MSE/ECE 310
Electrical
Prop of Matls
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MSE/ECE 310
Electrical
Prop of Matls
Knowlton
Thermodynamics: A Brief Introduction
Thermodynamics: A Brief Introduction
‰ Brief review or introduction to Classical
Thermodynamics…
‰ Hopefully you remember this equation from chemistry.
The Gibbs Free Energy (G) as related to enthalpy (H)
and entropy (S) and temperature (T).
‰ What is our approach to study thermodynamics?
‰ Study a system of interest through the variables that:
9 mathematically describe the system
9 Are relatively easy to measure by experiments.
ΔG = ΔH − T ΔS
‰ How is the system of interest studied?
‰ We perturb the system and see what outcome results.
‰ One way of doing this is the following:
‰ We are going to expand upon this a bit.
9 Determine potentially how a particular system depends on
energy.
9 This can be done with mathematics.
9 That is, write an equation in terms of energy.
9 Then, mathematically perturb the energy of the system by
changing one of the variables on which the system energy
depends.
9 Examine how the equation of energy changes when a
variable is changed.
‰ Question: Why study thermodynamics?
‰ One Answer: Thermodynamics allows us to describe
changes in a system from an initial state to a final state.
‰ Applications: primarily kinetics
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Phase equilibria
Defect equilibrium
Phase transformations
Heat transport
Mass transport
Growth of materials
Deposition of materials
Etching of materials
Many more …
‰ If this is done by keeping in mind what variables are
easiest to measure during an actual experiment, then
thermodynamics is a very useful tool.
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MSE/ECE 310
Electrical
Prop of Matls
Knowlton
MSE/ECE 310
Electrical
Prop of Matls
Knowlton
Thermodynamics: A Brief Introduction
9 2nd term on right: chemical work done by the system
(hence, the positive sign) where μ & N are the chemical
potential and number of particles in the system.
9 The last term allows for other forms of work performed
on or by the system.
9 The full form of the FEOT is:
‰ Brief review or introduction to Classical
Thermodynamics:
9 Central equation of thermodynamics is the fundamental
equation of thermodynamics (FEOT).
(FEOT)
9 Describes change of internal energy of a system that is:
o initially in one state of equilibrium
o and is perturbed to another state of equilibrium.
9 In its simplest form, FEOT is given by a change in
internal energy of a system as described by the sum of the
change in heat, Q, and work, W:
+ dW
dU = dQ
(1)
dU = TdS − PdVo + μ dN + ∑ YdX
( 2)
o 1st point: FEOT Variables – differentials and their prefactors:
1st Law
• Differential Variables:
» Called extensive or state variables.
» Directly measurable quantities & describe the state of the
system prior to (initial state) and following (final state) the
system change.
» Changes in state variables are path independent (do not depend
on past history).
2nd Law
• Pre-factor variables to the differential variables:
» Called intensive or field variables.
» Measurable indirectly only, by means of the response of an
extensive variable to a perturbed system.
9 Where T and S are the temp and entropy of the system.
9 Similarly, the work differential is given by:
= − PdV + μ dN + YdX
dW
∑
o
( 4)
9 Two important points concerning the FEOT:
9 Heat and work differentials are inexact since they are path
dependent (depend on past history).
9 Heat differential is described mathematically as:
= TdS
dQ
Thermodynamics: A Brief Introduction
• Work performed on or by system = product between an
extensive or state variable and an intensive or field variable.
• This can be seen upon examination of the FEOT.
( 3)
9 1st term on right: hydrostatic work on system (hence, the
negative sign) where P and V are the pressure & volume.
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MSE/ECE 310
Electrical
Prop of Matls
Knowlton
MSE/ECE 310
Electrical
Prop of Matls
Knowlton
Thermodynamics: A Brief Introduction
‰ How does this apply to diffusion?
‰ Examples of Work as a product of extensive and
intensive variables:
9 Two important points concerning the FEOT (cont.):
o 2nd point: follows from the 1st point = intensive or “field
field”
variable.
∑ YdX ( 5)
• The force acting on the system or particles in the system may
be described by the negative gradient of a field variable.
• The force is given by:
Table 2.1: Types of thermodynamic work in differential form
Thermodynamic
Work
Work
Differential
(dW)
thermal
T•dS
hydrostatic
P•dV
Intensive or
Field Variable
F = −∇V
Extensive or State
Variable
T = temperature
S = entropy
P = pressure
V = volume
N = number of
particles
μ = chemical
chemical
μ•dN
strain –
mechanical
electrical
σ •dε
σ = stress
ε = strain
E•dP o
E = electric field
P o = polarization
potential
magnetic
H•dM g
H = magnetic field
M g = magnetization
time
dependent
P w•dt
P w = power
t = time
Momentum
transfer
ma•dr
Mass
acceleration
gravitational
mg•dr
mg = mass &
gravitational
acceleration
&
( 6)
where V is the potential variable.
• Negative gradient of a field variable = field potential.
