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Answer:
The answer to the question is
If an enzyme increases the rate of the hydrolysis reaction by a factor of 1 million the activation barrier has to be 34.229 kJ lower when sucrose is in the active site of the enzyme
Explanation:
To solve the question we note that Arrhenius's equation for the rate constant is given by
[tex]k = Ae^{\frac{-E_{a} }{RT} }[/tex]
Therefore we have for the un-catalyzed reaction
[tex]k = Ae^{\frac{-108000 }{(8.314)(298)} } =[/tex] A ×1.171 ×10⁻¹⁹ s⁻¹
When an enzyme is added the rate of hydrolysis increases by a factor of 1 million therefore
new rate = initial rate × 1000000 = A ×1.171 ×10⁻¹⁹ s⁻¹ ×100000 = A ×1.171 ×10⁻¹³ s⁻¹
Hence we have the new given by
[tex]k_{new} = Ae^{\frac{-E_{a(new)} }{RT} }[/tex] since A, R and T remain constant
The above relation becomes
A ×1.171 ×10⁻¹³ s⁻¹ = [tex]Ae^{\frac{-E_{a(new)} }{(8.314)(298)} }[/tex] which is the same as
1.171 ×10⁻¹³ s⁻¹ = [tex]e^{\frac{-E_{a(new)} }{(8.314)(298)} }[/tex] that is ㏑(1.171 ×10⁻¹³ ) = [tex]{\frac{-E_{a(new)} }{(8.314)(298)} }[/tex]
or Eₐ = 29.8 × 8.314 × 298 = 73771.1 J = 73.7711 kJ
This means that the activation energy barrier needs to be 73.7711 kJ or (108000 J - 73771.1 J), 34228.9 J lower
The activation energy needs to be 34.229 kJ lower.
If an enzyme increases the rate of the hydrolysis reaction by a factor of 1 million the activation barrier has to be 34.229 kJ.
Arrhenius Equation:
[tex]K=Ae^{\frac{-E_a}{RT} }[/tex]
For uncatalysed reaction,
[tex]K=Ae^{\frac{-E_a}{RT}\\\\A*1.171 *10^{-19} s^{-1}[/tex]
When an enzyme is added the rate of hydrolysis increases by a factor of 1 million therefore
New rate = initial rate × 1000000 = A ×1.171 ×10⁻¹⁹ s⁻¹ ×100000 = A ×1.171 ×10⁻¹³ s⁻¹
A ×1.171 ×10⁻¹³ s⁻¹ = [tex]{\frac{-E_a(new)}{e^{(8.314)(298)}}[/tex] which is the same as
or Eₐ = 29.8 × 8.314 × 298 = 73771.1 J = 73.7711 kJ
This means that the activation energy barrier needs to be 73.7711 kJ or (108000 J - 73771.1 J), 34228.9 J lower
The activation energy needs to be 34.229 kJ lower.
Find more information about Activation energy here:
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