,ring)=y(:,ring-1)+vy*tres;

xtemp=x(:,ring);
ytemp=y(:,ring);

ind=find(xtemp>L);
vx(ind)=-abs(vx(ind));
xtemp(ind)=L;

ind=find(xtemp<0);
vx(ind)=abs(vx(ind));
xtemp(ind)=0;
x(:,ring)=xtemp;

ind=find(ytemp>L);
vy(ind)=-abs(vy(ind));
ytemp(ind)=L;

ind=find(ytemp<0);
vy(ind)=abs(vy(ind));
ytemp(ind)=0;
y(:,ring)=ytemp;

T =[T, m*mean(vx.^2+vy.^2)/k/2];

% P = [P,(N*k*T/((1-N*0.0391))-(a*N*N/(V*V)))];


for ring2=1:N %applying the interparticle forces. here, I'm taking a potential of the form V = a
^2 +
^4
vxtemp=vx+C_repulsive./((x(:,ring)-x(ring2,ring)).^2+(y(:,ring)-y(ring2,ring)).^2).^(3/2).*(x(:,ring)-x(ring2,ring))+C_attractive./((x(:,ring)-x(ring2,ring)).^2+(y(:,ring)-y(ring2,ring)).^2).^(5/2).*(x(:,ring)-x(ring2,ring)); %This calculates all x-component velocity shifts from particle "ring2" on other particles simultaneously including a spurious (infinite!) shift due to that particle interacting with itself (which obviously doesn;t actually happen). I fix this self-interaction issue below.
vytemp=vy+C_repulsive./((x(:,ring)-x(ring2,ring)).^2+(y(:,ring)-y(ring2,ring)).^2).^(3/2).*(y(:,ring)-y(ring2,ring))+C_attractive./((x(:,ring)-x(ring2,ring)).^2+(y(:,ring)-y(ring2,ring)).^2).^(5/2).*(y(:,ring)-y(ring2,ring));
ind=isfinite(vxtemp); %finding the infinity due to the spurious interaction of the particle with itself ("ind" is a set of indices of the non-infinite values).
vx(ind)=vxtemp(ind); %only updating the non-infinite valued velocities (i.e. all those except the self-interaction)
ind=isfinite(vytemp);
vy(ind)=vytemp(ind);

end



vx=gamma*vx; %damping
vy=gamma*vy;

if round(ring/50)==ring/50 %updating the particle positions only every 50th cycle of the loop
plot(x(:,ring), y(:,ring),'.');
axis([0 1 0 1])
f = getframe;
im(:,:,1,ring/50) = rgb2ind(f.cdata,map,'nodither'); %saving the time steps of the plot in a movie
end

ing/N_time; %counting up to one so we know how far through the loop is
end

imwrite(im,map,'test2.gif','DelayTime',0,'LoopCount',inf) %saving the movie as a gif
time=[1:N_time]*tres; %creating time axis for figure 2;
figure(2)
plot(time,T)% Plot between time vs temperature
ylim([-100,300])
ylabel('Temperature')
xlabel('Time')
Matlab_thermo_11062020/archive-5co5yn1p
asic_ideal_gas_model_assignment.m
%Basic simulation of trajectories of particles (and temperature) in a two
%dimensional
%ideal gas
%Saves postions and temperature of particles as a function of time, and a
%movie showing the trajectories
%Written by Warwick Bowen, April 2020. Can be used as a basis for the
%computational question in Assignment 3 of PHYS2020 in 2020.,
clear all
T_i=300; %Temperature in Kelvin
k=1.38e-23; %Boltzmann constant
m=1e-26; %Mass of a particle in kg
v_rms=sqrt(k*T_i/m); %Mean particle velocity expected
N=1000; %number of particles
L=1; %length of cube side in meters
%A=L^2; %area of cube
Tau=L/v_rms; %typical time for particle to traverse box
tmax=Tau*100; %simulation time. I.e. choosing a simulation time for which each particle collides many times with the walls.
tres=Tau/100; %time step. I.e. choosing a time resolution small compared to the time between collisions with walls
%particle position: randomly locating particles
x_i=rand(N,1);
y_i=rand(N,1);
%particle velocity: randomly selecting their vwlocity components from a Gaussian
%distribution
vx_i=v_rms*randn(N,1);
vy_i=v_rms*randn(N,1);
figure(1) %plotting the initial positions of the particles
plot(x_i, y_i,'.')
axis([0 1 0 1])
f = getframe;
[im,map] = rgb2ind(f.cdata,256,'nodither');
im(1,1,1,20) = 0;
N_time=round(tmax/tres); %number of time steps in simulation, dues to define length of loop
vx=vx_i; %Initialising velocities in loop
vy=vy_i;
x=zeros(N,N_time); %Initialising positions in loop
x(:,1)=x_i;
y=zeros(N,N_time);
y(:,1)=y_i;
T=[]; %Initialising temperature of ensemble (note, without any potential energy this will not change during the simulation)
for ring=2:N_time
x(:,ring)=x(:,ring-1)+vx*tres;
y(:,ring)=y(:,ring-1)+vy*tres;

ind=find(x(:,ring)>L);
vx(ind)=-vx(ind);

ind=find(x(:,ring)<0);
vx(ind)=-vx(ind);

ind=find(y(:,ring)>L);
vy(ind)=-vy(ind);

ind=find(y(:,ring)<0);
vy(ind)=-vy(ind);

