Which expression represents the integrated rate law for a second-order reaction?

Which expression represents the integrated rate law for a second-order reaction?

• A. [A]t = -kt + [A0]
• B. [A]t = [A0] / (1 + k t)
• C. 1/[A]t = kt + 1/[A0]
• D. ln[A]t = -kt + ln[A0]

Answer: (C)

For a reaction second order in A, the rate is -d[A]/dt = k[A]^2. Integrating d[A]/[A]^2 = k dt gives -1/[A] = kt + C. applying the initial condition t = 0, [A] = [A]0 sets C = -1/[A]0, leading to 1/[A] = kt + 1/[A]0. This is the standard integrated rate law for a second-order reaction, and it can be rearranged to [A]t = [A]0 / (1 + k[A]0 t). The other forms correspond to different orders or omit the [A]0 factor inside the second-order expression, so they don’t match the correct second-order integrated form.

For a reaction second order in A, the rate is -d[A]/dt = k[A]^2. Integrating d[A]/[A]^2 = k dt gives -1/[A] = kt + C. applying the initial condition t = 0, [A] = [A]0 sets C = -1/[A]0, leading to 1/[A] = kt + 1/[A]0. This is the standard integrated rate law for a second-order reaction, and it can be rearranged to [A]t = [A]0 / (1 + k[A]0 t). The other forms correspond to different orders or omit the [A]0 factor inside the second-order expression, so they don’t match the correct second-order integrated form.