During the hydrolysis of $\mathrm{Fe}^{3+}$, $\mathrm{FeOH}^{2+}$ and $\mathrm{Fe}_{2}(\mathrm{OH})_{2}^{4+}$ are produced. To investigate the dissociation of $\mathrm{Fe}_{2}(\mathrm{OH})_{2}^{4+}$, the following reaction mechanism is proposed:  
$$
\mathrm{Fe}_{2}(\mathrm{OH})_{2}^{4+} \xrightleftharpoons[k_{-1}]{k_1} 2 \mathrm{FeOH}^{2+} \\  
\mathrm{Fe}_{2}(\mathrm{OH})_{2}^{4+} +\mathrm{H^+}\xrightleftharpoons[k_{-2}]{k_2} \mathrm{Fe}^{3+}+\mathrm{FeOH}^{2+}+\mathrm{H_2O} \\  
\mathrm{Fe}^{3+} + \mathrm{H_2O}\rightleftharpoons \mathrm{FeOH}^{2+} + \mathrm{H}^+,\quad \beta  
$$  
The first two equations are elementary reactions, and the third is an equilibrium equation. Derive the apparent hydrolysis equilibrium constant $K = \frac{c_1^2}{c_2}$ expressed in terms of $k_{1}$, $k_{-1}$, $k_{2}$, $k_{-2}$, $c_H$, and $\beta$. Here, $c_H$ is the hydrogen ion concentration in the aqueous solution, $c_1$ is the total concentration of mononuclear ions, and $c_2$ is the concentration of the dimer $\mathrm{Fe}_{2}(\mathrm{OH})_{2}^{4+}$. The concentration of water is already included in the constants and need not be explicitly stated.
$\mathrm{Fe}^{3+}$水解过程中产生$\mathrm{FeOH}^{2+}$和$\mathrm{Fe}_{2}(\mathrm{OH})_{2}^{4+}$, 为研究 $\mathrm{Fe}_{2}(\mathrm{OH})_{2}^{4+}$ 的解聚, 并提出了以下的反应机理:
$$
\mathrm{Fe}_{2}(\mathrm{OH})_{2}^{4+} \xrightleftharpoons[k_{-1}]{k_1} 2 \mathrm{FeOH}^{2+} \\
\mathrm{Fe}_{2}(\mathrm{OH})_{2}^{4+} +\mathrm{H^+}\xrightleftharpoons[k_{-2}]{k_2} \mathrm{Fe}^{3+}+\mathrm{FeOH}^{2+}+\mathrm{H_2O} \\
\mathrm{Fe}^{3+} + \mathrm{H_2O}\rightleftharpoons \mathrm{FeOH}^{2+} + \mathrm{H}^+,\quad \beta 
$$
前两个方程为基元反应, 第三个为平衡方程, 推导用$k_{1}$, $k_{-1}$, $k_{2}$, $k_{-2}$, $c_H$和$\beta$表示的表观水解平衡常数$K = \frac{c_1^2}{c_2}$. 其中$c_H$为水溶液中氢离子浓度, $c_1$为单核离子总浓度, $c_2$为二聚体$\mathrm{Fe}_{2}(\mathrm{OH})_{2}^{4+}$的浓度, 水的浓度已经包含在各个常数中, 不必标出.
$K=\frac{k_1 + k_2 c_H}{\frac{c_H}{c_H +\beta}\left( k_{-1} \frac{c_H}{c_H +\beta}  + k_{-2} \frac{\beta}{c_H +\beta} \right)}$
$K=\frac{k_1 + k_2 c_H}{ k_{-1} \frac{c_H}{c_H +\beta}  + k_{-2} \frac{\beta}{c_H +\beta} }$
$K=\frac{k_1 + k_2 c_H}{\frac{c_H}{c_H +\beta}\left( k_{-1}  + k_{-2}  \right)}$
$K = \frac{(k_1 + k_2 c_H)(c_H + \beta)^2}{k_{-1} c_H^2}$
$K = \frac{(k_1 + k_2 c_H)(c_H + \beta)^2}{k_{-2} c_H \beta}$
$K=\frac{k_1 + k_2 c_H}{\frac{\beta}{c_H +\beta}\left( k_{-1} \frac{c_H}{c_H +\beta}  + k_{-2} \frac{\beta}{c_H +\beta} \right)}$
$K =\frac{k_1 + k_2 c_H}{\frac{\beta}{c_H +\beta}\left( k_{-1} \frac{\beta}{c_H +\beta}  + k_{-2} \frac{c_H}{c_H +\beta} \right)}$
$K=\frac{k_1 + k_2 c_H}{\frac{c_H}{c_H +\beta}\left( k_{-1} \frac{c_H}{c_H +\beta}  + k_{-2} \frac{\beta}{c_H +\beta} \right)}$
$K=\frac{k_1 + k_2 c_H}{k_{-1} \frac{c_H}{c_H +\beta}  + k_{-2} \frac{\beta}{c_H +\beta} }$
$K=\frac{k_1 + k_2 c_H}{\frac{c_H}{c_H +\beta} \left( k_{-1}  + k_{-2}  \right)}$
$K = \frac{(k_1 + k_2 c_H)(c_H + \beta)^2}{k_{-1} c_H^2}$
$K = \frac{(k_1 + k_2 c_H)(c_H + \beta)^2}{k_{-2} c_H \beta}$
$K=\frac{k_1 + k_2 c_H}{\frac{\beta}{c_H +\beta}\left( k_{-1} \frac{c_H}{c_H +\beta}  + k_{-2} \frac{\beta}{c_H +\beta} \right)}$
$K =\frac{k_1 + k_2 c_H}{\frac{\beta}{c_H +\beta}\left( k_{-1} \frac{\beta}{c_H +\beta}  + k_{-2} \frac{c_H}{c_H +\beta} \right)}$
The rate of the first depolymerization reaction is $v_{d1}=k_{1} c_2$  

