Graphs exponential

Personal Tutor For an online tutorial on exponential graphing, visit glencoe.com. 

Interpolation is a technique where a graph is used to determine data points between those at which you have taken measurements. Figure 11.34 is a graph of concentration of hydrogen peroxide against time. It is an exponential graph and the dotted construction lines are interpolation lines to prove that it is a first-order reaction (Chapter 16). The half-life of the reaction is approximately 25 seconds. 

FIG. 16-32 Exponentially modified Gaussian peak with Xq/Gq = 1.5. The graph also shows the definition of the peak asymmetry factor at 10 percent of peak height. 

The graphs of each of the species concentrations are plotted as a function of position along the tube z and time t. At the edges of the graphs for the concentrations of A and B we see the boundary and initial conditions. All values are unit or zero concentration as we had specified. As we move through time, we see the concentrations of both species drop monotonically at any position. Furthermore, if we take anytime slice, we see that the concentrations of reactants drop exponentially with position—as we know they should. At the longer times the profiles of... 

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 13.10 The graph of Ihe concentration of a reactant in a tirst-orrler reaction is an exponential decay, as shown here. The larger the rate constant, the faster the decav from the same initial concentration. 

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. 

Plot a graph of decay rate versus time and draw a smooth line through the data points. This curve is an example of an exponential decay curve. Label the graph Figure A. 

Figure 15.2 shows some typical hardness data for a typical metal (copper) as a function of temperature. It indicates that there are usually two regimes one above about half the melting temperature and one below. Both tend to be exponential declines, so they are linear on semi-logrithmic graphs. The temperature at which the break occurs is not strictly fixed, but varies from one metal to another, with the purity of a metal, with grain size, and so on. 

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.

If the product distribution follows a perfect ASF product distribution, then the graph in Figure 10.3 should fit an exponential function. However, this is not the case and can only be done if the production is separated in two product spectra, one being described by al and consisting mostly of <05, and the other one C1-C100+, with the products >05 being mostly from the spectrum described by a.2. 

FIGURE 10.4 Molar production with an exponential function fitted to the C >15 part of the graph and extrapolated down to Cl. 

Figure 5.19 Arrhenius plot of diffusion data, In D versus 1/T. The slope of the straight-line graph allows the activation energy of diffusion, a, to be determined, and the intercept at 1/T = 0 gives a value for the pre-exponential factor.

Calculate the activation energy Ea and pre-exponential factor A by plotting an Arrhenius graph. 

Figure 12 Damping coefficient yr 1(.0 = F/Av obtained from simulating two atomically flat surfaces separated by a simple fluid consisting of monomers at constant temperature and normal pressure. Different coverages were investigated. The numbers in the graph denote the ratio of atoms contained in the fluid Ng relative to the atoms contained per surface layer of one of the two confining walls Nw. The walls are (111) surfaces of face-centered-cubic solids. They are rotated by 90° with respect to each other in the incommensurate cases. Full circles represent data for which Nt-]/Nw is an integer. The arrow indicates the point at which the damping coefficients for commensurate walls increases exponentially.

Consider where a curve actually starts on the graph you are drawing. Does it begin at the origin or does it cross the y axis at some other point If so, is there a specific value at which it crosses the y axis and why is that the case Some curves do not come into contact with either axis, for example exponentials and some physiological autoregulation curves. If this is the case, then you should demonstrate this fact and be ready to explain why it is so. Consider what happens to the slope of a curve at its extremes. It is not uncommon for a curve to flatten out at high or low values, and you should indicate this if it is the case. 

The middle section of a curve may cross some important points as previously marked on the graph. Make sure that the curve does, in fact, cross these points rather than just come close to them or you lose the purpose of marking them on in the first place. Always try to think what the relationship between the two variables is. Is it a straight line, an exponential or otherwise and is your curve representing this accurately ... 

The semi-log transformation again makes the rise and fall of the graph linear. Note that this time there is no recirculation hump. As the fall on the initial plot was exponential, so the curve is transformed to a linear fall by plotting it as a semi-log. The AUC is still used in the calculations of cardiac output. 

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. 

---