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Relations I Week 13. Thermodynamic Property

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Week 13. Thermodynamic Property

Relations I

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Objectives

1. Develop fundamental relations between commonly encountered

thermodynamic properties and express the properties that cannot be measured directly in terms of easily measurable properties

2. Develop the Maxwell relations, which form the basis for many thermodynamic relations

3. Develop the Clapeyron equation and determine the enthalpy of vaporization from P,v, and T measurements alone

4. Develop general relations for C

v

, C

p

, du, dh, and ds that are valid for all pure substances

5. Discuss the Joule-Thomson coefficient

6. Develop a method of evaluating the ∆h, ∆u, and ∆s of real gases through the use of generalized enthalpy and entropy departure charts

GENESYS Laboratory

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Partial Differentials

• The variation of z(x, y) with x when y is held constant is called the partial derivative of z with respect to x, and it is expressed as

0 0

( , ) ( , )

lim lim

x x

y y

z z z x x y z x y

x

 

x

 

x

    

     

      

   

• The symbol  represents differential changes, just like the symbol d.

They differ in that the symbol d represents the total differential change of a function and reflects the influence

of all variables, whereas  represents the partial differential change due to the variation of a single variable.

  x

y

dx   z

y

dz

• The changes indicated by d and  are identical for independent variables, but not for dependent variables.

Geometric representation of partial derivative (z/x)y.

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Partial Differentials

• 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) shown in Figure

• The dependent variable z changes by , which can be expressed as

( , ) ( , )

Adding and substracting ( , ), we get

( , ) ( , ) ( , ) ( , )

using the definitions of part the fundamental relation for the

ial derivatives, total diff z z x x y y z x y

z x y y

z x x y y z x y y z x y y z x y

z x y

x y

      

 

         

    

 

erential of a dependent variable in terms of its partial derivatives with respect to the independent variables is

y x

z z

dz x y

x y

 

 

 

       

GENESYS Laboratory

z

Geometric representation of total derivative dz for a function z(x, y)

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Partial Differentials Relations

The order of differentiation is immaterial for properties since they are continuous point functions and have exact differentials. Thus,

Reciprocity relation Cyclic relation

Demonstration of the reciproci ty relation for the function z + 2xy 3y2z = 0.

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Ex 3) Verification of Cyclic and Reciprocity Relations

Using the ideal-gas equation of state, verify (a) the cyclic relation, and (b) the reciprocity relation at constant P.

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The Maxwell Relations

The equations that relate the partial derivatives of properties P, v, T, and s of a simple compressible system to each

GENESYS Laboratory

Helmholtz function Gibbs function

Maxwell relations

Maxwell relations are extremely valua ble in thermodynamics because they p rovide a means of determining the cha nge in entropy, which cannot be meas ured directly, by simply measuring the changes in properties P, v, and T.

These Maxwell relations are limited to simple compressible systems.

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The Maxwell Relations

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Ex 4) Verification of the Maxwell Relations

GENESYS Laboratory

Verify the validity of the last Maxwell relation for steam at 250oC and 300 kPa.

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The Clapeyron Equation

The Clapeyron equation enables us to determine the enthalpy of vapor ization 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.

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Ex 5) Evaluating the h

fg

of a Substance from the P-v-T Data

GENESYS Laboratory

Using the Clapeyron equation, estimate the value of the enthalpy of vaporization of refrigerant-134a at 20oC, and compare it with the tabulated value.

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The Clapeyron Equation

At low pressures Treating vapor as an ideal gas

• The Clapeyron equation can be simplified for liquid–vapor and solid–vapor phase changes by utilizing some approximations.

Substituting these equations into the Clapeyron equation

Integrating between two saturation states

The Clapeyron–Clausius equation 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.

Clapeyron–Clausius equation

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General Relations For du, dh, ds, C v , And C p

GENESYS Laboratory

• The state postulate established that the state of a simple compressible system is completely specified by two independent, intensive properties.

• Therefore, we should be able to calculate all the properties of a system such as internal energy, enthalpy, and entropy 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.

• General relations could be developed for changes in internal energy, enthalpy, and entropy in terms of pressure, specific volume, temperature, and specific heats alone.

The relations developed will enable us to determine the changes in these properties.

• The property values at specified states can be determined only after the selection of a reference state, the choice of which is quite arbitrary.

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Internal Energy Changes

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Internal Energy Changes

GENESYS Laboratory

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Enthalpy Changes

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Enthalpy Changes

GENESYS Laboratory

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Entropy Changes

The first relation is

where

and

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Entropy Changes

GENESYS Laboratory

The second relation is

and

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Specific Heats c

v

and c

p

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Specific Heats c

v

and c

p

GENESYS Laboratory

(23)

Mayer

Specific Heats c

v

and c

p

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Ex 9) The Specific Heat Difference of an Ideal Gas

GENESYS Laboratory

Show that cp-cv=R for an ideal gas.

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