THERMODYNAMIC RELATIONS

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Total DeferentialIf we have a function f=f(x) The derivative of a function f(x) with respect to x represents the rate of change of f with x.

Partial DifferentialsThe 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

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The Clapeyron Equation

Consider a Carnot heat engine operating across a small temperature difference . The corresponding saturation pressures are as shown the figures (a) and (b).

TTandT PPandP

fgLfgH sTTqandTsqThen )(

Note that, ΔT is infinitesimal and soΔ P is infinitesimal also.

Hq

Lq

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The net work is:

The four processes is steady state, hence, the work is expressed as:

And so,

Then:

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And so,

Then:

ThsThen fgfg /As:

Sustituting in dPsat/dT:This equation is called Clapeyron equation

The importance of this equation is that, we can experimentaly know dPsat/dT Also. We can measure both vf and vg

Then, we can calculate hfg and sfg

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Then:

So:

Sustituting in dPsat/dT:

For vapor-solid phase, we can use the same equation:

For very low temperature and pressure

phasesolidvaprinvvOR

phaseliquidvaprinvv

fg

fg

The vapor can be considered as ideal gas, Then: PRTv /

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Example:

Solution:

Clapeyron equation:

Arranging the equation and integrating:

Let state (1) is T=-60 C=213.2 K State (2) is T=-40 C=233.3 K

From ice-vapor table P2=0.0129 kPa and hig=2838.9 kJ/kg.

We can note that, at this range of temperature hig can be assumed constant.

Solving the integration:

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Then:

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Mathematical Relations for Homogenous PhaseFor a function f = z(x,y). The total differential of z is written as:

Where:

Taking the partial derivative of M with respect to y and of N with respect to x yields

Then:

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The equations that relate the partial derivatives of properties P, v, T, and s of a simple compressible substance to each other are called the Maxwell relations. They are obtained from the four Gibbs equations.

du T ds Pdvdh T ds v dP

The Maxwell relations

The second two Gibbs equations result from the definitions of the Helmholtz function a

And the Gibbs function g defined as

The first two of the Gibbs equations are those resulting from the internal energy u and the enthalpy h.

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Setting the second mixed partial derivatives equal for these four functions yields the Maxwell relations

Another useful relations can be derived:

v

s

T

P

a

u

h

g

These relations can be summarized by the following

figure

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Now we develop two more important relations for partial derivatives—the reciprocity and the cyclic relations. Consider the function z = z(x,y) expressed as x = x(y,z). The total differential of x is

Now combine the expressions for dx and dz.

Rearranging,

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Since y and z are independent of each other, the terms in each bracket must be zero. Thus, we obtain the reciprocity relation that shows that the inverse of a partial derivative is equal to its reciprocal.

or

The second relation is called the cyclic relation.

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Example:

Solution:

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v

s

T

P

a

u

h

g

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The constant pressure lines intersection point can be illustrated in the following figure.

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Example:

Solution:

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v

s

T

P

a

u

h

g

Substituting in 3, yields:

v

s

T

P

a

u

h

g

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Example:

Solution:

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v

s

T

P

a

u

h

g

v

s

T

P

a

u

h

g

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Example:

Solution:

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v

s

T

P

a

u

h

g

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Example:

Solution:

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Example:

Solution:

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Example:

Solution:

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v

s

T

P

a

u

h

g

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v

s

T

P

a

u

h

g

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Example:

Solution:

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v

s

T

P

a

u

h

g

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Volume Expansibility, Isothermal Compressibility and Adiabatic Compressibility

Coefficient of linear expansion: The change in length as the temperature changes, while pressure remains constant

Coefficient of volume expansion: The changes in volume as the temperature changes, while pressure remains constant

The isothermal compressibility: The change in volume as the pressure changes, while temperature remains constant

The isothermal bulk modulus: It is the reciprocal of the isothermal compressibility.

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The adiabatic compressibility: The change in volume as the pressure changes, while entropy remains constant

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Example:

Solution:

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v

s

T

P

a

u

h

g

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