<Checkpoint>[$v_{d1}=k_{1} c_2$](1)</Checkpoint>  

The rate of the second depolymerization reaction is $v_{d2}=k_{2} c_2 c_H$  

<Checkpoint>[$v_{d2}=k_{2} c_2 c_H$](1)</Checkpoint>  

The total depolymerization rate is $v_d = v_{d1} + v_{d2} = \left( k_1 + k_2 c_H \right) c_2$  

<Checkpoint>[$v_d = \left( k_1 + k_2 c_H \right) c_2$](0.5)</Checkpoint>  

The equilibrium concentrations of mononuclear species are derived using the distribution coefficient:  

$$
c\left( \mathrm{Fe}^{3+} \right) = c_1 \frac{c_H}{c_H +\beta} \\
c\left(\mathrm{FeOH}^{2+}\right) = c_1 \frac{\beta}{c_H +\beta}
$$  

<Checkpoint>[$c\left( \mathrm{Fe}^{3+} \right) = c_1 \frac{c_H}{c_H +\beta}$](0.5)</Checkpoint>  

<Checkpoint>[$c\left(\mathrm{FeOH}^{2+}\right) = c_1 \frac{\beta}{c_H +\beta}$](0.5)</Checkpoint>  

The rate of the first polymerization reaction is $v_{a1} = k_{-1} c\left( \mathrm{FeOH}^{2+} \right)^2 = k_{-1}c_1^2 \left( \frac{\beta}{c_H +\beta} \right)^2$  

<Checkpoint>[$v_{a1} = k_{-1}c_1^2 \left( \frac{\beta}{c_H +\beta} \right)^2$](1)</Checkpoint>  

The rate of the second polymerization reaction is $v_{a2} = k_{-2} c\left( \mathrm{Fe}^{3+} \right) c\left(\mathrm{FeOH}^{2+}\right) = k_{-2} c_1^2 \frac{c_H}{c_H +\beta}\frac{\beta}{c_H +\beta}$  

<Checkpoint>[$v_{a2} = k_{-2} c_1^2 \frac{c_H}{c_H +\beta}\frac{\beta}{c_H +\beta}$](1)</Checkpoint>  

The total polymerization rate is $v_a = v_{a1}+v_{a2} = \left( k_{-1} \left( \frac{\beta}{c_H +\beta} \right)^2 + k_{-2} \frac{c_H}{c_H +\beta}\frac{\beta}{c_H +\beta} \right) c_1^2$  

<Checkpoint>[$v_a = \left( k_{-1} \left( \frac{\beta}{c_H +\beta} \right)^2 + k_{-2} \frac{c_H}{c_H +\beta}\frac{\beta}{c_H +\beta} \right) c_1^2$](0.5)</Checkpoint>  

At equilibrium, the total depolymerization rate $v_d$ of the dimer equals the total polymerization rate $v_a$, i.e.,  

<Checkpoint>[$v_d=v_a$](1)</Checkpoint>  

$$
\left( k_1 + k_2 c_H \right) c_2 = \left( k_{-1} \left( \frac{\beta}{c_H +\beta} \right)^2 + k_{-2} \frac{c_H}{c_H +\beta}\frac{\beta}{c_H +\beta} \right) c_1^2
$$  

