Question

At 298K, k for the reaction that causes milk to sour is 2.70 x 10^-4 s^-1, and it takes 60.0 hours for the milk to spoil. How long does it take for milk to spoil at 275K, assuming that the concentrations of all pertinent species are constant at both temperatures? The activation energy for the reaction is 24.7kJ/mol

A) 26.0 hours

B) 117 hours

C) 270. hours

D) 17.4 hours

E) 138 hours

Answer 1

The given values are:

$k_{1} =2.7 \times 10^{-4}s^{-1}$

$T_{1} =298 K$

$T_{2} =275 K$

$Ea=24.7\frac{kJ}{mol}$

$k_{2} = ?$

Now, we can estimate k₂ using the following equation:

$ln\left ( \frac{k_1}{k_2} \right )=\left ( \frac{1}{T_1}-\frac{1}{T_2} \right )\cdot\frac{-Ea}{R}$

$ln(2.7\times10^{-4}s^{-1})-ln(k_2)=\left ( \frac{1}{298K}-\frac{1}{275K} \right )\cdot \frac{-24.7\times 10^{3}\frac{J}{mol}}{8.314\frac{J}{K\cdot mol}}$

$k_2=e^{-\left [\left ( \frac{1}{298K}-\frac{1}{275K} \right )\cdot \frac{-24.7\times 10^{3}\frac{J}{mol}}{8.314\frac{J}{K\cdot mol}}-ln(2.7\times10^{-4}s^{-1}) }\right ]}=1.17 \times10^{-4}s^{-1}$

Then,

$1.17 \times10^{-4}s^{-1}\times \frac{60h}{2.70\times 10^{-4}s^{-1}}=26.0h$

There fore, the correct answer is A