The inversion of cane sugar proceeds with $$half-life$$ of $$500$$ min at pH $$=$$ $$5$$ for any concentration of sugar. However, if pH $$=$$ $$6$$, the $$half-life$$ changes to $$50$$ min.Expression for the rate law for sugar inversion is

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At, pH $$=$$ $$5$$, the $$half-life$$ is $$500$$ min for all concentration of sugar, i.e. $$t1/2$$∝[ sugar ]. Thus, the reaction is of first order with respect to sugar.Now,           rate $$=k$$[ sugar ]$$1H+mAlso$$ for         $$H+t1/2$$∝$$H+1$$−m                 ∴ $$500$$∝$$10$$−$$51$$−m              …….(i)                          $$50$$∝$$10$$−$$61$$−m                …….(ii)∴ $$10=(10)(1$$−$$m)$$ or $$(1$$−$$m)=1$$          $$m=1$$−$$1=0$$     rate $$=k$$[ sugar ]$$1H+0$$

Experiment	$[X] mol L^{- 1}$	$[Y] mol L^{- 1}$	Initial rate $(mol L^{- 1} s^{- 1})$
I	$2.5 \times 10^{- 4}$	$6 \times 10^{- 5}$	$5 \times 10^{- 4}$
II	$5 \times 10^{- 4}$	$12 \times 10^{- 5}$	$4 \times 10^{- 3}$
III	$1 \times 10^{- 3}$	$12 \times 10^{- 5}$	$1.6 \times 10^{- 2}$

The inversion of cane sugar proceeds with half-life of 500 min at pH = 5 for any concentration of sugar. However, if pH = 6, the half-life changes to 50 min.Expression for the rate law for sugar inversion is

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