Rearranging gives  

$$
K =\frac{k_1 + k_2 c_H}{\frac{\beta}{c_H +\beta}\left( k_{-1} \frac{\beta}{c_H +\beta}  + k_{-2} \frac{c_H}{c_H +\beta} \right)}
$$  

<Checkpoint>[$ K =\frac{k_1 + k_2 c_H}{\frac{\beta}{c_H +\beta}\left( k_{-1} \frac{\beta}{c_H +\beta}  + k_{-2} \frac{c_H}{c_H +\beta} \right)} $](0.5)</Checkpoint>
第一个反应解聚的速率为 $v_{d1}=k_{1} c_2$

<Checkpoint>[$v_{d1}=k_{1} c_2$](1)</Checkpoint>

第二个反应解聚的速率为 $v_{d2}=k_{2} c_2 c_H $

<Checkpoint>[$v_{d2}=k_{2} c_2 c_H $](1)</Checkpoint>

总解聚速率 $v_d = v_{d1} + v_{d2} = \left( k_1 + k_2 c_H \right) c_2$

<Checkpoint>[$v_d = \left( k_1 + k_2 c_H \right) c_2$](0.5)</Checkpoint>

利用分布系数推导单核形态平衡浓度:

$$
c\left( \mathrm{Fe}^{3+} \right) = c_1 \frac{c_H}{c_H +\beta} \\
c\left(\mathrm{FeOH}^{2+}\right) = c_1 \frac{\beta}{c_H +\beta}
$$

<Checkpoint>[$c\left( \mathrm{Fe}^{3+} \right) = c_1 \frac{c_H}{c_H +\beta}$](0.5)</Checkpoint>

<Checkpoint>[$c\left(\mathrm{FeOH}^{2+}\right) = c_1 \frac{\beta}{c_H +\beta}$](0.5)</Checkpoint>  

第一个反应聚合速率为$v_{a1} = k_{-1} c\left( \mathrm{FeOH}^{2+} \right)^2 = k_{-1}c_1^2 \left( \frac{\beta}{c_H +\beta} \right)^2$

<Checkpoint>[$v_{a1} = k_{-1}c_1^2 \left( \frac{\beta}{c_H +\beta} \right)^2$](1)</Checkpoint>

第二个反应聚合速率为 $v_{a2} = k_{-2} c\left( \mathrm{Fe}^{3+} \right) c\left(\mathrm{FeOH}^{2+}\right) = k_{-2} c_1^2 \frac{c_H}{c_H +\beta}\frac{\beta}{c_H +\beta}$

<Checkpoint>[$v_{a2} = k_{-2} c_1^2 \frac{c_H}{c_H +\beta}\frac{\beta}{c_H +\beta}$](1)</Checkpoint>

总聚合速率 $v_a = v_{a1}+v_{a2} = \left( k_{-1} \left( \frac{\beta}{c_H +\beta} \right)^2 + k_{-2} \frac{c_H}{c_H +\beta}\frac{\beta}{c_H +\beta} \right) c_1^2$

<Checkpoint>[ $v_a = \left( k_{-1} \left( \frac{\beta}{c_H +\beta} \right)^2 + k_{-2} \frac{c_H}{c_H +\beta}\frac{\beta}{c_H +\beta} \right) c_1^2$](0.5)</Checkpoint>

平衡时, 二聚体总解聚反应速率$v_d$与总聚合反应速率$v_a$满足$v_d=v_a$, 即

<Checkpoint>[$v_d=v_a$](1)</Checkpoint>

$$
\left( k_1 + k_2 c_H \right) c_2 = \left( k_{-1} \left( \frac{\beta}{c_H +\beta} \right)^2 + k_{-2} \frac{c_H}{c_H +\beta}\frac{\beta}{c_H +\beta} \right) c_1^2
$$
变形得
$$
K =\frac{k_1 + k_2 c_H}{\frac{\beta}{c_H +\beta}\left( k_{-1} \frac{\beta}{c_H +\beta}  + k_{-2} \frac{c_H}{c_H +\beta} \right)}
$$

<Checkpoint>[$ K =\frac{k_1 + k_2 c_H}{\frac{\beta}{c_H +\beta}\left( k_{-1} \frac{\beta}{c_H +\beta}  + k_{-2} \frac{c_H}{c_H +\beta} \right)} $](0.5)</Checkpoint>