T=[T, m*mean(vx.^2+vy.^2)/k/2];

if round(ring/50)==ring/50 %updating the particle positions only every 50th cycle of the loop
plot(x(:,ring), y(:,ring),'.');
axis([0 1 0 1])
f = getframe;
im(:,:,1,ring/50) = rgb2ind(f.cdata,map,'nodither'); %saving the time steps of the plot in a movie
end


end
imwrite(im,map,'particle_trajectories.gif','DelayTime',0,'LoopCount',inf) %saving the movie as a gif
time=[2:N_time]*tres; %creating a time axis for figure 2
figure(2)
plot(time,T)
hold on
plot([min(time) max(time)], [T_i T_i])
hold off
ylabel('Temperature')
xlabel('Time')

Matlab_thermo_11062020/archive-5co5yn1p/Lectures/Lecture 11-12 R(slides).pdf
CRICOS code 00025B
Ebinazar Namdas
The University of Queensland
Brisbane, Australia
Critical points,
continuous phase transitions
superconductor and liquid helium
CRICOS code 00025BCRICOS code 00025B
Main ideas
Critical points are ubiquitous: they occur in liquids, binary
mixtures, magnets, superfluids, superconductors, ….
Universality near critical points: law of co
esponding states
Critical exponents define thermodynamic properties near the
critical point.
Broken symmetry and order parameters
continuous phase transitions
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At the critical point the phase transition is not first orde
At the critical point the
volume and entropy are
continuous
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[Entity Name]
Phase transitions give discontinuities in
thermodynamic functions
First orde
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[Entity Name]
Critical opalescence at the critical point of CO2
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[Entity Name] PVT diagram of pure substance in 3D
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[Entity Name]
T-V diagram for liquid-vapour transition
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Continuous phase transition Reading quiz
Describe two phenomena or characteristics associated with continuous
phase transitions
• no latent heat (or other discontinuities in first-order derivatives of G)
• associated with a critical point
• large fluctuations (e.g. “critical opalescence”) and divergence of
compressibility or analogous quantities
• emergence of new order (“order parameter”, symmetry
eaking)
• more advanced: power-law divergences (critical exponents),
universality, power-law decay of co
elations
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[Entity Name]
Universality & law of co
esponding states for liquid-vapour transition
Reduced temperature vs. reduced density
Independent of chemical
& physical details
Guggenheim (1945)
Critical exponent β
ρ/ρc
ρ/ρc = A(1-T/Tc)β
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[Entity Name]
Reduced temperature vs. density
T/Tc
ρ/ρc
liquid-vapou
mixture liquid
vapou
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[Entity Name]
Phase diagram for partially miscible liquids
Temperature vs. Composition Mixed vs unmixed phases
Immiscible region: unmixed combination of an
A-rich phase (ax) and a B-rich phase (xb)
G = xGA + (1-x)GB + ∆Hmixing - T∆Smixing
Critical point
17
Immiscible liquids
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[Entity Name]
Universality and law of co
esponding states
ecall: van der Waals gas
educed variables: p = P/Pc, v = V/Vc, t = T/Tc
P =
NkT
V �Nb �
aN2
V 2
p =
8t
3v � 1 �
3
v2
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[Entity Name]
Critical points in magnetic phase diagrams Magnetic field vs. temperature
Magnetic Gi
s free energy G(T,H) = U - TS - MH
H is external magnetic field, M is magnetisation
dG= - S dT - μ
0
M dH
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[Entity Name]
Critical point for a fe
omagnetic phase
transition in zero magnetic field
Demos 19-23 & 19-24
Magnetisation when H=0
M(T) = 0 T>Tc
≠ 0 TCRICOS code 00025BCRICOS code 00025B
Fe
omagnet-paramagnet transition in Nickel Tc=627 K
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[Entity Name]
Fe
omagnet-paramagnet transition in Dysprosium, Tc=85K
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Phase diagram of a superconductor:
magnetic field vs. temperature
Tc is the transition temperature. Hc is
the upper critical field.
For H > 0, transition is first-orde
H=0, T=Tc is a critical point
Hc
H
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[Entity Name]
Superconductor critical point
Specific heat capacity vs.
temperature
Appears to be discontinuous
at Tc
Phillips, Phys. Rev.
114, 676 (1959)
ΔCp
Tc
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[Entity Name] Superfluid helium
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Superfluids
Have zero viscosity
Ca
y no entropy
Infinite thermal conductivity
In contrast to normal fluids that have finite viscosity, entropy, and
thermal conductivity.
Search “superfluid helium” on Youtube
Can be also be viewed as Bose-Einstein condensates (BEC’s)
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Superfluid transition in 4He is a continuous transition
Heat capacity vs.
temperature
No discontinuity,
so not a second-orde
transition
Enthalpy and entropy are
continuous at Tλ
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[Entity Name]
Specific heat capacity near the superfluid transtion: 4He vs. 3He
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[Entity Name] Order parameters and
oken symmetry
An order parameter is a thermodynamic state variable which is non-zero
in a new phase of matter.
Example: the magnetization M in a fe
omagnet
eaks rotational
symmetry
Order parameters are usually associated with a
oken symmetry, i.e., the
new phase has less symmetry than the higher temperature phase.
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Simple example of
oken symmetry
Find shortest network of roads joining fou
cities at the corners of a square
* *
* *
Square has four-fold rotational symmetry
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[Entity Name] Soap films illustrate
oken symmetry
Film area tends to an
extremum
Demos 13-24 & 13-
22
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[Entity Name] Order parameters
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[Entity Name]
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[Entity Name]
Universality & law of co
esponding states for liquid-vapour transition
Reduced temperature vs. reduced density
Independent of chemical
& physical details
Guggenheim (1945)
Critical exponent β
ρ/ρc
ρ/ρc = A(1-T/Tc)β
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[Entity Name]
Key ideas
The order parameter is a thermodynamic state variable that
quantifies the amount of symmetry
eaking in an ordered
phase.
Landau theory is the simplest way to describe a continuous
transition.
It predicts universal critical exponents: but the values differ from
experimental ones.
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Nobel prizes in physics related to
oken symmetry & order
parameters
Neel (1970): antife
omagnetism
Josephson (1973): tunneling between superconductors
Wilson (1982): renormalisation group
de Gennes (1991): polymers and liquid crystals
Leggett (2002): order parameter superfluid 3He
Ginzburg (2002): Ginzburg-Landau theory of superconductivity
Nambu (2008), Higgs (2013): symmetry
eaking in elementary particle
physics
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[Entity Name]
Nobel prizes in physics related to
oken symmetry & order
parameters
Neel (1970): antife
omagnetism
Josephson (1973): tunneling between superconductors
Wilson (1982): renormalisation group
de Gennes (1991): polymers and liquid crystals
Leggett (2002): order parameter superfluid 3He
Ginzburg (2002): Ginzburg-Landau theory of superconductivity
Nambu (2008), Higgs (2013): symmetry
eaking in elementary particle
physics
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[Entity Name]
Order parameters and
oken symmetry
An order parameter is a thermodynamic state variable which
is non-zero in a new phase of matter.
Example: the magnetization M in a fe
omagnet
eaks
otational symmetry
Order parameters are usually associated with a
oken
symmetry, i.e., the new phase has less symmetry than the
higher temperature phase.
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[Entity Name]
Lev Landau (1908-1968)
Nobel prize (1962) for theory of superfluidity
Continuous phase transitions and symmetry
eaking (1937)
Ginzburg-Landau theory ofsuperconductivity
(1950)
Fermi liquid theory (1957)
Founded School in Soviet Union
10 volume Course in Theoretical
Physics (Landau & Lifschitz)
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[Entity Name]
Landau theory of continuous phase transitions
Also called mean-field theory [related to van der Waals theory of
critical point]
Assumptions
Gi
s free energy is an analytic function of the order paramete
G[ ] =
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Derivations
Temperature dependence of
Order paramete
Specific heat capacity
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[Entity Name] Course summary (second half)
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Quantification of chemical Energy: Gi
s Free Energy)
Gi
s free energy is a measure of chemical energy
It is the energy which is available to do work
All chemical systems tend naturally toward states of minimum
Gi
s free energy
G = H - TS
Where:
G = Gi
s Free Energy
H = Enthalpy (heat content)
T = Temperature in Kelvins
S = Entropy (can think of as randomness)
Free Energy is TS !!
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CaChemical Reaction of electrolysis
222 O2
1H OH +®
Calculate: ∆G = ∆H - T∆S
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A battery
The voltage V is related to the change in Gi
s free energy by:
Electrochemistry provides a simple way to accurately measure the change of Gi
s
free energy in chemical reactions
Example: lead-acid cell (car battery)
V = -ΔG/nF
• n = number of electrons transfe
ed
• F = Faraday constant = 96485 C/mol
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[Entity Name] Phase diagram of ca
on dioxide
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Clausius-Clapeyron equation
• Illustrates the power of thermodynamics.
• Applicable to any phase boundary.
• Relates line slope to volume change.
• Measurements at a single pressure can be used to determine how the melting
or boiling temperature varies with pressure.
• Can provide a consistency check on measurements.
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van der Waals isotherms are unstable for TCRICOS code 00025BCRICOS code 00025B
[Entity Name] Entropy of mixing of 2 ideal gases
dU = TdS - PdV
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[Entity Name] Dilute limit of an ideal mixture
x
Gi
s energy of an ideal mixture
G = xG˚A + (1-x)G˚B + NkT[xln(x) + (1-x)ln(1-x)] 

