The exponential graph

Take the exponential logarithm (Ine) of one set of the values, and plot them against the other set of values without alteration. Exponential logs are taken in a way similar to logsjo, except using the In key. 

The result will be a straight line if the original plot was exponential. Note that when deciding which set of values to take lUg of, it will almost certainly not be time. 

These exponential curves are frequently found in experiments on kinetics, measuring the speed of reactions. Here is an example. (To understand the chemistry behind these figures, see Module 5 we are just using a real life example to practise plotting a graph.) 

A radioactive isotope of phosphorus has a half-life of 14 days (that means it takes 14 days for the amount of to fall to half its original value, and another 14 days to fall to a quarter, and so on). We can plot a graph of amount of ip against time in days from these figures  

To convert this to a straight fine, take the lUc of the amounts of P remaining. To find the values of lue, follow the instructions as above, but using the Irtg button. This gives these results. 

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Once there is an appreciable amount of cells and they are growing very rapidly, the cell number exponentially increases. The optical cell density of a culture can then be easily detected that phase is known as the exponential growth phase. The rate of cell synthesis sharply increases the linear increase is shown in the semi-log graph with a constant slope representing a constant rate of cell population. At this stage carbon sources are utilised and products are formed. Finally, rapid utilisation of substrate and accumulation of products may lead to stationary phase where the cell density remains constant. In this phase, cell may start to die as the cell growth rate balances the death rate. It is well known that the biocatalytic activities of the cell may gradually decrease as they age, and finally autolysis may take place. The dead cells and cell metabolites in the fermentation broth may create... 

FIGURE 13.12 Thu ohange in concentration of the reactant in two first-order reactions plotted on the same graph When the first-order rate constant is large, the half-life of the reactant is short, because the exponential decay of the concentration of the reactant is then fast. 

Figure 15.2 Log H plotted versus temperature for copper a typical metal. The graph indicates the exponential decline of the hardness with increasing temperature, and the change in behavior at about half the melting point, Tm.

Draw and label the axes as shown. At a pH of 6, 7 and 8, [H+] is 1000, 100 and lOnmol.l-1, respectively. Plot these three points on the graph and join them with a smooth line to show the exponential relationship between the two variables. 

The argument of the exponential is x. Therefore, molar distributions of oligomeric species leached from cured resins will be presented on semilogarlthmlc graphs. 

Because of the exponential growth of the Indian companies, a logarithmic scale has been adopted for the Y-axis. The graph shows first of all the different growth patterns of the Western and Asian companies. Sales revenues of all three European companies had peaked in 2001, went through a trough between 2001 and 2005 and have not fully recovered. 

We illustrate the behavior of the line g(y, K) on the left-hand side of (4.87) and the right-hand-side exponential function f(y, K) of equation (4.87) in Figure 4.29, which includes five versions of the topmost graph in Figure 4.28 for varying feedback values K = 0,2,4,6, and 10 in varying colors. 

Fig. 4.29. Normalized integrated intensities (left) of substrate core levels in dependence on deposition time for the spectra shown in Fig. 4.26. The deposition rate is estimated to be 2nmmin 1. The lines in the left graph are obtained by curve fitting of the data to an exponential decay. The derived attenuation times are displayed in the right graph in dependence on electron kinetic energy together with theoretical energy-dependent escape depth calculated using the formula by Tanuma, Powell, and Penn [37] and using a y/ E law [38]...

In adsorption with redistribution in which there is no functional relationship between E and Q, it is assumed that the velocity of redistribution is larger than the velocity of adsorption. Consequently the various sections of the surface will be occupied in the order of decreasing heats of adsorption and at all times all sections will be involved in the adsorption irrespective of the magnitude of their activation energy. However, due to the exponential dependence of the velocity of adsorption upon the activation energy the character of the adsorption will be determined principally by the processes occurring upon sections with minimal values of E. These sections form on the graph p(E), as shown in Fig. 8, a relatively narrow vertical band the position of which determines the... 

The graph in Fig. 21.2 shows the behaviour of the function /(x) = e, which is known as exponential decay . Note that the value of the exponential becomes ever closer to zero but never actually reaches it. 

Figure 5 is a graph of these results showing that there is obviously a very good correlation between the two machines. A calculated correlation equation using all the individual test results, except for a few to be discussed later, gave the equation P = 0.88/V + 7.07 (r = 0.99). This is an excellent correlation, and the slope 0.88 is very close to the theoretical value of 0.857 calculated as explained eariier. This curve is shown as a solid line in Fig. 5. Theoretically, the relationship between the two testers should be a straight line. Experiments show that a slightly better correlation can be achieved by using the exponential formula for a curvilinear correlation.

Figure 5 is a graph of E versus Pe calculated according to Equation 22. There was some concern about the sensitivity of the calculated value for R to the choice of E. The asymptotic form for the boundary-layer modulus. Equation , was used for Eyi even though that form is truly exact at E>>E for a given wall Pecldt number. Locally, the exponential form for at E>E... 

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