For a reaction that is second order with respect to a reactant, the rate law expression is Rate = k[A]², where [A] is the concentration of the reactant. When the concentration is doubled (2[A]), the new rate becomes k(2[A])² = 4k[A]², which is 4 times the original rate. When the concentration is reduced to half ([A]/2), the new rate becomes k([A]/2)² = k[A]²/4, which is one-fourth of the original rate. This demonstrates that for second order reactions, the rate changes by the square of the factor by which concentration changes.