Factor, integrating

In the example of the previous section, the release of the stop always leads to the motion of the piston in one direction, to a final state in which the pressures are equal, never in the other direction. This obvious experimental observation turns out to be related to a mathematical problem, the integrability of differentials in themiodynamics. The differential Dq, even is inexact, but in mathematics many such expressions can be converted into exact differentials with the aid of an integrating factor. 

Bom [2] cites as an example of a simple expression for which no integrating factor exists... 

If an integrating factor exists (h and From the first of these relations one... 

There are an infinite number of other integrating factors X with corresponding fiinctions ( ) the new quantities T and. S are chosen for convenience.. S is, of course, the entropy and T, a fiinction of 0 only, is the absolute temperature , which will turn out to be the ideal-gas temperature, 0jg. The constant C is just a scale factor detennining the size of the degree. 

Equation 3-133 is a first order linear differential equation of the form dy/dx -i- Py = Q. The integrating factor is IF = and... 

This simple steady-state form is amenable to analytical solution and the application of an integrating factor followed by subsequent integration yields... 

To find the dependence of Cb on time we use the integrating factor method. Write Eq. (3-22) as... 

By multiplying through with the integrating factor eJ the solution is... 

It follows that 1/T is the integrating factor of SQ. Now since SQ is a function of two variables (in the simple case of a homogeneous fluid), and since the integrating factor of such a magni-" tude is usually also a function of the same two variables, we must regard the proposition that the integrating factor of SQ is a function of one variable only as expressing a physical, not a mathematical, truth. 

When the Pfaffian expression is inexact but integrable, then an integrating factor A exists such that AbQ = d5, where dS is an exact differential and the solution surfaces are S = constant. While solution surfaces do not exist for the inexact differential 8Q, solution curves do exist. The solution curves to dS = 0 will also be solution curves to bQ = 0. Since solution curves for dS on one surface do not intersect those on another surface, a solution curve for 8Q — 0 that lies on one surface cannot intersect another solution curve for bQ = 0 that lies on a different surface. 

For the two-dimensional Pfaffian, condition (c) above is no longer applicable. That is, the Pfaffian differential expression is either (a), exact, or (b), integrable with an integrating factor. 

Schottky effect in solids 580-5 Second law of thermodynamics 56-90 absolute temperature, identification of as integrating factor 71-8 Caratheodory principles differentials 63-7 and inaccessible states 68-71... 

When you come across the term convolution integral later in Eq. (4-10) and wonder how it may come about, take a look at the form of Eq. (2-3) again and think about it. If you wonder about where (2-3) comes from, review your old ODE text on integrating factors. We skip this detail since we will not be using the time domain solution in Eq. (2-3). 

We can find the solution to Eq. (4-1), which is simply a set of first order differential equations. As analogous to how Eq. (2-3) on page 2-2 was obtained, we now use the matrix exponential function as the integration factor, and the result is (hints in the Review Problems)... 

Laplace transforms 279-286 mapping of a function 271 integrating factor 56-59, 86, 360 integration 43-61 along a curve 51-52 by parts 46 by substitution 45-46 chemical kinetics 47-48 constant of integration 43,49, 89, 124... 

The differential equations (36) and (37) are solved by the integrating factor method, with the boundary condition C = C0 at x = 0. 

This first-order linear differential equation may be solved using an integrating factor approach to give... 

This equation is linear first-order and may be solved in a variety of fashions. One may use an integrating factor approach, Laplace transforms, or rearrange the equation and obtain the sum of the homogeneous and particular solutions. The solution is... 

This equation may be solved using an integrating factor approach to determine the absolute yield of V as a function of the fraction A reacted in an isothermal reactor. For the case where no V or W are present in the original gas stream, integration gives ... 

The integrating factor for this first-order linear equation is dB = B ... 

This Is a linear equation whose integrating factor is 1/ Tb and whose solution is... 

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