Download Introduction into thermodynamics Thermodynamic variables

Introduction into thermodynamics
Solid-state thermodynamics, J. Ma jzlan
Chemical thermodynamics deals with reactions between substances and species. Mechanical thermodynamics, on the other hand, works with engines and their performance. In this class, we will
explore chemical thermodynamics. We will ask and nd the answer to questions such as why
does ice melt at room temperature, why can diamond burn, and why some substances dissolve
but others apparently do not.
It is very important to keep in mind that thermodynamics deals with models
reality itself. Reality is too complicated and therefore we need such models.
of reality, not
First, we need a few denitions.
System : A part of the universe we consider. For example, water in the bottle, a piece of rock.
Surrounding : Everything outside of the system.
Equilibrium : A state of a system in which none of the properties of the system change with
time.
Phase : A mechanically separable part of a system. Minerals are phases ; water is also a phase.
Sodium chloride dissolved in water is not a phase because it cannot be mechanically separated
(for example, by ltration) from the water.
Component : An algebraic operator. The composition of any phase in the system must be
expressed as a linear combination of the components. One should try to select the smallest
number of components.
Example. A system is made of K-feldspar (KAlSi3 O8 ), sillimanite (Al2 SiO5 ), and quartz (SiO2 ).
Dene the components.
Components for this system are K2 O, Al2 O3 , and SiO2 . Each phase can be expressed as a linear
combination of these components :
1
1
K2 O + Al2 O3 + 3SiO2 = KAlSi3 O8
2
2
0K2 O + 1Al2 O3 + 1SiO2 = Al2 SiO5
0K2 O + 0Al2 O3 + 1SiO2 = SiO2
Thermodynamic variables
1
Temperature. In everyday life, temperature is measured in o C. In thermodynamics, however,
we always express temperature in Kelvin, K. The relationship between these two units is
T [K] = T [o C] + 273.15
Pressure. The SI unit for pressure is Pascal, Pa. One Pascal is a small pressure ; the atmospheric
pressure is approximately 105 Pa. Variables that were previously in use are also 1 bar (= 105
Pa) and 1 atmosphere (1 atm ≈ 1 bar).
Internal energy. Internal energy U is a fundamental property of every system. It can be related
to the rest energy Er dened by the well-known equation
Er = mc2
as
Er =
U + constant
where the value of the constant is unknown. The consequence is that the internal energy has no
absolute scale. All variables which are dened via internal energy have no absolute scale. The
importance of this statement will become soon obvious.
Internal energy can be transferred in two ways, by transfer of heat (q ) and transfer of work (w).
Mathematically, this is expressed in the equation
∆
U=q+w
Under work, we usually understand mechanical work in thermodynamics. Without going into
detail, the mechanical work is the "pressure-volume work", that is
w = −P ∆V
Heat and entropy :
There is another thermodynamic function which relates heat and temperature.
q = T ∆S
The equation for internal energy can be rewritten as
∆
U = T ∆S − P ∆V
∆U = T ∆S − P ∆V
2
if we consider the extensive or intensive internal energy of a system, respectively. The dierential
form of this equation
dU = T dS − P dV
is a fundamental equation of thermodynamics.
Thermodynamic variables in equilibrium
Entropy is a variable that will be maximized in a system in equilibrium.
Internal energy will be minimized in systems held at constant entropy and volume :
dU = T dS − P dV
Such conditions, that is, constant entropy and volume, are impossible to maintain in the laboratory or expect in the nature. Therefore, although the internal energy if a fundamental variable
used to derive other important thermodynamic functions, it is of little importance for practical
use. How about enthalpy ? Let's recall the denition of the enthalpy
H = U + PV
and dierentiate to
dH = dU + P dV + V dP
Inserting the fundamental equation of thermodynamics, we get
dH = dU + P dV + V dP = T dS − P dV + P dV + V dP = T dS + V dP
Therefore, enthalpy is a variable that will be minimized in a system in equilibrium at constant
entropy and pressure. Such conditions are also dicult to maintain or imagine. We need yet
another thermodynamic variable.
Gibbs energy. We can dene another variable
G = U + PV − TS
and dierentiate to
dG = dU + P dV + V dP − T dS − SdT = T dS − P dV + P dV + V dP − T dS − SdT = −SdT + V dP
3
This variable, Gibbs energy or Gibbs free energy, will be minimized in equilibrium in a system
at constant temperature and pressure. Such conditions can be maintained in a laboratory or
attained in nature. Gibbs free energy is therefore of central importance in thermodynamics. It is
derived from internal energy and therefore has no absolute scale.
Gibbs free energy is dened as
G = U + PV − TS
which can be reduced to
G = H − TS
Because we do not know the absolute values of those energies, this equation can be actually used
in the form
∆G = ∆H − T ∆S
Let's consider a general chemical equation
mM + nN = xX + yY
If the phases involved in this reaction are pure solids or liquids in their standard state, the change
in the Gibbs free energy associated with that reaction is
∆r Go = xGoX + yGoY − mGoM − nGoN
This is, however, a problem, because we do not know the absolute values of G. We solve these
problems by introducing reference state and write
∆r Go = x∆f GoX + y∆f GoY − m∆f GoM − n∆f GoN
What happens, when the phases that participate in this reaction are not pure solids and liquids ?
We have to know how does the Gibbs free energy of these phases change with their composition.
We dene a new variable closely related to Gibbs free energy, the chemical potential µ
µi =
G
∆
∆ni
or for small (innitesimal) changes in ni
4
T,P,nj
µi =
G
∂
∂ni
T,P,nj
For our chemical reaction
mM + nN = xX + yY
The change of chemical potential for this reaction, ∆r µ, is
∆r µ = x∆µX + y∆µY − m∆µM − n∆µN
∆r µ = x(∆µoX + RT lnaX ) + y(∆µoY + RT lnaY ) − m(∆µoM + RT lnaM ) − n(∆µoN + RT lnaN )
This equation can be re-arranged
∆r µ = ∆r µo + RT ln
axX ayY
n
am
M aN
This expression is valid for the general reaction above. For any chemical reaction, the equation
can be further generalized
∆r µ = ∆r µo + RT lnΠaνi i = ∆r µo + RT lnQ
Greg Anderson (2005 : Thermodynamics of Natural Systems) calls this equation "the most
useful equation in thermodynamics". It relates the change in Gibbs free energy, via the chemical
potential, to the thermodynamic activities of the species involved.
In chemical equilibrium, ∆r µ is equal to zero. The equation reduces to
∆r µo = −RT lnQ
In this, and only in this situation, the stoichiometric quotient Q gets a special name equilibrium
constant and the symbol K
∆r µo = −RT lnK
Variation of G with temperature
Heat capacity. Heat capacity is the amount of heat that has to be added to 1 mole of a
substance to increase its temperature by 1 K. It is designated as Cp and dened
5
Cp =
dH
dT
P
Heat capacity of substances is measured and then tabulated as polynomials which were tted to
the data. A common Cp function is the Maier-Kelley heat capacity polynomial
Cp = a + bT + cT −2
Other functions were also proposed and some will be discussed in the class.
The change in entropy can be also dened via heat capacity as
Z
∆S =
6
Cp
dT
T