= NAµ˚A + NBµ˚B + kT[NAln(x) + NBln(1-x)]!
But G = NAµA + NBµB!
µA = µ˚A + kTln(x)!
µB = µ˚B + kTln(1-x)!
If the mixture is mostly A, (x ≈ 1; 1-x ≈ NB/NA):!
µA ≈ µ˚A - kTNB/NA!
µB ≈ µ˚B + kTln(NB/NA)
ln(x) ≈ x-1
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[Entity Name] Dilute solutions
Solution= mixture with solute B (minority species)
+ solvent A (majority)
Dilute means NA
NB
Chemical potentials
dilute solution
mostly solvent, with a tiny bit of solute!
solute: µB ≈ µ˚B + kTln(NB/NA)

= µ˚ + kT ln(mB)







!
!
solvent: µA ≈ µ˚A - kTNB/NA
steep: a little impurity
has a big effect
molality: 

moles of solute !
per kg of solvent
edefine
eference state: 

1 mole of solute
per kg of solvent
dilute solution
mostly solvent, with a tiny bit of solute!
solute: µB ≈ µ˚B + kTln(NB/NA)

= µ˚ + kT ln(mB)







!
!
solvent: µA ≈ µ˚A - kTNB/NA
steep: a little impurity
has a big effect
molality: 

moles of solute !
per kg of solvent
edefine
eference state: 