• Hence: FORCE FIELD!
FIELD
• As shown in previous table, the force (potential
potential) field may
be:
»
»
»
»
»
»
»
r = distance
Thermal field
Chemical potential
Hydrostatic field
Electrical field
Magnetic field
Stress field
Gravitational field
• Although the superposition principle does NOT apply to
field/intensive variables, it does to FORCES.
• Thus, gradients of the potential fields are additive.
r = distance
Note: The intensive and extensive variable
pairs are known as conjugate variables. For
example, T and S are conjugate variables.
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Thermodynamics: A Brief Introduction
o Key Point: Therefore, the forces driving system change,
change
such as diffusion,
diffusion are additive as well.
o This is the field of KINETICS!
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MSE/ECE 310
Electrical
Prop of Matls
Knowlton
MSE/ECE 310
Electrical
Prop of Matls
Knowlton
Thermodynamics: A Brief Introduction
‰ Example: How to use the FEOT to obtain important
thermodynamical relationships via mathematics.
‰ Mathematical Properties of the FEOT
9 FEOT is an exact differential:
9 The chemical potential of atoms A is given by the
thermodynamic relation:
⎛ ∂Ε A ⎞
⎟
⎝ ∂N A ⎠T , P
μA = ⎜
o Definition of an Exact differential:
z = z ( x, y )
(7)
⎛ ∂z ⎞
⎛ ∂z ⎞
dz = ⎜ ⎟ dx + ⎜ ⎟ dy
⎝ ∂x ⎠ y
⎝ ∂y ⎠ x
where ΕA is a free energy of the A atoms in B.
9 Examples of Free Energy, E:
o
o
o
o
⎛ ∂z ⎞
⎛ ∂z ⎞
Let: M = ⎜ ⎟ & N = ⎜ ⎟
⎝ ∂x ⎠ y
⎝ ∂y ⎠ x
Gibbs, G = G(T, P, N)
Helmhotz, F = F(T, V, N)
Enthalpy, H = H(S, P, N)
Omega potential or Grand Potential, Ω = Ω(T, V, μ)
⎛ ∂M ⎞
∂2 z
∂2 z
⎛ ∂N ⎞
Then: ⎜
& ⎜
⎟ =
⎟ =
⎝ ∂x ⎠ y dydx
⎝ ∂y ⎠ x dxdy
9 Since the order of differentiation is immaterial because dz
is an exact differential, then:
Question 1: How are relationships between extensive and
intensive variables, such as μ, obtained in
thermodynamics?
Question 2: If we want to examine some Free Energy w.r.t.
certain variables, how do we do so?
⎛ ∂M ⎞ ⎛ ∂N ⎞
⎜
⎟ =⎜
⎟
⎝ ∂y ⎠ x ⎝ ∂x ⎠ y
9 Apply this to FEOT.
To answer these question, we need to know several
mathematical properties of the FEOT.
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Thermodynamics: A Brief Introduction
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MSE/ECE 310
Electrical
Prop of Matls
Knowlton
MSE/ECE 310
Electrical
Prop of Matls
Knowlton
Thermodynamics: A Brief Introduction
‰ Mathematical Properties of the FEOT
Thermodynamics: A Brief Introduction
‰ Mathematical Properties of the FEOT
9 FEOT:
9 Defining Free Energies using:
dU = TdS − pdV + μ dN (1)
o FEOT &
o Properties of Exact Differentials:
But FEOT is an exact differential.
Thus:
9 Example 1:
⎛ ∂U ⎞
⎛ ∂U ⎞
⎛ ∂U ⎞
dU = ⎜
⎟ dS + ⎜
⎟ dV + ⎜
⎟ dN (2)
⎝ ∂S ⎠V , N
⎝ ∂V ⎠ S , N
⎝ ∂N ⎠ S ,V
So, FEOT is: U = U ( S , V , N ) the Internal FREE ENERGY
o Enthalpy (Free Energy): H=H(S,p,N)
H = U + pV
Differentiate H :
9 Coefficient And Maxwell Relations:
Relations
dH = dU + pdV + Vdp
o Using the property below of an exact differential, we can
obtain relationships between extensive & intensive
variables.
Substitute FEOT: dU = TdS − pdV + μ dN
dH = TdS − pdV + μ dN + pdV + Vdp
9 Coefficient Relations:
∴ dH = TdS + Vdp + μ dN (1)
o Compare equations (1) and (2), we see that:
⎛ ∂U ⎞
T=⎜
⎟
⎝ ∂S ⎠V , N
⎛ ∂U ⎞
p =- ⎜
⎟
⎝ ∂V ⎠ S , N
Thus: H = H ( S , p, N )
⎛ ∂U ⎞
μ=⎜
⎟
⎝ ∂N ⎠ S ,V
Using (2), we can write the differential form of dH :
⎛ ∂H ⎞
⎛ ∂H ⎞
⎛ ∂H ⎞
dH = ⎜
⎟ dp + ⎜
⎟ dS + ⎜
⎟ dN (3)
⎝ ∂S ⎠ p , N
⎝ ∂N ⎠ S , p
⎝ ∂p ⎠ S , N
Coefficient Relations: obtained by comparing (1) and (3)
