The rate of production of ethanol in a fermentor is directly...

The rate of production of ethanol in a fermentor is directly proportional to the number of cells of yeast present. At a cell mass of $14 \mathrm{~g}$ yeast/liter, ethanol production rate is $18 \mathrm{~g}$ ethanol/ $(\mathrm{L} \cong \mathrm{h}$ ). If a batch fermentor is inoculated with $0.1 \mathrm{~g}$ of yeast $/ \mathrm{L}$, and the generation time of the yeast is 1.5 hours, what will be the ethanol production after 10 hours of fermentation. Assume no alcohol-induced inhibition of yeast growth and that yeast doubles in cell mass with each generation time.

Key Concepts

Direct Proportionality

This concept captures the idea that one variable changes at the same rate as another. In the context of fermentation, the rate of ethanol production is directly proportional to the yeast cell mass, meaning that if the cell mass increases, the production rate increases at a constant multiplier. This relationship helps in determining unknown rates if a reference point is available.

Exponential Growth

Exponential growth describes a process where the increase in quantity is proportional to the current amount, leading to growth that accelerates over time. For microorganisms like yeast, under ideal conditions without inhibition, the cell population doubles at regular intervals, causing an exponential increase in biomass.

Generation Time

Generation time is the period required for a microbial population to double in number. It is a key parameter in exponential growth calculations, enabling the modeling of how quickly a yeast culture increases in cell mass, which in turn affects production rates in a biological process such as fermentation.

Integration Over Time

When a rate of production depends on a quantity that changes over time, the total production is found by integrating the rate function over the time interval of interest. In fermentation, because the yeast grows exponentially, integrating the changing production rate yields the cumulative amount of ethanol produced over the process duration.

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