1 mole of solute
per kg of solvent
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Osmotic pressure
van’t Hoff equation
nB/V = concentration of
solute B in an ideal solution in
solvent A
(moles B/volume of A)
P2 – P1 = (nB/V) R T
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[Entity Name] Osmotic pressure
Semi permeable mem
ane
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Solutes change freezing and boiling points of solutions
Boiling point increases
Freezing point decreases
The solute reduces the chemical potential
(Gi
s free energy) of the solution
Boiling and Freezing Points
adding a small amount of solute to a solvent will lower
the Gi
s free energy of the liquid!
liquid becomes more thermodynamically favourable!
oiling point raised!
freezing point lowered
G
T
TbTf
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[Entity Name] Continuous phase transition
Describe two phenomena or characteristics associated with continuous
phase transitions
• no latent heat (or other discontinuities in first-order derivatives of G)
• associated with a critical point
• large fluctuations (e.g. “critical opalescence”) and divergence of
compressibility or analogous quantities
• emergence of new order (“order parameter”, symmetry
eaking)
• more advanced: power-law divergences (critical exponents),
universality, power-law decay of co
elations
Matlab_thermo_11062020/archive-5co5yn1p/Lectures/Lecture 1-2 (R).pdf
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Ebinazar Namdas
The University of Queensland
Brisbane, Australia
Condensed Matter Physics
Chemical Reactions and
Gi
s Free Energy
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[Entity Name] Condensed Matter Physics
Solids and liquids
Superconductors and Superfluids
Magnets
Semiconductor (organic and inorganic)
Polymers
Condensed matter physics involves most fields of physics:
o Thermodynamics
o Statistical Mechanics
o Quantum Mechanics
o Electromagnetism and Optics
o Quantum Field Theory
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[Entity Name] Condensed matter physics
has strong overlaps with:
o Chemistry and Chemical Engineering
o Materials Science and Engineering
o Biophysics
o Atom optics (Bose-Einstein Condensates)
o Quantum field theory and elementary particle physics
o Presents some of the most challenging problems in theoretical and experimental physics
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o In the past 30 years the Nobel Prize in Physics has been awarded 13 times
for work on condensed matter.
o Since 1998 seven condensed matter physicists have received the Nobel
Prize in Chemistry!
o Of the 144 physicists who were most highly cited in 2014 for papers in the
period 2002-2012 (ISI highlycited.com), more than one half are condensed
matter physicists.
Condensed matter physics
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Condensed Matter Physics underpins:
Modern electronics, computer, TV/mobile screen, lasers, advanced materials and biological
systems
Condensed matter physics
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Photovoltaic
cells
Sensors
Integrated
circuits
New Generation of Organic Opto-electronics
OPD
Organic Lasers
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[Entity Name] Cyclic Process
• Previous chapters applied to cyclic process
• A cyclic process is a sequence of processes that leaves the system
in the same state in which it started
• Operation of heat engine and refrigerators– Energy and entropy are
unchanged over the long term
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• Chemical Reactions – not cyclic
• Laws of thermodynamics to chemical Reactions
• What is chemical Reactions ?
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[Entity Name] Basics of Chemical Reactions
1. Color changes
2. A solid is formed (precipitation)
3. Bu
les form (gas)
4. Change of temperature (Heat is produced/ abso
ed)
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A + B ® C + D
Reactants Products
Chemical Reactions
Chemical Equation
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[Entity Name] Chemical Reactions
Products contain the same atoms as reactants.
Rea
angement of atoms
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Balance a chemical equation
• Law of conservation of mass
• Atoms are neither destroyed nor created.
• They shift from one substance to another.
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Fe(s) + O2(g) à Fe2O3(s)
1 2 2 3
x2 1.5
2 Fe(s) + 1.5 O2(g) à Fe2O3(s)
2 x {2 Fe(s) + 1.5 x O2(g)} à 2 x { Fe2O3(s) }
4 Fe(s) + 3 O2(g) à 2 Fe2O3(s)
Examples
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Chemical Reactions– Bond Energy
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DH reaction = DH products - DH reactants
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Energy must be SUPPLIED to
eak chemical bonds of
eactants
Energy is RELEASED when new
chemical bonds are made in the
products
A reaction is EXOTHERMIC if more energy is RELEASED than SUPPLIED. If more
energy is SUPPLIED than is RELEASED then the reaction is ENDOTHERMIC
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Based on the type of energy (heat) change involved, chemical
eactions are classified as either exothermic or endothermic.
– Exothermic: energy is released
• Exo- = “exit”
• Burning of gasoline
– Endothermic: energy is abso
ed
• Endo- = “into”
• Cooking / ba
ecue
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Energy
abso
e
d in
eaction
Activation
Energy
Energy used
in bond
eaking
Endothermic – more energy is taken in to
eak the bonds in the reactants than released by
the bonds being formed in the products. Therefore, energy is abso
ed.
Energy
eleased in
ond making
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Energy
elease
d in
ond
making
Activation
Energy
Energy used in
ond
eaking
Exothermic – More energy is released when the products where formed than energy was
used to
eak bonds in the reactants. Therefore, a net release of energy.
