In the reaction a b c, doubling the concentration of a doubles the reaction rate and doubling the concentration of b does not affect the reaction rate. what is the rate law for this reaction?

The reaction must be a + b - - > c

Then you can predict a reaction rate, r o the type r = k * a^n * b^m

Given that the reaction rate is not affected by the concentration of b you can state that m = 0 and r = k * a^n.

Now given, that there is a proportional relation between the reaction rate and a (double a gives double rate), then n = 1 and r = k*a. You can verify that if you dobule a r also doubles.

Answer: r = k*a