thermodynamic relation,lausius Clapreyon equation , joule thomson coefficient, specific heat relations ,

1. Thermodynamic Relations

2. Some Mathematical Theorem
• Theorem 1- If the relation exists among relations x, y and z, then z may be
expressed as a function of x and y.
Where
Where z, M and N are functions of x and y. Differentiating M partially with respect to y ,
and N with respect to x.
The order of differentiation is immaterial for properties since they are continuous point
functions and have exact differentials. Therefore, the two relations above are identical:
This is the condition of exact differential.
In thermodynamics, this relation forms the basis for
the development of the Maxwell relations discussed
in the next section.

3. THE MAXWELL RELATIONS
• The equations that relate the partial derivatives of properties P, v, T,
and s of a simple compressible system to each other are called the
Maxwell relations.
Since U,H ,F and G are thermodynamics properties
and exact differential of the type
dz = Mdx + Ndy, then

4. Helmholtz and Gibbs Function
• The Gibbs free energy or Gibbs function is a thermodynamic function of a
certain system , it is equal to the enthalpy H of the system minus the
product of the entropy S of this system and its thermodynamic
temperature T .

5. Helmholtz and Gibbs Function
• The Helmholtz free energy or Helmholtz function is a thermodynamic
function of a certain system equal to internal energy U of the system
minus the the entropy S of this system multiplied by its thermodynamic
temperature T .

6. THE MAXWELL RELATIONS - continued
Applying this to all four equations
Apply
•These are called the Maxwell relations
•They are extremely valuable in thermodynamics because they provide a means of
determining the change in entropy, which cannot be measured directly, by simply
measuring the changes in properties P, v, and T.
•Note that the Maxwell relations given above are limited to simple compressible systems.
However, other similar relations can be written just as easily for non simple systems such as
those involving electrical, magnetic, and other effects.

7. TDS Equations
Two general relations for the entropy change of a simple
compressible system.
• First TDS Relation –
• Let entropy S be imagined as a function of T and V. Then
• S = f(T,V)
Multiply T , temperature to all equation
Equation Number 1

8. Internal Energy relations
• We choose the internal energy to be a function of T and v; that is, u=
u(T, v) and take its total differential
Now we choose the entropy to be a function of T and v; that is, s s(T, v) and take
its total differential,
Using the definition of cv, we have
Substituting this into the T ds relation, du = (T ds - P dv), yields
Equating the coefficients of dT and dv
Equation Number 2

9. TDS Equationscontinued
Equation Number 1
Equation Number 2
Equation Number 3
Substituting 2nd and 3rd equation in first
This is First TDS
Equation

10. TDS Equationscontinued
• Second TDS Relation –
• Let entropy S be imagined as a function of T and P. Then
• S = f(T,P)
Multiply T , temperature to all equation
Equation Number 4

11. Enthalpy Change Relations
• This time we choose the enthalpy to be a function of T and P, that
is, h h(T, P), and take its total differential,
Using the definition of cp, we have
Now we choose the entropy to be a function of T and P; that is, we take
s s(T, P) and take its total differential,
Substituting this into the T ds relation, dh = (T ds + v dP) gives
Equating the coefficients of dT and dP Equation
Number 5

12. TDS Equationscontinued
Equation Number 4
Equation
Number 5
Maxwell Fourth relation, Equ 6
Substituting 5 and 6 in equation number 4
This is Second TDS
Equation
Either relation (First TDS or Second TDS)
can be used to determine the entropy
change. The proper choice depends on the
available data.

13. Specific Heats cv and cp
• The specific heats of an ideal gas depend on temperature only.
• For a general pure substance, however, the specific heats depend
on specific volume or pressure as well as the temperature.
• Below we develop some general relations to relate the specific
heats of a substance to pressure, specific volume, and temperature.
A general relation involving specific heats is one that relates the two specific heats
cp and cv.
The advantage of such a relation is obvious:
We will need to determine only one specific heat (usually cp) and calculate the
other one using that relation and the P-v-T data of the substance. We start the
development of such a relation by equating the two Tds relations
Equation Number 7

14. Specific Heats cv and cp - continued
Choosing T T(v, P) and differentiating, we get
Equation Number 7
Equation Number 8
Equating the coefficient of either dv or dP of the above two equations (Equn 7 and
8) gives the desired result:
Use any relation both will give same result
Equation Number 9