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Quantification of chemical Energy: Gi
s Free Energy)
Gi
s free energy is a measure of chemical energy
It is the energy which is available to do work
All chemical systems tend naturally toward states of minimum
Gi
s free energy
G = H - TS
Where:
G = Gi
s Free Energy
H = Enthalpy (heat content)
T = Temperature in Kelvins
S = Entropy (can think of as randomness)
Free Energy is TS !!
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[Entity Name]
Gi
s Free Energy
Josiah Willard Gi
s (1839-1903)
Gi
s free energy, originally called available energy, was developed in
1873, in a footnote, Gi
s defined what he called the “available
energy” of a body as such:
“ The greatest amount of mechanical work which can be obtained from a given
quantity of a certain substance in a given initial state, without increasing its total
volume or allowing heat to pass to or from external bodies, except such as at the
close of the processes are left in their initial condition.”
Prof at Yale University
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key ideas
Gi
s free energy, G = U + PV – TS, is an extremely useful
thermodynamic potential.
In an electrochemical cell (battery) ΔG is related to the voltage of the
attery
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What is the minimum energy cost to make a baby out of nothing?
The system can spontaneously abso
heat from the environment to produce its entropy
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[Entity Name] Thermodynamic potentials
U (or E) F (or A)
= U - TS
(Constant Volume)
H
= U + PV
G
= U + PV - TS
= H-TS
- TS
+PV
Constant V Constant P
Constant S U H
Constant T F G
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DG reaction = DG products - DG reactants
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Gi
s Free Energy, G
∆G = ∆H - T∆S
Determine ∆Hrxn and ∆Srxn and use Gi
s equation.
∆Grxn = S ∆G (products) - S ∆G (reactants)
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[Entity Name] Gi
s Free Energy
If DG is a negative number, the reaction will be spontaneous.
If DG is a positive number, the reaction will be nonspontaneous.
If DG is a zero, the reaction is at equili
ium.
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A rock naturally rolls down a hill – spontaneous
It must be pushed back up – nonspontaneous
A hot object naturally cools – spontaneous
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Gi
s Free energy -- Spontaneity
• The tendency for a process to advance to equili
ium without external influence
• Something that happens naturally is spontaneous
• Any process will be spontaneous in one direction
• The reverse is non-spontaneous
• If work needs to be done, it is not spontaneous
• A rock naturally rolls down a hill - spontaneous
• It must be pushed back up - nonspontaneous
• A hot object naturally cools - spontaneous
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[Entity Name]
Some examples of spontaneous processes
eactions:
1. The co
osion of iron is spontaneous, but the reverse is not:
4Fe(s) + 3 O2(g) ® 2Fe2O3(s)
2. Ice melts spontaneously at room temperature, but water will not freeze
at room temperature:
H2O(s) ® H2O(l)
3. Heat spontaneously flows from a hot object to a cold object if the
objects are in contact.
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[Entity Name] Examples
A spontaneous but very slow process
The conversion of ca
on from the diamond to graphite spontaneous process
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[Entity Name] Example -- Electrolysis of water
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CaChemical Reaction of electrolysis
222 O2
1H OH +®
Calculate: ∆G = ∆H - T∆S
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[Entity Name] Hydrogen fuel cell - electrolysis in reverse
Honda FCX Clarity
Hyundai ix35
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[Entity Name] Example -- Fuel Cells
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http:
hyperphysics.phy-
astr.gsu.edu/hbase/thermo/electrol.html
• H2 and O2 be combined in a fuel cell to produce electrical energy.
• H2 and O2 is continuously supplied.
• Calculate Enthalpy change
• Entropy change =
• Electric Energy output
• Efficiency
Fuel cell
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Hydrogen Fuel Cell
What voltage does a single cell
produce?
Does the use of a hydrogen fuel cell
avoid ca
on emissions?
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A battery (Nobel prize Li ion battery – 2019)
The voltage V is related to the change in Gi
s free energy by:
Electrochemistry provides a simple way to accurately measure the change of Gi
s
free energy in chemical reactions
Example: lead-acid cell (car battery)
V = -ΔG/nF
• n = number of electrons transfe
ed
• F = Faraday constant = 96485 C/mol
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[Entity Name] Example
What is the standard reaction Gi
s energy for the
combustion of CH4(g) (methane)?
CH4[g] + 2 O2[g] -> CO2[g] + 2 H2O[g]
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Why should “free energy” really be called “costly energy”?
“The free energy is not actually the energy you gain for free; the component TS is the energy that is
gained for free (heat transfe
ed into the system from its environment). The remainder of the work (G
= H-TS) needed to create a system out of nothing must still be supplied, and hence is better called
the 'costly' energy.”
“Because it's how much energy you still need to put into the system, after subtracting the energy that
the environment gives you. The reason that they call it free energy is that when it was
developed, they were thinking of how much energy you would 'free up' when you annihilated an
object.“
“In the analogy of creating something out of the blue used in the textbook, the "free energy" actually
efers to the energy one would need to create the system minus the "free as in free beer" energy one
could extract from the su
