SMES1202

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Partial Derivatives and associated relations
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Derivatives

• Steepness of the curve is a measure of the degree
  of dependence of f on x
• Steepness of a curve at a point is measured by
  the slope of a line tangent to the curve at that
  point

the derivative of the function at that point
The derivative of a function f(x) with respect to
x represents the rate of change of f with x
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Partial Derivatives
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At point 1, slope of tangent
the partial derivative of V with respect to T at
constant P
For an ideal gas
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Slope of chord from 1 to 2
Suppose the volume varies with temperature, not
along the actual curve but along the tangent at
point 1, then the increase in volume when the
temperature was increased by ?T is
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Coefficient of volume expansivity (or
expansivity)
Or in specific volumes
For ideal gas
Unit K-1
For two closely adjacent states of a system at
the same pressure
The mean expansivity
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Isothermal compressibility
Or in specific volumes
Negative sign included because the volume always
decreases with increasing pressure at constant
temperature so that is inherently
negative.
For ideal gas
Unit kPa-1
The mean expansivity
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ß and ? are in general functions of both T and P
Copper at 1 atm
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Mercury, at T0oC
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Calvin and Hobbes
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State 1 at P1 and T1 State 2 at P3 and T2
Different P and T
The volume difference between the states depends
only on the states and is independent of any
particular process
In terms of ß and ?,
If ß and ? can be measured experimentally, the
equation of state can be found.
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Suppose that it was found experimentally, for a
gas at low pressure, that
Then from
Integrating
And
If integration is carried out from (Vo,Po,To)
?(V,P,T), then
If ß and ? can be considered constant,then
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Relations between partial derivatives
Eliminating dP
or
Recriprocity relation
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Combining with the reciprocity relation
Cyclic relation
In general, if any three variables satisfy the
equation
Then
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Example
Function
Thus,
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Example
At constant volume
The pressure change in a finite change
intemperature at constant volume is
If ?T is small, ß and ? can be considered
constant.
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Exact differentials
Mixed second partial derivatives
The value is independent of the order of
differentiation
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Have shown before that
In general, if for any three variables x,y, and
z, we have a relation of the form
The differential is exact if
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FIGURE 11-6Demonstration of the
reciprocityrelation for the functionz 2xy -
3y2z 0.
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Partial differentials are powerfultools that are
supposed to makelife easier, not harder.