15. Specific Heats cv and cp - continued
Equation Number 9
From Theorem 2
Equation
Number 10

16. Volume expansivity- Co-efficient of cubical expansion.
When the experimental results are plotted as a series of constant pressure
lines on a v-T diagrams
The slope of a constant pressure line
at any given state is
If the gradient is divided by the volume at that
state, we have a value of a property of the
substance called its co-efficient of cubical
expansion.
For solids and liquids over the normal working range of pressure and temperature, the variation
of β is small and can often be neglected. In tables of physical properties β is usually quoted as an
average value over a small range of temperature, the pressure being atmospheric. This average
co-efficient may be symbolised by β and it is defined by
Equation Number 11

17. the isothermal compressibility
• To show the variation of volume with pressure for various constant values of
temperature. In this case, the gradient of a curve at any state is
When this gradient is divided by the volume at that state, we have a property
known as the compressibility of the substance
Since this gradient is always negative, i.e., the
volume of a substance always decreases with
increase of pressure when the temperature is
constant, the compressibility is usually made a
positive quantity by inserting negative sign.
Equation
Number 12

18. Specific Heats cv and cp - continued
Equation Number 10
Equation Number 11
Equation Number 12
From Equation Number 10, 11 and 12
It is called the Mayer relation in honor of the German
physician and physicist J. R. Mayer (1814–1878). We can
draw several conclusions from this equation:

19. Some Conclusions From the Mayer relation
The isothermal compressibility a is a positive quantity for all substances in all phases.
The volume expansivity could be negative for some substances (such as liquid water below
4°C), but its square is always positive or zero.
The temperature T in this relation is thermodynamic temperature, which is also positive.
Therefore we conclude that the constant-pressure specific heat is always greater than or equal
to the constant-volume specific heat:
The difference between cp and cv approaches zero as the absolute temperature approaches
zero.
The two specific heats are identical for truly incompressible substances since v= constant. The
difference between the two specific heats is very small and is usually disregarded for substances
that are nearly incompressible, such as liquids and solids.

20. Prove Cp and Cv are function of temperature
only.

21. Joule Thomson Coefficient
When a fluid passes through a restriction
such as a porous plug, a capillary tube, or
an ordinary valve, its pressure decreases.
The enthalpy of the fluid remains
approximately constant during such a
throttling process.
The temperature of a fluid may increase,
decrease, or remain constant during a
throttling process.
The temperature behavior of a fluid during a
throttling (h constant) process is described
by the Joule-Thomson coefficient, defined
as

22. Joule Thomson Coefficient
Some constant-enthalpy lines on the T-P
diagram pass through a point of zero slope
or zero Joule-Thomson coefficient. The
line that passes through these points is
called the inversion line, and
the temperature at a point where a
constant-enthalpy line intersects the
inversion line is called the inversion
temperature.
The temperature at the intersection of the
P = 0 line (ordinate) and the upper part of
the inversion line is called the maximum
inversion temperature.
It is clear from this diagram that a cooling effect cannot be achieved by throttling
unless the fluid is below its maximum inversion temperature.

23. Joule Thomson Coefficient

24. Joule Thomson coefficient in terms of Volume
expansivity and Maximum Cooling

25. The Clausius -Clapeyron Equation
It enables us to determine the enthalpy change associated with a phase change
(such as the enthalpy of vaporization hfg) from a knowledge of P, v, and T data
alone.
Consider the third Maxwell relation,
During a phase-change process, the pressure
is the saturation pressure, which depends on
the temperature only and is independent of
the specific volume.
Psat = f (Tsat).
Therefore, the partial derivative (P/T )v can be expressed as a total derivative (dP/dT
)sat, which is the slope of the saturation curve on a P-T diagram at a specified
saturation state

26. The Clausius -Clapeyron Equation
which is called the Clapeyron
equation after the French
engineer and physicist E.
Clapeyron (1799–1864).

27. The Clausius -Clapeyron Equation
The Clapeyron equation can be simplified for liquid–vapor and solid–vapor phase
changes by utilizing some approximations. At low pressures vg vf , and thus vfg vg.
By treating the vapor as an ideal gas, we have vg RT/P. Substituting these
approximations
For small temperature intervals hfg can be treated as a constant at some average
value. Then integrating this equation between two saturation states yields
This equation is called the Clapeyron–Clausius equation, and it can be used to
determine the variation of saturation pressure with temperature. It can also be used
in the solid–vapor region by replacing hfg by hig (the enthalpy of sublimation) of the
substance.

28. • Thank you.
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