oundings (due to constant temperature of su
oundings). So the "free
energy" actually refers to the "costly" energy (the energy one cannot get for free) in this analogy.”
Matlab_thermo_11062020/archive-5co5yn1p/Lectures/lecture 3-4-r.pdf
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Ebinazar Namdas
The University of Queensland
Brisbane, Australia
Gi
s Free Energy -II
Maxwell relations
Thermodynamics of Ru
e
Magnets
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[Entity Name] Basics of Chemical Reactions
1. Color changes
2. A solid is formed (precipitation)
3. Bu
les form (gas)
4. Change of temperature (Heat is produced/ abso
ed)
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Chemical Reactions– Bond Energy
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[Entity Name]
Energy must be SUPPLIED to
eak chemical bonds of
eactants
Energy is RELEASED when new
chemical bonds are made in the
products
A reaction is EXOTHERMIC if more energy is RELEASED than SUPPLIED. If more
energy is SUPPLIED than is RELEASED then the reaction is ENDOTHERMIC
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[Entity Name]
Based on the type of energy (heat) change involved, chemical
eactions are classified as either exothermic or endothermic.
– Exothermic: energy is released
• Exo- = “exit”
• Burning of gasoline
– Endothermic: energy is abso
ed
• Endo- = “into”
• Cooking / ba
ecue
CRICOS code 00025BCRICOS code 00025B
Quantification of chemical Energy: Gi
s Free Energy)
Gi
s free energy is a measure of chemical energy
It is the energy which is available to do work
All chemical systems tend naturally toward states of minimum
Gi
s free energy
G = H - TS
Where:
G = Gi
s Free Energy
H = Enthalpy (heat content)
T = Temperature in Kelvins
S = Entropy (can think of as randomness)
Free Energy is TS !!
CRICOS code 00025BCRICOS code 00025B
[Entity Name]
Gi
s Free Energy
Josiah Willard Gi
s (1839-1903)
Gi
s free energy, originally called available energy, was developed in
1873, in a footnote, Gi
s defined what he called the “available
energy” of a body as such:
“ The greatest amount of mechanical work which can be obtained from a given
quantity of a certain substance in a given initial state, without increasing its total
volume or allowing heat to pass to or from external bodies, except such as at the
close of the processes are left in their initial condition.”
Prof at Yale University
CRICOS code 00025BCRICOS code 00025B
[Entity Name]
CaChemical Reaction of electrolysis
222 O2
1H OH +®
Calculate: ∆G = ∆H - T∆S
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[Entity Name] Hydrogen fuel cell - electrolysis in reverse
Honda FCX Clarity
Hyundai ix35
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[Entity Name] Example (Gi
s Energy) - 1
What is the standard reaction Gi
s energy for the combustion of CH4(g) (methane)?
CH4[g] + 2 O2[g] -> CO2[g] + 2 H2O[g]
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attery example
The voltage V is related to the change in Gi
s free energy by:
Electrochemistry provides a simple way to accurately measure the change of Gi
s
free energy in chemical reactions
Example: lead-acid cell (car battery)
V = -ΔG/nF
• n = number of electrons transfe
ed
• F = Faraday constant = 96485 C/mol
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[Entity Name] Example 1 (battery)
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V = -ΔG/nF
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Example 2 (lemon battery)
Zn à Zn+2 + 2e-
2H+ + 2 e- à H2
V ~ 1 V
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Thermodynamic identities express the changes in a thermodynamic
potential in terms of its ‘natural’ variables. They can be used to
derive Maxwell relations.
Thermodynamic Identities
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“Natural” Variables for U
What are the “natural” variables for U?
dU = TdS – PdV (thermodynamic identity)
a) p and V
) S and T
c) T and V
d) V and S
e) T and P
f) S and P
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[Entity Name]
“Natural” Variables for H
H= U+PV
dH= dU+PdV+VdP (1)
dU = TdS – PdV (thermodynamic identity) (2)
dH= TdS-pdV+pdV+Vdp
dH=TdS+Vdp
a) p and V
) S and T
c) T and V
d) V and S
e) T and p
f) S and p
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[Entity Name]
F = U - TS
dF = dU - d(TS) = dU-TdS-SdT
dF= -SdT - pdV
T and V are the natural variables for F
dU = TdS – PdV
dF= TdS-PdV-TdS-SdT
“Natural” Variables for F
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[Entity Name]
G = H - TS
dG = dH - d(TS) = VdP – SdT
dH=TdS+VdP
dG = TdS+VdP-TdS-SdT
dG= VdP-SdT
p and T are the natural variables for G
“Natural” Variables for G
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[Entity Name]
Potential Variables Differential
Internal Energy U(S, V) dU = TdS - PdV
Enthalpy H(S, P) TdS+VdP
Helmhotz
Energy
F(T, V) dF = -PdV - SdT
Gi
s Energy G(T,P) dG = VdP - SdT
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[Entity Name]
• Some variables are hard to measure experimentally such as entropy
• Variables like Pressure, Temperature are easy to measure
• Maxwell relations provide a way to exchange variables
Born in Edinburgh, Scotland
Well-known for electromagnetism and field theory
Also known for thermodynamics and kinetic theory of gases
Maxwell relations in Thermodynamic Potentials
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Deriving Maxwell Relations Continued
The second partial derivative of the potentials with respect to the
independent variables does not depend on the order of derivatives
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Deriving Maxwell Relations Continued
T
S
U
V
=÷
ø
ö
ç
è
æ
¶
¶ P
V
U
S
-=÷
ø
ö
ç
è
æ
¶
¶
Take derivative with respect to its natural variables, we can refer back to the original
equation of state and define, in this example, T and P.
dV
V
UdS
S
UdU
SV
÷
ø
ö
ç
è
æ
¶
¶
+÷
ø
ö
ç
è
æ
¶
¶
=
dU = TdS - PdV
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G = H - TS
dG = Vdp - SdT
