1. THERMODYNAMIC PROPERTY RELATIONS
ERT206 THERMODYNAMICS
MISS WAN KHAIRUNNISA BINTI WAN RAMLI

2. OBJECTIVE
Develop fundamental relations between commonly encountered thermodynamic properties and express the
properties that cannot be measured directly in terms of easily measurable properties.
Develop the Maxwell relations, which form the basis for many thermodynamic relations.
Develop the Clapeyron equation and determine the enthalpy of vaporization from P, v, and T measurements alone.
Develop general relations for cv, cp, du, dh, and ds that are valid for all pure substances.
Develop a method of evaluating the Δh, Δu, and Δs of real gases through the use of generalized enthalphy & entrophy departure chart

3. PARTIAL DERIVATIVES & ASSOCIATED RELATIONS
The state postulate  The state of a simple, compressible substance is completely specified by any two independent, intensive properties.
All other properties at that state can be expressed in terms of those
two properties.
The derivative of a function f(x) with respect to x represents the rate of
change of f with x.

4. PARTIAL DERIVATIVES
PARTIAL DERIVATIVES of z  The variation of z(x, y) with x when y is held constant with respect to x, and it is expressed as
The symbol δ  differential changes
Difference:
Symbol d  the total differential change of a function and reflects
the influence of all variables,
Symbol δ  the partial differential change due to the variation of a single variable.
The changes indicated by d and δ
are identical for independent
variables, but not for dependent
variables.

5. Adding & subtracting z(x, y + Δy), we get
To obtain a relation for the total differential change in z(x, y) for simultaneous changes in x and y, consider a small portion of the surface z(x, y) [Fig 3]. When the independent variables x and y change by Δx and Δy, respectively, the dependent variable z changes by Δz, which can be expressed as
Adding & subtracting z(x, y + Δy), we get or
Taking the limits as Δx  0 & Δy  0, using definitions of partial derivatives, we obtain
Fig 3: Geometric representation of total derivative dz for a function z(x, y). E1
This is the fundamental relation for the total differential of a dependent variable in terms of its partial derivatives with respect to the independent variables.

6. PARTIAL DIFFERENTIAL RELATIONS
Same form as E1
The order of differentiation is immaterial for properties since they are continuous point functions and have exact differentials. Thus,
Demonstration of the reciprocity relation for the function z+2xy – 3y2z = 0
RECIPROCITY RELATION
CYCLIC RELATION

7. THE MAXWELL RELATIONS
MAXWELL RELATIONS  The equations that relate the partial derivatives of properties P, v, T, and s of a simple compressible system to each other.
Obtained from the four Gibbs equations by exploiting the exactness of the differentials of thermodynamic properties
Maxwell relations provide means of determining the change in entropy, which cannot be measured directly, by simply measuring the changes in properties P, v, and T, limited to simple compressible systems.

8. THE CLAPEYRON EQUATION
3rd Maxwell relation
During a phase-change process, P = Psat, which depends on T only, is independent of the specific volume. Psat = f(Tsat).
The partial derivative (δP/δT)v can be expressed  total derivative (dP/dT)sat = SLOPE of the saturation curve on a P-T diagram at a specified saturation state [Fig 4].
This slope is independent of the specific volume,
thus it can be treated as a constant during the
integration of above Eq. between two saturation states
at the same T for an isothermal liquid-vapor phase change
process. For example, the integration yields Or
The slope of the saturation curve on a P-T diagram is constant at a constant T or P.

9. During this process the P also remains constant. Therefore, from Eq
Substituting this result into Eq , we obtain
CLAPEYRON EQUATION
The Clapeyron equation enables us to determine the enthalpy of vaporization hfg at a given temperature by simply measuring the slope of the saturation curve on a P-T diagram and the specific volume of saturated liquid and saturated vapor at the given temperature.
General form of the Clapeyron
equation when the subscripts 1
and 2 indicate the two phases.

