Question

A sample of milk splits after 60 min. at 300K and after 40 min at 400K when the population of lactobacillus acidophilus in it doubles . The activation energy (in kJ/mol) for this process is closest to

(Given, $R=8.3{\text{J mol}}^{-1}{K}^{-1}),\ln \left(\frac{2}{3}\right)=0.4,e^{-3}=4.0)$

Answer 1

The generation time can be utilized to get an indication of the rate ratio. Let the amount generated be (x).
$\text{Rate}=\frac{\text{Amount generated}}{\text{Time Taken}}$
${\text{Rate}}_{300K}=\frac{\left(x\right)}{60}$
${\text{Rate}}_{400K}=\frac{\left(x\right)}{40}$
$\frac{{\text{Rate}}_{300K}}{{\text{Rate}}_{400K}}=\frac{40}{60}$
For the same concentration (which is applicable here), the rate ratio can also be equaled to the ratio of rate constants.
$\ln \left[\frac{K_{\text{at 400K}}}{K_{\text{at 300K}}}\right]=\frac{E_a}{R}[\frac{1}{T_1}-\frac{1}{T_2}]$
$\ln \frac{60}{40}=\frac{E_a}{8.3}[\frac{1}{300}-\frac{1}{400}]$
$E_a=0.4\times 8.3\times 1200=3984\text{J/mol}=3.98\text{kJ/mol}$