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Question: The inversion of cane sugar proceeds with the half-life of 600 minutes at pH=5 for any concentration...

The inversion of cane sugar proceeds with the half-life of 600 minutes at pH=5 for any concentration of sugar. However at pH=6, the half life changes to 60 minute. The rate law expression for sugar inversion can be written as: (if it is first order with respect to sugar):
A.r=k[sugar]2[H+]0r = k{\left[ {sugar} \right]^2}{\left[ {{H^ + }} \right]^0}
B.r=k[sugar]1[H+]0r = k{\left[ {sugar} \right]^1}{\left[ {{H^ + }} \right]^0}
C.r=k[sugar]2[H+]1r = k{\left[ {sugar} \right]^2}{\left[ {{H^ + }} \right]^1}
D.r=k[sugar]1[H+]−1r = k{\left[ {sugar} \right]^1}{\left[ {{H^ + }} \right]^{ - 1}}

Explanation

Solution

We have to know that the first order reaction is the reaction that goes through at a rate that is dependent straightly on concentration of one reactant. The half-life of reaction in which the initial population is reduced by half of its original value and half-life of a first order reactant is constant.

Complete step by step answer:
We know that order of the chemical reaction is the sum of the power of concentration of reactants in the rate law expression.
For a reaction, xA+yB→PxA + yB\xrightarrow{{}}P
Rate=k[A]x[B]yk{\left[ A \right]^x}{\left[ B \right]^y}
By summing up the power of the concentration of reactants we give the order of the reaction.
Order=x+yx + y
When the pH of the solution is raised by one unit from five to six the concentration of the hydrogen ion decreases to one-tenth. Therefore, the half-life period also changes to one-tenth. The reaction rate increases ten times and the order of the reaction with respect to the concentration of the hydrogen ion  - 1{\text{ - 1}}. Therefore, the formula for the rate of the reaction is written as,
r=k[sugar]1[H]−1r = k{\left[ {sugar} \right]^1}{\left[ H \right]^{ - 1}}
Therefore, the option (D) is correct.

Note:
We have to remember that the rate constant of the forward reaction divided by the rate constant of the reverse reaction gives the equilibrium constant of a reaction. The rate constant of a first order reaction contains time as its unit and not concentration unit. From this we could relate that for a first order reaction, the numerical value of k is independent of the unit that expresses the concentration. Even if we change the concentration unit, the numerical value of k for a first order reaction would not change.