10. CLAPEYRON-CLAUSIUS EQUATION
The Clapeyron equation can be simplified for liquid–vapor and solid–vapor phase changes by utilizing some approximations.
At low pressures
Treating vapor as an ideal gas
Substituting into Clapeyron Equation
Integrating between 2 saturation states
The Clapeyron–Clausius eq.
can be used to determine the variation of saturation pressure with temperature.
can also be used in the
solid–vapor region by replacing hfg by hig (the
enthalpy of sublimation) of the substance
CLAPEYRON-CLAUSIUS EQUATION

11. GENERAL RELATIONS FOR du, dh, ds, cv & cp
The state of a simple compressible system is completely specified by two independent, intensive properties.
Able to calculate all the properties of a system (u, h, and s) at any state once two independent, intensive properties are available.
The calculation of these properties from measurable ones depends on the availability of simple and accurate relations between the two groups.
Development of general relations for Δu, Δh, Δs in terms of P, v, T, and cv, cp.
The property values at specified states can be determined only after the selection of a reference state, the choice of which is quite arbitrary

12. INTERNAL ENERGY CHANGES
Consider u = u(T, v) and take its total differential (Eq. 12-3)
Using the definition of cv, we have
Consider entrophy to be a function of T and v  s = s(T,v) and take its total differential
Substituting this into the T ds relation du = T ds – P dv yields

13. Equating the coefficients of dT and dv in Eqn. 12-25 & 12-27 gives
Using the 3rd Maxwell relation (Eq ), we get
Substituting this into Eq , we obtain the desired relation for du:
The change in internal energy of a simple compressible system associated with a change of state from (T1, v1) to (T2,v2) is determined by integration of Eqn.12-29:
Δu general relation

14. ENTHALPHY CHANGES
Consider the enthalpy to be a function of T and P  h = h(T,P), and take its total differential,
Using the definition of cp, we have
Consider again the entrophy to be a function of T and P  s = s(T,P) and take its total differential,
Substituting this into the T ds relation dh = T ds + v dP gives

15. Equating the coefficients of dT and dP in Eq. 12-31 & 12-33, we obtain
Using the 4th Maxwell relation, we have
Substituting this into Eq , we obtain the desired relation for dh:
The change in enthalpy of a simple compressible system associated with a change of state from (T1,P1) to (T2, P2) is determined by integration:
The other case can easily be determined by using the definition of enthalpy h = u + Pv:
Δh general relation

16. ENTROPHY CHANGES
The first relation is obtained by replacing the 1st partial derivative in the total differential ds (Eq ) by Eq and the second partial derivative by 3rd Maxwell relation (Eq ), yielding
And
The 2nd relation is obtained by replacing the 1st partial derivative in the total differential of ds (Eq ) by Eq and the 2nd partial derivative by the 4th Maxwell relation (Eq ), yielding
Either relation can be used to determine the entrophy change. The proper choice depends on the available data
Δs general relation

17. EXAMPLE
Determine the change in the enthalpy of carbon dioxide (CO2), in kg/kg, as it undergoes a change of state from 100 kPa and 20° C to 600 kPa and 300 ° C using the equation of state P(v-a)=RT where a = 0.01 m3/kg, and compare the result to the value obtained by using the ideal gas equation
of state.
SOLUTION:
Using Eq ,
Assumptions: Constant specific heats for CO2 can be used
Properties: For CO2, the average T  (20+300)/2 = 160 C = 433K
Cp = kJ/kg.K
Analysis: Solving the equation of state for v gives

18. Substituting,
Integrating this result between the 2 states gives
For an ideal gas,
Which when integrated gives

19. Cp, Cv general relation I
SPECIFIC HEATS cv & cp
At low pressures gases behave as ideal gases, and their specific heats depend on T only  zero pressure or ideal gas, specifics heats (denoted cv0 and cp0)and they are relatively easier to determine.
It is desirabe to have some general relations for determining the specific heats at higher P (or lower specific volumes) from a knowledge of cv0 or cp0 and P-v-T behaviour of the substance.
Such relations are obtained by applying the test of exactness (Eq. 12-5) on Eq and Eq , which yields
The deviation of cp from cp0 with increasing pressure is determined by integrating Eq from zero pressure to any pressure P along an isothermal path:
Cp, Cv general relation I