p and T are the natural variables for G
Maxwell relation:
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÷
ø
ö
ç
è
æ
¶
¶
÷
ø
ö
ç
è
æ
¶
¶
÷
ø
ö
ç
è
æ
¶
¶
÷
ø
ö
ç
è
æ
¶
¶
÷
ø
ö
ç
è
æ
¶
¶
÷
ø
ö
ç
è
æ
¶
¶
÷
ø
ö
ç
è
æ
¶
¶
÷
ø
ö
ç
è
æ
¶
¶
==
-=-=
-=-=
==
P
G
P
H
V
F
V
U
T
G
T
F
S
H
S
U
TS
TS
PV
PV
V
P
S
T
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[Entity Name] Free energy as a force towards equili
ium
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Extensive
-Properties that do depend on the amount of matter present.
Mass, weight, volume, length
Intensive and Extensive Properties
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Intensive
•Properties that do not depend on the amount of the matter present.
-Color , odor, hardness, melting/freezing point, boiling point,
density, malleability, luste
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[Entity Name] Rods, Ru
er & Magnets
Non-PV systems
dU = TdS - YdX
X: generalised displacement (extensive)
Y: generalised force (intensive)
Y X W
fluid -P V -PdV
elastic rod f L fdL
liquid film ɣ A ɣdA
dielectric E PE E∙dPE
magnetic B M B∙dM
ɣ:surface tension
f: tension
electric dipoles
magnetic dipoles
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u
er band heat engine
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[Entity Name] Example (ru
er)
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[Entity Name] Thermodynamics of magnetic materials
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[Entity Name] Paramagnetism
dU = TdS + B·dM
dG = -SdT - M·dB
new Maxwell relation:
Define: magnetic susceptibility:
Curie’s Law:
M: total magnetic moment
B: magnetic field
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[Entity Name] Chemical Potential
Potential Variables Differential
Gi
s Energy G(T,P, N ?) dG = VdP - SdT
• Chemical potential is a energy (Joule/mole)
• Gi
s Free Energy /mole – Energy required to add a particle into the
system
• It measure how far we have to go to come to equili
ium.
• Direction of mass transfer from high chemical potential to low chemical
potential (imagine heat transfer / charge transfer due to temp/ potential
gradient
• Chemical potential is similar to gravitational potential energy
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[Entity Name]Chemical potential is the Gi
s free energy per particle
dG = -SdT + Vdp + µdN -
G = Nµ (only because we can add particles without changing T or P)
i.e. chemical potential is Gi
s energy per particle
For more than one kind of particle:
G = N1µ1 + N2µ2 + ...
ut! µ1, µ2, ... depend on the composition
intensive
Matlab_thermo_11062020/archive-5co5yn1p/Lectures/lecture 5-6(1).pdf
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Ebinazar Namdas
The University of Queensland
Brisbane, Australia
Maxwell relations
Phase transition
Clausius-Clapeyron equation and
Van-der Walls Forces
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[Entity Name]
• Some variables are hard to measure experimentally such as entropy
• Variables like Pressure, Temperature are easy to measure
• Maxwell relations provide a way to exchange variables
Born in Edinburgh, Scotland
Well-known for electromagnetism and field theory
Also known for thermodynamics and kinetic theory of gases
Maxwell relations in Thermodynamic Potentials
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Deriving Maxwell Relations Continued
The second partial derivative of the potentials with respect to the
independent variables does not depend on the order of derivatives
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Deriving Maxwell Relations Continued
T
S
U
V
=÷
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¶ P
V
U
S
-=÷
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Take derivative with respect to its natural variables, we can refer back to the original
equation of state and define, in this example, T and P.
dV
V
UdS
S
UdU
SV
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æ
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+÷
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=
dU = TdS - PdV
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G = H - TS
dG = Vdp - SdT
p and T are the natural variables for G
Maxwell relation:
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[Entity Name]
÷
ø
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è
æ
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¶
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==
-=-=
-=-=
==
P
G
P
H
V
F
V
U
T
G
T
F
S
H
S
U
TS
TS
PV
PV
V
P
S
T
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[Entity Name]
CRICOS code 00025BCRICOS code 00025B
[Entity Name] Chemical Potential
Potential Variables Differential
Gi
s Energy G(T,P, N ?) dG = VdP - SdT
• Chemical potential is a energy (Joule/mole)
• Gi
s Free Energy /mole – Energy required to add a particle into the
system
• It measure how far we have to go to come to equili
ium.
• Direction of mass transfer from high chemical potential to low chemical
potential (imagine heat transfer / charge transfer due to temp/ potential
gradient
• Chemical potential is similar to gravitational potential energy
CRICOS code 00025BCRICOS code 00025B
[Entity Name]Chemical potential is the Gi
s free energy per particle
dG = -SdT + Vdp + µdN -
G = Nµ (only because we can add particles without changing T or P)
i.e. chemical potential is Gi
s energy per particle
For more than one kind of particle:
G = N1µ1 + N2µ2 + ...
ut! µ1, µ2, ... depend on the composition
intensive
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[Entity Name] Key ideas: phase diagrams
•There are many possible phases of matter!
•A phase diagram tells us which phase is the most
thermodynamically stable at a given set of state variables
(e.g., pressure and temperature).
•If two distinct phases co-exist at a given pressure and
temperature then they must have the same Gi
s free
energy.
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[Entity Name] Phase change
Gas
Solid Liquid
Melting
Sublim
ation
Ev
ap
o
at
ion

