Partial Derivatives and associated relations SMES1202 khkwek

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 SMES1202 khkwek

Partial Derivatives SMES1202 khkwek

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 SMES1202 khkwek

At point 1, slope of tangent = the partial derivative of V with respect to T at constant P For an ideal gas: SMES1202 khkwek

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: SMES1202 khkwek

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: SMES1202 khkwek

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: SMES1202 khkwek

β and κ are in general functions of both T and P Copper at 1 atm SMES1202 khkwek

Mercury, at T=0oC SMES1202 khkwek

Calvin and Hobbes SMES1202 khkwek

β- β+ β=0 SMES1202 khkwek

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. SMES1202 khkwek

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: SMES1202 khkwek

Relations between partial derivatives Eliminating dP: or Recriprocity relation SMES1202 khkwek

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: SMES1202 khkwek

Example: Function: Thus, SMES1202 khkwek

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. SMES1202 khkwek

Exact differentials Since: In the limit Mixed second partial derivatives The value is independent of the order of differentiation SMES1202 khkwek

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 SMES1202 khkwek

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 SMES1202 khkwek

FIGURE 11-6 Demonstration of the reciprocity relation for the function z + 2xy - 3y2z = 0. SMES1202 khkwek

Partial differentials are powerful tools that are supposed to make life easier, not harder. SMES1202 khkwek