Answer:
![r = k [A]^{2}[B]^{2}](https://tex.z-dn.net/?f=r%20%3D%20k%20%5BA%5D%5E%7B2%7D%5BB%5D%5E%7B2%7D)
Explanation:
A + B + C ⟶ D
![\text{The rate law is } r = k [A]^{m}[B]^{n}[C]^{o}](https://tex.z-dn.net/?f=%5Ctext%7BThe%20rate%20law%20is%20%7D%20r%20%3D%20k%20%5BA%5D%5E%7Bm%7D%5BB%5D%5E%7Bn%7D%5BC%5D%5E%7Bo%7D)
Our problem is to determine the values of m, n, and o.
We use the method of initial rates to determine the order of reaction with respect to a component.
(a) Order with respect to A
We must find a pair of experiments in which [A] changes, but [B] and C do not.
They would be Experiments 1 and 2.
[B] and [C] are constant, so only [A] is changing.
![\begin{array}{rcl}\dfrac{r_{2}}{r_{1}} & = & \dfrac{ k[A]_{2}^{m}}{ k[A]_{1}^{m}}\\\\\dfrac{2.50\times 10^{-2}}{6.25\times 10^{-3}} & = & \dfrac{0.100^{m}}{0.0500^{m}}\\\\4.00 & = & 2.00^{m}\\m & = & \mathbf{2}\\\end{array}\\\text{The reaction is 2nd order with respect to A}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Brcl%7D%5Cdfrac%7Br_%7B2%7D%7D%7Br_%7B1%7D%7D%20%26%20%3D%20%26%20%5Cdfrac%7B%20k%5BA%5D_%7B2%7D%5E%7Bm%7D%7D%7B%20k%5BA%5D_%7B1%7D%5E%7Bm%7D%7D%5C%5C%5C%5C%5Cdfrac%7B2.50%5Ctimes%2010%5E%7B-2%7D%7D%7B6.25%5Ctimes%2010%5E%7B-3%7D%7D%20%26%20%3D%20%26%20%5Cdfrac%7B0.100%5E%7Bm%7D%7D%7B0.0500%5E%7Bm%7D%7D%5C%5C%5C%5C4.00%20%26%20%3D%20%26%202.00%5E%7Bm%7D%5C%5Cm%20%26%20%3D%20%26%20%5Cmathbf%7B2%7D%5C%5C%5Cend%7Barray%7D%5C%5C%5Ctext%7BThe%20reaction%20is%202nd%20order%20with%20respect%20to%20A%7D)
(b) Order with respect to B
We must find a pair of experiments in which [B] changes, but [A] and [C] do not. There are none.
They would be Experiments 2 and 3.
[A] and [C] are constant, so only [B] is changing.
![\begin{array}{rcl}\dfrac{r_{3}}{r_{2}} & = & \dfrac{ k[B]_{3}^{n}}{ k[B]_{2}^{n}}\\\\\dfrac{1.00\times 10^{-1}}{2.50\times 10^{-2}} & = & \dfrac{0.100^{n}}{0.0500^{n}}\\\\4.00 & = & 2.00^{n}\\n & = & \mathbf{2}\\\end{array}\\\text{The reaction is 2nd order with respect to B}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Brcl%7D%5Cdfrac%7Br_%7B3%7D%7D%7Br_%7B2%7D%7D%20%26%20%3D%20%26%20%5Cdfrac%7B%20k%5BB%5D_%7B3%7D%5E%7Bn%7D%7D%7B%20k%5BB%5D_%7B2%7D%5E%7Bn%7D%7D%5C%5C%5C%5C%5Cdfrac%7B1.00%5Ctimes%2010%5E%7B-1%7D%7D%7B2.50%5Ctimes%2010%5E%7B-2%7D%7D%20%26%20%3D%20%26%20%5Cdfrac%7B0.100%5E%7Bn%7D%7D%7B0.0500%5E%7Bn%7D%7D%5C%5C%5C%5C4.00%20%26%20%3D%20%26%202.00%5E%7Bn%7D%5C%5Cn%20%26%20%3D%20%26%20%5Cmathbf%7B2%7D%5C%5C%5Cend%7Barray%7D%5C%5C%5Ctext%7BThe%20reaction%20is%202nd%20order%20with%20respect%20to%20B%7D)
(c) Order with respect to C
We must find a pair of experiments in which [C] changes, but [A] and [B] do not.
They would be Experiments 1 and 4.
[A] and [B] are constant, so only [C] is changing.
![\begin{array}{rcl}\dfrac{r_{4}}{r_{1}} & = & \dfrac{ k[C]_{4}^{o}}{ k[C]_{1}^{o}}\\\\\dfrac{6.25\times 10^{-3}}{6.25\times 10^{-3}} & = & \dfrac{0.0200^{o}}{0.0100^{o}}\\\\1.00 & = & 2.00^{o}\\o & = & \mathbf{0}\\\end{array}\\\text{The reaction is zero order with respect to C.}\\\text{The rate law is } r = k [A]^{2}[B]^{2}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Brcl%7D%5Cdfrac%7Br_%7B4%7D%7D%7Br_%7B1%7D%7D%20%26%20%3D%20%26%20%5Cdfrac%7B%20k%5BC%5D_%7B4%7D%5E%7Bo%7D%7D%7B%20k%5BC%5D_%7B1%7D%5E%7Bo%7D%7D%5C%5C%5C%5C%5Cdfrac%7B6.25%5Ctimes%2010%5E%7B-3%7D%7D%7B6.25%5Ctimes%2010%5E%7B-3%7D%7D%20%26%20%3D%20%26%20%5Cdfrac%7B0.0200%5E%7Bo%7D%7D%7B0.0100%5E%7Bo%7D%7D%5C%5C%5C%5C1.00%20%26%20%3D%20%26%202.00%5E%7Bo%7D%5C%5Co%20%26%20%3D%20%26%20%5Cmathbf%7B0%7D%5C%5C%5Cend%7Barray%7D%5C%5C%5Ctext%7BThe%20reaction%20is%20zero%20order%20with%20respect%20to%20C.%7D%5C%5C%5Ctext%7BThe%20rate%20law%20is%20%7D%20r%20%3D%20k%20%5BA%5D%5E%7B2%7D%5BB%5D%5E%7B2%7D)