9 Maxwell Relations: here we just derive one of three.
⎛ ∂M ⎞ ⎛ ∂N ⎞
Using the exact differential relationship ⎜
⎟ =⎜
⎟ :
⎝ ∂y ⎠ x ⎝ ∂x ⎠ y
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⎛ ∂ 2U ⎞
⎛ ∂ 2U ⎞
∂ ⎛ ∂U ⎞
⎛ ∂T ⎞
⎛ ∂p ⎞
⎟ & -⎜ ⎟ = ⎜
⎟
⎜
⎟ =
⎜
⎟ =⎜
∂
∂
∂
∂
∂
∂
V
V
S
V
S
S
⎝
⎠S ,N
⎝
⎠V , N ⎝
⎝ ⎠V , N ⎝ ∂S ∂V ⎠ N
⎠N
⎛ ∂T ⎞
⎛ ∂p ⎞
∴⎜
⎟ = -⎜ ⎟
⎝ ∂V ⎠ S , N
⎝ ∂S ⎠V , N
(2)
⎛ ∂H ⎞
∴ T =⎜
⎟
⎝ ∂S ⎠ p , N
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⎛ ∂H ⎞
V =⎜
⎟
⎝ ∂p ⎠ S , N
⎛ ∂H ⎞
⎟
⎝ ∂N ⎠ S , p
μ =⎜
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MSE/ECE 310
Electrical
Prop of Matls
Knowlton
MSE/ECE 310
Electrical
Prop of Matls
Knowlton
Thermodynamics: A Brief Introduction
‰ The chemical potential is very important to mass
transport of particles such as:
‰ Mathematical Properties of the FEOT
9 Now that we derived the Coefficient relations…
⎛ ∂H ⎞
T =⎜
⎟
⎝ ∂S ⎠ p , N
⎛ ∂H ⎞
V =⎜
⎟
⎝ ∂p ⎠ S , N
Thermodynamics: A Brief Introduction
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⎛ ∂H ⎞
⎟
⎝ ∂N ⎠ s , p
μ =⎜
Atoms and molecules
Electrons
Phonons
Etc.
‰ Remember that from equation 7, the chemical potential
is given by:
⎛ ∂Ε A ⎞
⎟
⎝ ∂N A ⎠T , P
μA = ⎜
9 …we can develop Maxwell Relations:
9 We do so using the exact differential relation:
(7)
‰ We have examined two free energies, U and H.
‰ For U and H, does equation 7 hold?
⎛ ∂M ⎞ ⎛ ∂N ⎞
⎜
⎟ =⎜
⎟ :
⎝ ∂y ⎠ x ⎝ ∂x ⎠ y
⎛ ∂ 2U ⎞
⎛ ∂ 2U ⎞
⎛ ∂T ⎞
∂ ⎛ ∂H ⎞
⎛ ∂V ⎞
&
=
=
⎜
⎟
⎜
⎟
⎜
⎟ = ⎜
⎟
⎜
⎟
⎝ ∂S ⎠ p , N ⎝ ∂S ∂p ⎠ N
⎝ ∂p ⎠ S , N ∂p ⎝ ∂S ⎠ p , N ⎝ ∂p∂S ⎠ N
‰ Prove to yourself that it does.
‰ The particles that we deal with in this class are electrons
and holes which are Fermions which are particle with
spin ½.
‰ The chemical potential is defined as the Fermi Energy
level, Ef, as Temperature approaches absolute zero.
⎛ ∂T ⎞
⎛ ∂V ⎞
∴⎜
⎟ =⎜
⎟
∂
p
⎝
⎠ S , N ⎝ ∂S ⎠ p , N
9 …here we just derive one of three.
9 What are the other two?
E f =μ as T → 0.
‰ This is the primary thermo concept to understand for
this class.
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MSE/ECE 310
Electrical
Prop of Matls
Knowlton
Thermodynamics: A Brief Introduction
‰ What is thermodynamic equilibrium?
‰ Let’s define them.
‰ Consider two phases in physical contact in a system.
The two phases could be:
9 Ice and water
9 α and β in a two phase region
9 Etc.
‰ The specific types of equilibrium that can exist for the
two phases are:
Tphase1 = Tphase 2 Thermal Equilibrium
p phase1 = p phase 2 Mechanical Equilibrium
μ phase1 = μ phase 2 Mechanical Equilibrium
‰ Note that all of these variables are intrinsic variables.
‰ If one of these conditions do not exist, then equilibrium
for that specific type will not be equal and a gradient
(difference) exists.
‰ This will then be the driving force towards equil (equal)
librium.
‰ The driving force for diffusion is when chemical
equilibrium does not exist:
μ phase1 ≠ μ phase 2 i.e. Δμ ≠ 0 or ∇μ ≠ 0
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