Freezing
Deposition
Co
nd
en
sa
tio
n
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Sublimation/ deposition of ca
on dioxide (dry ice)
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[Entity Name] Phase diagram of a pure substance
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e.g., at 1 atm and -5 degrees C,
water must be solid
Phase diagram of H2O
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Phase change of wate
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Boiling point of H2O at mount Everest is ?
1. 100 C
2. Less than 100
3. Higher than 100
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[Entity Name] Triplet point of water
Courtesy University of California, USA
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Regelation of ice
On the solid-liquid boundary dP/dT<0
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[Entity Name] Phase diagram of ca
on dioxide
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Liquid-vapour boundary disappears at the critical point
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Critical point of ca
on dioxide
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Ca
on
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Phase diagram of ca
on
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Clicker Quiz
Graphite is more stable than diamond at room temperature and pressure because
1. It is more compressible
2. It has a larger volume per mole
3. It has a larger entropy
4. It has a lower Gi
s free energy
5. It has stronger chemical bonding
6. Don’t know
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[Entity Name]
At what pressure will graphite be converted to
diamond?
Schroeder, pp. 170-171
Graphite Diamond
ΔfG (kJ/mol) 0 +2.90
Vs (cm3/mol) 5.30 3.42
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[Entity Name]
Gi
s free energy vs. pressure
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[Entity Name] Superconductivity Discovery
http:
hyperphysics.phy-astr.gsu.edu/hbase/Solids/scdis.html#c1
Boiling point of liquid Helium = 4.22 K
Superconductivity in Hg = 4.2 K
Heike Kamerlingh Onnes
Resistivity R=0 below TC; (R<10-23 W×cm, 1018 times smaller than for Cu)
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Superconductors
A new phase of matter.
A macroscopic quantum state!
Conduct electricity with exactly zero electrical resistivity.
Expel magnetic flux (Meissner effect).
Used to levitate magnets
Quantumlevitation.org
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Nobel Prizes in Physics for discoveries about superconductivity
2002 – A
ikosov, Ginzburg
1987 – Bednorz, Mulle
1973 – Giaver, Josephson
1972 – Bardeen, Cooper, Schrieffe
1913 – Kamerlingh Onnes
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Effect of Magnetic Field
Critical magnetic field (HC) –
Minimum magnetic field required to
destroy the superconducting property
at any temperature
H0 – Critical field at 0K
T - Temperature below TC
TC - Transition Temperature
Superconducting
Normal
T (K) TC
H0
HC
2
0 1C
C
TH H
T
é ùæ ö
ê ú= -ç ÷
ê úè øë û
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[Entity Name] Meissner effect
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Pressure dependence
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Key ideas
Clausius-Clapeyron relation describes the slope of the
oundary between two phases in a phase diagram.
At a first-order phase transition there are discontinuities
in the first-order derivatives of the Gi
s free energy.
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Application to
Liquid-solid phase boundary
What determines the sign of the slope?
Liquid-vapour boundary
Why is the magnitude of the slope much less than that
for the solid-liquid boundary?
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Clausius-Clapeyron equation
• Illustrates the power of thermodynamics.
• Applicable to any phase boundary.
• Relates line slope to volume change.
• Measurements at a single pressure can be used to determine how the melting
or boiling temperature varies with pressure.
• Can provide a consistency check on measurements.
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[Entity Name] Slope of phase boundary
Gi
s free energy
(chemical potential)
of two phases is equal
on the phase boundary
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Derivation of Clapeyron equation
Schroeder, pp. 172-173
Starts with dG=-SdT+VdP
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Phase diagram
e.g., at 1 atm and -5 degrees C,
water must be solid
For H2O Typical Phase diagram
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Ice bom
dP/dT <0 for solid-liquid transition in
wate
Vsolid > Vliquid
Demo 15-15
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Ice bom
Source: Royal Society Of Chemistry, London
https:
www.youtube.com/channel/UCRaqrYgbZAdqCl-_tpG150Q
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First-order phase
transitions involve discontinuities in
the first-order derivatives of the
Gi
s free energy.
Entropy of
gas
liquid
solid
Why?
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Review: First-order phase transitions
Clausius-Clapeyron relation describes the slope of the
oundary between two phases in a phase diagram
At a first-order phase transition there are discontinuities in
the first-order derivatives of the Gi
s free energy.
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Continuity of gas and liquid
Phase transition (real gas) Ideal gas (PV=nRT)
van der Waals model
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Various Bonds
Covalent Bond (100 -500 kJ/mole) -- Si, C
Ionic Bond (100- 500 kJ/mole) -- NaCl, TiO2
Metallic Bond (80 -600 kJ/mole) -- metals
Dipole-dipole (40 kJ/mole) -- molecules
Hydrogen bond (30 kJ/mole)– water
Van der Waals bond/forces (0.1- 20 kJ/mole) -- inert gas
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Various Bond
Covalent bond
Ionic bond (NaCl)
Hydrogen bonding
Dipole-dipole interaction
Diamond
Graphite
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Bonding between nonpolar atoms/molecules :van der Waals forces
electrons in atoms/molecules are moving at high speeds in o
itals
In a given time more electrons to be on one side of an atom/molecule
a dipole forms when one side is slightly negative and other slightly positive
a dipole in one atom/molecule can then induce a dipole in a neighboring molecule
Van der Waals attractions work at very close range
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Gecko.
Geckos can walk to walls due to intermolecular interactions called van
der Waals forces.
The fine hairs form many contacts with any surface. Due to van der
Waals forces geckos can hang to walls
PNAS April 16, 2013 110 (16) 6340-6345;
https:
doi.org/10.1073/pnas.1219317110
https:
doi.org/10.1073/pnas.1219317110
Matlab_thermo_11062020/archive-5co5yn1p/Lectures/lecture 7-8 (slides).pdf
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Ebinazar Namdas
The University of Queensland
Brisbane, Australia
van der Waals equation of state
liquid-vapour transition
Phase diagrams of mixtures
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[Entity Name] Phase change
Gas
Solid Liquid
Melting
Sublim
ation
Ev
ap
o
at
ion

Freezing
Deposition
Co
nd
en
sa
tio
n
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Boiling point of H2O at mount Everest is ?
1. 100 C
2. Less than 100
3. Higher than 100
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[Entity Name] Phase diagram of ca
on dioxide
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Liquid-vapour boundary disappears at the critical point
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Critical point of ca
on dioxide
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Critical pressures and temperatures for different
substances vary by two orders of magnitude
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First-order phase
transitions involve discontinuities in
the first-order derivatives of the
Gi
s free energy.
Entropy of
gas
liquid
solid
Why?
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Phase transitions give discontinuities in
thermodynamic functions
First orde
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Main ideas
Derivatives of free energy are continuous at the critical point.
van der Waals equation of state can describe critical point of the
liquid-vapour transition.
With the Maxwell equal area construction vdW can describe
liquid-vapour mixtures.
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