I came across a question worded as follows- "The inversion of cane sugar proceeds with half life of 50 minutes at pH=5 at any concentration of sugar. However, if the pH=6, the half life changes to 500 minutes at any concentration of sugar, the rate law expression for inversion of sugar can be written as?"
"at any concentration of sugar" suggests time period is independent of concentration of sugar and therefore of order 1 with respect to sugar. Reducing H+ concentration by 10 increases half life by 10, so half life is inversely proportional to [H+], also,as a rule- half life is inversely proportional to [a]^n-1 for any order n and concentration of reactant [a], comparing exponents of a with [H+]- we get n=2, but this is clearly not the correct way to go about it since the rate law is actually given by k(sugar)(H+) while we get k(sugar)(h+)^2; that is order 1 and 2 w.r.t to reactants- So my question is what is the correct method to find rate law expression in such conditions?