1. Partial Derivatives
and
associated relations
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2. 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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3. Partial Derivatives
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4. Total differential change
When the independent variables x and y change by Δx and Δy respectively, the dependent variable z changes by Δz
Adding and subtracting
Total differential
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5. 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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6. 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 ΔT is:
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7. 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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8. 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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9. β and κ are in general functions of both T and P
Copper at 1 atm
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10. Mercury, at T=0oC
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11. Calvin and Hobbes
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12. β- β+
β=0
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13. 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 β and κ,
If β and κ can be measured experimentally, the equation of state can be found.
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14. If integration is carried out from (Vo,Po,To) (V,P,T), then:
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 β and κ can be considered constant,then:
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15. Relations between partial derivatives
Eliminating dP: or
Recriprocity relation
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16. In general, if any three variables satisfy the equation:
Combining with the reciprocity relation:
Cyclic relation
In general, if any three variables satisfy the equation:
Then:
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17. Example:
Function:
Thus,
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18. Example: At constant volume:
The pressure change in a finite change intemperature at constant volume is:
Can be integrated if β(T) and κ(T) are known at constant volume
If ΔT is small, β and κ can be considered constant.
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19. Exact differentials Since: In the limit
Mixed second partial derivatives
The value is independent of the order of differentiation
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20. Differentials for which this is true is called an exact differential
Differentials of all properties of a system (e.g. V, P, T, etc…) are exact.
A quantity whose differential is not exact is not a thermodynamic property .
If V1 and V2 are the volumes in two states:
value independent of path
If path is cyclic : V2 = V1, and V2-V1=0
Thus, if the integral of a differential between two arbitrary states is independent of the path, the integral around any closed path is zero, and the differential is exact.
True only if dV between states 1 and 3 is the same for all processes between the states
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21. coefficients of dT and dP
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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22. FIGURE 11-6 Demonstration of the reciprocity relation for the function z + 2xy - 3y2z = 0.
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23. Partial differentials are powerful tools that are supposed to make life easier, not harder.
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