Problem: The activation barrier for the hydrolysis of sucrose into glucose and fructose is 108 kJ/mol. If an enzyme increases the rate of the hydrolysis reaction by a factor of 1 million, how much lower does the activation barrier have to be when sucrose is in the active site of the enzyme? (Assume that the frequency factors for the catalyzed and uncatalyzed reactions are identical and that the temperature is 298 K.)

FREE Expert Solution
91% (347 ratings)
FREE Expert Solution

Without the enzyme, we have:

$\overline{){{\mathbf{k}}}_{{\mathbf{1}}}{\mathbf{=}}{\mathbf{A}}{\mathbf{·}}{{\mathbf{e}}}^{\frac{\mathbf{-}{{\mathbf{E}}_{\mathbf{a}}}_{\mathbf{1}}}{\mathbf{RT}}}}$

In the presence of the enzyme, we have:

$\overline{){{\mathbf{k}}}_{{\mathbf{2}}}{\mathbf{=}}{\mathbf{A}}{\mathbf{·}}{{\mathbf{e}}}^{\frac{\mathbf{-}{{\mathbf{E}}_{\mathbf{a}}}_{\mathbf{2}}}{\mathbf{RT}}}}$

Getting the ratio of k2/k1:

$\frac{{\mathbf{k}}_{\mathbf{2}}}{{\mathbf{k}}_{\mathbf{1}}}\mathbf{=}\frac{{{\mathbf{e}}}^{\frac{\mathbf{-}{{\mathbf{E}}_{\mathbf{a}}}_{\mathbf{2}}}{\mathbf{RT}}}}{{{\mathbf{e}}}^{\frac{\mathbf{-}{{\mathbf{E}}_{\mathbf{a}}}_{\mathbf{1}}}{\mathbf{RT}}}}\mathbf{=}{\mathbf{e}}^{\frac{{\mathbf{E}}_{\mathbf{a}\mathbf{1}}\mathbf{-}{\mathbf{E}}_{\mathbf{a}\mathbf{2}}}{\mathbf{RT}}}$

Getting ln of both sides:

$\mathbf{ln}\left(\frac{{k}_{2}}{{k}_{1}}\right)\mathbf{=}\mathbf{ln}\left({e}^{\frac{{E}_{a1}-{E}_{a2}}{\mathrm{RT}}}\right)$

$\left[\mathrm{ln}\left(\frac{{k}_{2}}{{k}_{1}}\right)=\frac{{E}_{a1}-{E}_{a2}}{\overline{)\mathrm{RT}}}\right]\overline{)\mathbf{RT}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\overline{){{\mathbf{E}}}_{\mathbf{a}\mathbf{1}}{\mathbf{-}}{{\mathbf{E}}}_{\mathbf{a}\mathbf{2}}{\mathbf{=}}{\mathbf{RTln}}\mathbf{\left(}\frac{{\mathbf{k}}_{\mathbf{2}}}{{\mathbf{k}}_{\mathbf{1}}}\mathbf{\right)}}$

91% (347 ratings)
Problem Details

The activation barrier for the hydrolysis of sucrose into glucose and fructose is 108 kJ/mol. If an enzyme increases the rate of the hydrolysis reaction by a factor of 1 million, how much lower does the activation barrier have to be when sucrose is in the active site of the enzyme? (Assume that the frequency factors for the catalyzed and uncatalyzed reactions are identical and that the temperature is 298 K.)

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Arrhenius Equation concept. You can view video lessons to learn Arrhenius Equation Or if you need more Arrhenius Equation practice, you can also practice Arrhenius Equation practice problems .

What is the difficulty of this problem?

Our tutors rated the difficulty of The activation barrier for the hydrolysis of sucrose into gl... as medium difficulty.

How long does this problem take to solve?

Our expert Chemistry tutor, Sabrina took 12 minutes to solve this problem. You can follow their steps in the video explanation above.

What professor is this problem relevant for?

Based on our data, we think this problem is relevant for Professor Kuebler's class at UCF.