Question

Suppose $50$ bacteria are placed in a flask containing nutrients for the bacteria so that they can multiply.
A study at ${35}^{o}C$ gave the following results:
$\begin{array}{cccccc}Time\left(minute\right)& 0& 15& 30& 45& 60\\ Numberofbacteria& 50& 100& 200& 400& 800\end{array}$
Show that the rate of production of bacteria is of first-order. How many bacteria will be there after $3$ hours?

Answer 1

For bacterial growth,
$k=-\frac{2.303}{t}{\log }_{10}\left(\frac{a}{a+x}\right)$
At $15$ minutes: $k=-\frac{2.303}{15}{\log }_{10}\left(\frac{50}{100}\right)$
$=0.0462{min}^{-1}$
At $30$ minutes: $k=-\frac{2.303}{30}{\log }_{10}\left(\frac{50}{200}\right)$
$=0.0462{min}^{-1}$
At $45$ minutes: $k=-\frac{2.303}{45}{\log }_{10}\left(\frac{50}{400}\right)=0.0462{min}^{-1}$
Same values of rate constants show that the process corresponds to first order.
Let there be '$n$' bacteria after $3$ hrs.
$k=-\frac{2.303}{t}{\log }_{10}\left(\frac{a}{a+x}\right)$
$0.0462=-\frac{2.303}{180}{\log }_{10}\frac{50}{n}$
$n=2.04\times {10}^5$