Determine reaction order from data.
The method of initial rates finds the order in a reactant by changing only that reactant's concentration between two trials. Because the rate is proportional to concentration raised to the order, dividing one trial by the other cancels the rate constant and leaves the concentration ratio raised to the order — so taking logarithms of both ratios and dividing gives the order directly. Doubling a concentration and seeing the rate quadruple means second order.
Method of initial rates
order = ln(rate2 / rate1) / ln([A]2 / [A]1); k = rate1 / [A]1 ^ order
Because the derivation cancels the rate constant and every other concentration term. If a second reactant also changes, the ratio no longer isolates the order in A and the answer is meaningless.
Usually. Elementary steps give integer or half-integer orders, so an experimental 1.94 almost certainly means second order and the deviation is measurement error. Genuinely fractional orders occur in chain and enzyme-catalysed mechanisms and are informative rather than errors.
Because the rate is always in M/s while the concentration term carries a different power. Zero order gives k in M/s, first order in 1/s, and second order in 1/(M s).