Exponential growth rate constant

The exponential growth rate constant k is equal to the number of doublings per unit time. Thus, k is the reciprocal of the doubling time. It is easy to show that the number of bacteria present at time t will be given by the following equation. 

Here, W0 is the initial weight of tissue, and constants a and b are determined by fitting the equation with data. At early time intervals, this equation reduces to the well-known exponential growth rate ... 

Constant growth Rate, Exponential growth rate,... 

Asymptotic formulas for the growth-rate constant and the age distribution in exponential growth are given in the example following. 

As a subpostulate, Koch and Schaechter assume that the growth of individual cells of a culture in exponential growth is also exponential, with the same growth rate constant as that of the culture. That is... 

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... 

It has been suggested that fungi grow in filamentous form at an exponential rate with a constant specific growth rate (ji) until some substrate becomes growth limiting, according to the Monod equation 4 6... 

Thus, the exponential growth constant of the pressure oscillation is directly related to the acoustic admittance of the propellant. Hence, the acoustic admittance can be evaluated directly from the growth rate of the pressure amplitude. Ryan (R5) has also desired this espression on the basis of acoustic-energy considerations. 

Let us now reconsider our nucleation models of 4.4.1., specifically Models B, D and E. These are examples of phase-boundary controlled growth involving random nucleation. We now assume an exponential embryo formation law (see 4.4.7), with isotopic growth of nuclei in three dimensions and k2 as the rate constant. By suitable manipulation of 4.4.6.,... 

Chain-growth polymerizations are diffusion controlled in bulk polymerizations. This is expected to occur rapidly, even prior to network development in step-growth mechanisms. Traditionally, rate constants are expressed in terms of viscosity. In dilute solutions, viscosity is proportional to molecular weight to a power that lies between 0.6 and 0.8 (22). Melt viscosity is more complex (23) Below a critical value for the number of atoms per chain, viscosity correlates to the 1.75 power. Above this critical value, the power is nearly 3 4 for a number of thermoplastics at low shear rates. In thermosets, as the extent of conversion reaches gellation, the viscosity asymptotically increases. However, if network formation is restricted to tightly crosslinked, localized regions, viscosity may not be appreciably affected. In the current study, an exponential function of degree of polymerization was selected as a first estimate of the rate dependency on viscosity. 

The formation of the products [cyclic ketemimine (30) and/or triplet nitrene (33t)] was monitored at 380 nm. The decay of singlet phenylnitrene and the growth of the products are exponential and can be analyzed to yield an observed rate constant An Arrhenius treatment of these data (open circles), is presented in... 

When the characteristic time constants of decay of nitrene 16a and growth of triplet nitrene 20a and azepine 18a are significantly different a factor of 10) the kinetics of decay (a, c,e) and growth (b, d,f) could be fitted to simple mono exponential functions. The kinetics were analyzed to yield observed rate constants of decay (kdec) and growth (kgr). 

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