The Derivation of Equation

One has again been left with (1 - 0) on the right-hand side and thus one again substitutes from Eq. (A4.4.2). This process of substitution can be repeated ad infinitum to give the result 

The unit flux has been used in an attempt to simplify the solution of the partial-derivative equation of Pick s second law. (a) Calculate, for a univalent ion, what current density this flux will cause. Is this current density achievable 

Estimate the electrophoretic velocity of a sodium ion in 0.01 M NaCl solution under an electric field of 0.1 V cm . The viscosity of the solution is 0.00895 poise. (Kim). 

Ohm s law is generally applicable to electrolytes in solution. A theory suggests that the observed current depends on the difference of two exponential terms. 

If a current of 10 A is passed between two electrodes, each of 10 cm, and one is depositing Cu metal from a CUSO4 solution, calculate the thickness of this deposit after 6 h. 

---

Using a procedure similar to the derivation of Equation (4.13) the working equations of the U-V-P scheme for steady-state Stokes flow in a polar (r, 6) coordinate system are obtained on the basis of Equations (4.5) and (4.6) as... 

Finding the Estimated Slope andy-Intercept The derivation of equations for calculating the estimated slope and y-intercept can be found in standard statistical texts and is not developed here. The resulting equation for the slope is given as... 

The derivation of equation 7.31 is considered in problem 33 in the end-of-chapter problem set. 

In the derivation of equations 24—26 (60) it is assumed that the cylinder is made of a material which is isotropic and initially stress-free, the temperature does not vary along the length of the cylinder, and that the effect of temperature on the coefficient of thermal expansion and Young s modulus maybe neglected. Furthermore, it is assumed that the temperatures everywhere in the cylinder are low enough for there to be no relaxation of the stresses as a result of creep. 

Figure 4.28 shows the derivation of equation 4.38 from the algebra of random variables. (Note, this is exaetly the same approaeh deseribed in Appendix VIII to find the probability of interferenee of two-dimensional variables.)... 

Of course it should always be remembered that the solutions obtained in this way are only approximate since the assumptions regarding linearity of relationships in the derivation of equation (2.64) are inapplicable as the stress levels increase. Also in most cases recovery occurs more quickly than is predicted by assuming it is a reversal of creep. Nevertheless this approach does give a useful approximation to the strains resulting from complex stress systems and as stated earlier the results are sufficiently accurate for most practical purposes. 

The importance of the Gibbs free energy and the chemical potential is very great in chemical thermodynamics. Any thermodynamic discussion of chemical equilibria involves the properties of these quantities. It is therefore worthwhile considering the derivation of equation 20.180 in some detail, since it forms a prime link between the thermodynamics of a reaction (AG and AG ) and its chemistry. 

Differentiation and substitution into equations (7.93) and (7.94) in a manner similar to that described in the derivation of equations (5.41) and (5.42) gives... 

While true, this result is not helpful. The derivation of Equation (1.6) used the entire reactor as the control volume and produced a result containing the average reaction rate, In piston flow, a varies with z so that the local reaction rate also varies with z, and there is no simple way of calculating a-Equation (1.6) is an overall balance applicable to the entire system. It is also called an integral balance. It just states that if more of a component leaves the reactor than entered it, then the difference had to have been formed inside the reactor. 

Appendix A Derivation of Equations for Polymer Concentration This Appendix shows the derivation of Equations (1) and (3) in the text. 

FIGURE 18.4 Concerning the derivation of equations for nonuniform current distribution (a) in a flat electrode (b) in a cylindrical pore. 

The quantities RME and (Sheldon environmental impact factor or -factor based on mass) ° are related by a simple expression given by equation (4.4) which allows easy calculation of either parameter once one of them is known. It is often simpler to determine Em first and then to use equation (4.4) to calculate RME. The derivation of equation (4.4) is linked to that of equation (4.1) and is also given in Appendix B. 

The derivation of equation 12.8 is given by Kern (1950). The equation for a 1 shell 2 tube pass exchanger can be used for any exchanger with an even number... 

The derivation of Equations 15.62 to 15.64 is as follows1. From Bowman2, the value of P over Nshells number of 1-2 shells in series, Pn-2n, can be related to the value of P for each 1-2 shell, Pi 2, according to ... 

The simplest method of estimating Q)L from experimental tracer data is based on an evaluation of the slope of the F(t) curve at t =7. If we evaluate the derivative of equation 11.1.33 at this point,... 

We wish to develop an analytic expression for this situation. To do so, we will follow the same steps used in the standard development, but use the rigorously correct equation (i.e., equation 42-37 instead of the approximate equation previously used. The steps are the standard ones used for finding a minimum (or maximum) of a function take the derivative of equation 42-37, then set that derivative equal to zero. Since the derivative of interest is the derivative with respect to T, in preparation for this we reorganize equation 42-37 as follows we substitute equation 41-6 (reference [2], reorganized to Es = TEX (since Er is a constant) into equation 42-37 this enables us to eliminate Es from the equation ... 

To continue, therefore, we proceed in the usual manner for finding a minimum we take the derivative of equation 48-108 and then set the derivative equal to zero. Since equation 48-108 is complicated, and the derivative more so, we will generate the derivative in several steps ... 

The time at which the concentration of the adsorbate is maximum, t is found by setting the derivative of Equation (2) equal to zero. The solution is given by... 

The development of a quantitative theory of a free-radical copolymerization implies the derivation of equations for the rate of the monomers depletion and the statistical characteristics of the chemical structure of macromolecules present in the reaction system at the given conversion p of monomers. Elaborating such a theory one should take into account a highly important peculiarity inherent to any free-radical copolymerization. This peculiarity is that the characteristic time of a macroradical life is appreciably less than the time of the process duration. Consequently, its products represent definitely... 

The kinetic analysis of the mechanism 6a-6e,2 is more complicated than that of the mechanism 4a-4e because of the external reaction 6e but nevertheless is feasible using the steady state approximation. By a procedure similar to the derivation of equation 5 the following equation can be derived ... 

These boundary conditions and the derivation of equation 20.2-2 were first obtained by Langmuir (1908). See discussion by Weller (1994). 

The evolution of a concentration profile depends on how the velocity changes with concentration. It is left to the reader to show by taking the derivative of equation (8.3.17) at constant

trace element i, the last expression is the derivative of its... 

---