 In this module we will talk about again the batch culture kinetics which is the continuation of the previous module as I already told you that there is a relationship linear relationship or we can say that the biomass is directly proportioned with the time in fermentation process. So if the x is directly proportioned to t and then the change in biomass is directly proportioned with the change in time. So if there is a biomass at time t and the x naught is the biomass which we use as in in inoculation at the time of inoculation as an inoculum. So n is the number of generations so we can write the change of this in biomass with respect to that so that is the x naught is 2 powered by t over td so td here is the doubling time. So as in the previous slide you see that the dx change in biomass is directly related directly proportioned to the time when we change this proportion with the constant then we can write here the dx is equal to mu that is known as specific growth constant and x is the biomass that is the you can say that associated with the biomass this whole growth and dt is the change in time. So the change in biomass with respect to dt dx over dt that is equal to the mu dot x so here you can easily see that x is the concentration of the microbial biomass t is the time that mostly we noted in term of hours and then mu is the specific growth rate per unit cell mass mean in grams in kilogram in milligram etc. So that should be a very specific unit which we use for that. So specific growth rate that how much cell mass produce in unit time in a unit scale. So when this previous equation that dx over dt is equal to mu dot x on applying this equation the integration equation so this equation can be transformed into this equation x t is equal to x naught e mu t but here we can say here that x naught is the original biomass concentration which we add at the time of inoculum x t is the biomass concentration after time t or we can say that when we estimate after time t interval then e is the base of natural logarithm. So if this equation can further transformed into the natural log so you can see here what will be the equation. So you can say that if we transform this equation into the natural log so then you can easily understand here that by the logarithmic principle those figures which have the multiplication that become in sum up and those who as the division they come in a minus and those who are in a power then they are in a multiplication. So you can easily see here so here when there is a log of nature of e that is automatically understand that is equal to 1. So this equation is known as ln x t ln x naught plus mu t so that is the very fit into the linear regression equation you know very familiar equation which linear regression equation y is equal to a plus b x where the y is the vertical axis mostly we have a dependent variable on y and then is x that is known as horizontal axis mostly we plot independent variable on that a is the intercept and b is the slope of the graph. So if the ln x is taken along the y axis and t time on x axis then there is a plot. So in this here you can see that if we plot between t and then ln x t then there is a straight line so intercept will be the a and the slope of this will be the b so by this we can easily calculate the mu so the mu become the b that the slope of this line will be the actually the specific growth rate constant but in term of as I know already in previous module we talk about the doubling time by this equation we can easily calculate the doubling time just by putting the value x t is equal to 2 x naught. Simply you can see here that when we put this 2 x naught and then there is a equation of this and we already know that ln x 2 is 0.693 so if the doubling time and we know the specific growth rate constant of any organism then we can easily calculate the doubling time by using this equation.