Theoretical versus actual values

Figure 3.60 Plots of E° 2 (left ordinate, solid circles) and / im (right ordinate, open circles) versus pH for 02 reduction obtained with a Co2FTF 4-2, 1N-H coated rotating graphite disk electrode in buffered 02-saturated solutions as a function of pH (see original reference for actual compositions). The dashed line at / iim = 1.3 mAcm-2 represents the theoretical value for the four-electron reduction of 02.

Now return to the problem of the predicted 15-30-s pumpdown time versus the hour or longer that we actually pump. The discrepancy between our theoretical value and experiment is that the walls of our dewar are not perfectly clean. 

Note that the calculated distribution shows some deviation from the measured values because of the small departure of the actual sample from the theoretical coefficient of variation (i.e., 47.5 versus 52 percent). 

Look at Figure 14-3, which includes a graph of volume versus temperature for a gas sample kept at a constant pressure. Note that the resulting plot is a straight line. Note also that you can predict the temperature at which the volume will reach a value of zero liters by extrapolating the line at temperatures below which values were actually measured. The temperature that corresponds to zero volume is —273.15°C, or 0 on the kelvin (K) temperature scale. This temperature is referred to as absolute zero, and it is the lowest possible theoretical temperature. Theoretically, at absolute zero, the kinetic energy of particles is zero, so all motion of gas particles at that point ceases. 

Optimum reflux can be in principle determined by taking into account investment and operation costs. The column cost is roughly proportional with the number of stages N. The cost of utilities depends on the maximum vapour flow in the column, proportional with R+l. Consequently, the optimisation of the function N(R+l) versus R can supply a good estimation of actual optimum reflux, as shown in the Fig. 9.35-b. In practice, the optimal reflux takes values between 1.2-1.5 Rmm- Alternatively, the actual number of theoretical stages may be considered twice the minimum one. 

The dependence of X(T)/X(0) on T/Tq as calculated from Eqs. (12) and (13) is shown as the full curve in Figs. 4 and 5. For weak coupling this dependence is practically independent of NqV. For example, the values of X(T)/X(0) for NqV as large as 0.6 differ from those for NqV = 0.1 only in the third decimal place for most values of T/Tc. Thus X(T)/X(0) is essentially a universal function of T/Tc. In Fig. 4 this dependence is compared with the experimental dependence measured by Cooper et al. (14) for YBa2Cu307-6 while Fig. 5 shows the comparison with the data of Aeppll et al. (1 ) obtained for Lai.85Sro.15Cu0n. The agreement with the extensive data of Cooper et al. is particularly excellent. In Fig. 6 we have plotted the calculated values of X(T)/X(0) versus (T/Tc) in order to check the experimentally reported low temperature T dependence of X(T)-X(0). It is seen that the theoretical dependence is Indeed consistent with a T dependence, provided that T is not too low. We note that in the actual T/Tc 0 limit, Eqs. (12) and (13) predict... 

T , See melt temperature, theorem See mathematical theorem, theoretical versus actual properties See plastics, theoretical versus actual values of. theory A unifying principle that explains a body of facts and tbe laws that are based on those facts. See atomic theory design design-failure theory design theory and strength of material empirical engineering approach versus practical approach mathematical theorem mixing theory. 

It can be seen that the plot of E(9) versus 9 shifts away from the ideal CSTR plot and moves towards the ideal PFR plot as Pe increases. Given a plot of E(9) versus 9 obtained from the tracer experiment, the value of the parameter Pe is estimated as the value for which the experimental plot fits well with the theoretical plot of E(9) versus 9 shown in Figure 3.62. But one cannot derive a theoretical expression for E(9) as it is not possible to obtain an analytical solution to the model Equation 3.329 with Danckwarts boundary conditions (3.331) and (3.334). Flowever, an explicit equation relating the variance and mean 9 of the RTD to the Peclet number Pe has been derived using the method of moments without actually solving the model equation. This equation... 

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