20. Cp, Cv general relation II
Another desirable general relation involving specific heats which relates the two specific heats cp and cv.
The advantage of such a relation  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.
This development of such relation starts by equating the two ds relations (Eq & Eq ) and solving for dT:
Choosing T = T(v, P) and differentiating, we get
Equating the coefficient of either dv or dP of the above two equations gives the desired result:
Cp, Cv general relation II

21. Substituting the result into Eq. 12-45 gives
An alternating form of this relation is obtained by using the cyclic relation:
Substituting the result into Eq gives
This relation can be expressed in terms of two other thermodynamic properties called the volume expansivity β and the isothermal compressibility α, which are defined as [Fig 10]
And
Substituting these two relations into Eq , we obtain a third general relation for cp – cv
Volume expansivity
Isothermal compressibility
Cp, Cv general relation III

22. CONCLUSIONS FROM MAYER RELATION:
The right hand side of the equation is ALWAYS GREATER THAN/ EQUAL to zero. Therefore, we conclude that
The difference between cp and cv approaches zero as the absolute T 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 & solids
The volume expansivity (also called the coefficient of volumetric expansion) is a measure of the change in volume with T at constant P

24. THE ∆h, ∆u, AND ∆s OF REAL GASES
Gases at low pressures behave as ideal gases and obey the relation Pv = RT. The properties of ideal gases are relatively easy to evaluate since the properties u, h, cv, and cp depend on temperature only.
At high pressures, however, gases deviate considerably from ideal-gas behavior, and it becomes necessary to account for this deviation.
To evaluate the changes in the enthalpy, internal energy, and entropy of non ideal (real) gases, using the general relations for du, dh, and ds developed earlier.

25. ENTHALPHY CHANGES OF REAL GASES
The enthalpy of a real gas, in general, depends on P & T.
The enthalpy change of a real gas during a process can be evaluated from the general relation for dh
For an isothermal process dT = 0, and the first term vanishes.
For a constant-pressure process, dP = 0, and the second term vanishes.
An alternative process path to evaluate the enthalpy changes of real gases.

26. Using a superscript asterisk (
Using a superscript asterisk (*) to denote an ideal-gas state, we can express the enthalpy change of a real gas during process 1-2 as
ENTHALPHY DEPARTURE  The difference between h and h* is and it represents the variation of the enthalpy of a gas with pressure at a fixed temperature.
Can use the relation Pv = ZRT, where Z is the compressibility factor. Substituting,

27. INTERNAL ENERGY CHANGES OF REAL GASES
Enthalpy departure factor
The values of Zh are presented in graphical form as a function of PR (reduced pressure) and TR (reduced temperature) in the generalized enthalpy departure chart.
Zh is used to determine the deviation of the enthalpy of a gas at a given P and T from the enthalpy of an ideal gas at the same T.
For a real gas during a process 1-2
Taken from ideal gas tables
INTERNAL ENERGY CHANGES OF REAL GASES
Using the definition

28. ENTROPHY CHANGES OF REAL GASES
General relation for ds
Using the approach in the figure
During isothermal process
An alternative process path to evaluate the entropy changes of real gases during process 1-2.

29. Entrophy departure factor
The values of Zs are presented in graphical form as a function of PR (reduced pressure) and TR (reduced temperature) in the generalized entropy departure chart.
Zs is used to determine the deviation of the entropy of a gas at a given P and T from the entropy of an ideal gas at the same P and T.
For a real gas during a process 1-2
from the ideal gas relations

30. THANK YOU

31. TUTORIAL IV
A 0.05 m3 well-insulated rigid tank contains oxygen at 175K and 6 Mpa. A paddle wheel placed in the tank is turned on, and the temperature of the oxygen rises to 225K. Using the generalized charts, determine (a) the final pressure in the tank, (b) the paddle-wheel work done during this process.
SOLUTION: