Compound $\mathrm{A}$ can undergo dimerization to form $\mathrm{A}_{2}$, and this dimerization reaction constitutes the following equilibrium:

$$
2 \mathbf{A} \rightleftarrows \mathbf{A}_{2} \quad K_{c}^{\theta}
$$

The equilibrium constant of the reaction can be measured by polarimetry, which involves determining the specific optical rotation $[\alpha]_{\mathrm{D}}$ of the solution when the reaction reaches equilibrium and then calculating the equilibrium constant $K_{c}{ }^\circ$. It is known that when the reaction reaches equilibrium, $[\alpha]_{\mathrm{D}}$ satisfies the following relationship:

$[\alpha]_{\mathrm{D}}=x_{1}{ }^{2}[\alpha]_{1}+x_{2}{ }^{2}[\alpha]_{2}$

where $x_{1}$ and $x_{2}$ are the mole fractions of the corresponding species, and $[\alpha]_{1}$ and $[\alpha]_{2}$ are the specific optical rotations of the corresponding species when they exist independently.

1. The structure of compound $\mathbf{A}$ is shown in Figure 1. The task here is to deduce the structure of its dimer $\mathbf{A}_{2}$.

<MultiModal>![Fig. 1. The molecule is described by SMILES as: CC1(C)COC(=O)[C@@H]1O](/media/uploads/1c6370ec-eff9-4ddf-a523-6a560196d8e0.png)</MultiModal>

2. At room temperature, a solution of compound $\mathbf{A}$ with a total concentration of $8.0 \mathrm{mmol} / \mathrm{L}$ in $\mathrm{CCl}_{4}$ solvent reaches equilibrium, and the specific optical rotation of the system is measured as 

$[\alpha]_{\mathrm{D}}=-6~ \mathrm{deg}~ \mathrm{dm}^{-1} \mathrm{~g}^{-1} \mathrm{~mL}$. 

The specific optical rotations of compound $\mathbf{A}$ and dimer $\mathbf{A}_{2}$ when they exist independently are 

$[\alpha]_{1}=-1~ \mathrm{deg}~ \mathrm{dm}^{-1} \mathrm{~g}^{-1} \mathrm{~mL}$

$[\alpha]_{2}=-203$
$\mathrm{deg}~ \mathrm{dm}^{-1} \mathrm{~g}^{-1} \mathrm{~mL}$

Ignoring other possible species in the system, calculate the equilibrium constant $K_{c}^\circ$ for this dimerization reaction.

The following statements are given:
1. A covalent bond is formed during the dimerization of $\mathbf{A}$.
2. The dimer $\mathbf{A_2}$ does not contain a $C_2$ symmetry axis.
3. The equilibrium constant for the dimerization reaction is $K_{c}^\circ=33.5$.
4. The equilibrium constant for the dimerization reaction is $K_{c}^\circ=3.33$.
5. The equilibrium constant for the dimerization reaction is $K_{c}^\circ=28.7$.

Which of the statements above are correct?
化合物 $\mathrm{A}$ 可发生二聚形成 $\mathrm{A}_{2}$, 该二聚反应构成如下平衡:

$$
2 \mathbf{A} \rightleftarrows \mathbf{A}_{2} \quad K_{c}^{\theta}
$$

反应的平衡常数可通过偏振法测得, 其原理是测定溶液中反应达到平衡时的比旋光度 $[\alpha]_{\mathrm{D}}$,
以此计算得到反应平衡常数 $K_{c}{ }^\circ$ 。已知反应达到平衡时 $[\alpha]_{\mathrm{D}}$ 满足如下关系式:

$[\alpha]_{\mathrm{D}}=x_{1}{ }^{2}[\alpha]_{1}+x_{2}{ }^{2}[\alpha]_{2}$

其中 $x_{1}$ 和 $x_{2}$ 为对应物种的摩尔分数, $[\alpha]_{1}$ 和 $[\alpha]_{2}$ 为对应物种单独存在时的比旋光度。

1.化合物 $\mathbf{A}$ 的结构如图1所示, 本题需要推测其二聚体 $\mathbf{A}_{2}$ 的结构。

<MultiModal>![Fig. 1，该分子以SMILES描述为：CC1(C)COC(=O)[C@@H]1O](/media/uploads/1c6370ec-eff9-4ddf-a523-6a560196d8e0.png)</MultiModal>

2.在室温下, 总浓度为 $8.0 \mathrm{mmol} / \mathrm{L}$ 的化合物 $\mathbf{A}$ 在 $\mathrm{CCl}_{4}$ 溶剂中达到
平衡, 测得体系的比旋光度 

$[\alpha]_{\mathrm{D}}=-6~ \mathrm{deg}~ \mathrm{dm}^{-1} \mathrm{~g}^{-1} \mathrm{~mL}$ 。

化合物 $\mathbf{A}$ 和二聚
体 $\mathbf{A}_{2}$ 单独存在时的比旋光度分别为 

$[\alpha]_{1}=-1~ \mathrm{deg}~ \mathrm{dm}^{-1} \mathrm{~g}^{-1} \mathrm{~mL}$

$[\alpha]_{2}=-203$
$\mathrm{deg}~ \mathrm{dm}^{-1} \mathrm{~g}^{-1} \mathrm{~mL}$

忽略体系中可能存在的其他物种, 计算该二聚反应的平衡常数 $K_{c}^\circ$ 。

有以下几种说法：
1. $\mathbf{A}$二聚时形成了共价键 
2. 二聚体$\mathbf{A_2}$中不包含$C_2$对称轴
3. 二聚反应的平衡常数 $K_{c}^\circ=33.5$
4. 二聚反应的平衡常数 $K_{c}^\circ=3.33$
5. 二聚反应的平衡常数 $K_{c}^\circ=28.7$

选项中说法均正确的是？
1
2
3
4
5
1,3
1,4
1,5
2,3
2,4
2,5
1,2
1,2,3
1,2,4
1,2,5
2,3,4
2, 3, 5
3,4,5
1, 2, 3, 4
1, 2, 4, 5
2, 3, 4, 5
1,3,4,5
1, 2, 3, 4, 5
None of the above options is correct
1
2
3
4
5
1,3
1,4
1,5
2,3
2,4
2,5
1,2
1,2,3
1,2,4
1,2,5
2,3,4
2,3,5
3,4,5
1,2,3,4
1,2,4,5
2,3,4,5
1,3,4,5
1,2,3,4,5
以上选项均不正确
Observing the structure of compound $\mathbf{A}$, it contains:

Hydroxyl group ($\mathrm{-OH}$)  
Carbonyl group ($\mathrm{C=O}$)  

These two functional groups can interact through hydrogen bonding. The dimer $\mathrm{A_2}$ is most likely formed via intermolecular hydrogen bonds, where the ($\mathrm{-OH}$) of one molecule forms a hydrogen bond with the ($\mathrm{C=O}$) of another molecule, resulting in a cyclic dimer structure as shown below. The dimerization process does not involve covalent bond formation, so statement 1 is incorrect. The two monomers are connected by hydrogen bonds, forming two $\mathrm{C=O…H-O}$ hydrogen bonds in total. This structure exhibits $C_2$ symmetry, so statement 2 is incorrect. (An alternative form involves intermolecular transesterification to form a six-membered ring, with the product structure represented in SMILES as: CC(CO)([C@@H](OC([C@H](O1)C(CO)(C)C)=O)C1=O)C. However, this process is entropically unfavorable compared to intramolecular formation of a five-membered ring ester bond.)  

<Checkpoint>[Intermolecular transesterification to form a six-membered ring dimer is entropically unfavorable](1)</Checkpoint>  

<MultiModal>![The image shows the structure of the dimer $\mathbf{A_2}$, where the monomer $\mathbf{A}$ remains unchanged. $\mathbf{A}$ is described in SMILES as: CC1(C)COC(=O)[C@@H]1O. The two monomers are connected by hydrogen bonds between the hydroxyl ($\mathrm{-OH}$) and carbonyl ($\mathrm{C=O}$) groups, forming two $\mathrm{C=O…H-O}$ hydrogen bonds. The image is a schematic diagram of the chemical structure, featuring two mirror-image molecular structures interacting via two dashed lines. Each molecular structure centers on a five-membered ring containing two oxygen atoms, denoted by the letter "O." One of the ring oxygen atoms is adjacent to a carbonyl group (C=O), forming a lactone structure, with the carbonyl oxygen also labeled "O." On each five-membered ring, a carbon atom is connected to a hydroxyl group via a solid wedge bond, where the hydroxyl consists of an oxygen atom "O" and a hydrogen atom "H." Another carbon atom on the ring is bonded to two methyl groups, represented by short lines. The interaction at the center of the image is depicted by two dashed lines: one between the hydroxyl hydrogen "H" of the left molecule and the carbonyl oxygen "O" of the right molecule, and another between the hydroxyl hydrogen "H" of the right molecule and the carbonyl oxygen "O" of the left molecule. All chemical bonds in the diagram are represented by solid lines, except for the dashed lines indicating intermolecular interactions, wedge bonds for stereochemistry, and double lines for C=O bonds.](/media/uploads/6a2cd921-58a2-45a4-b715-f877cae4ff38.png)</MultiModal>  

<Checkpoint>[$\mathbf{A}$ dimerizes via hydrogen bonding](1)</Checkpoint>  

<Checkpoint>[The dimer exhibits $C_2$ symmetry.](1)</Checkpoint>  

Let the mole fractions of $\mathbf{A}$ and $\mathbf{A}_{2}$ be $x_{1}$ and $x_{2}$, respectively, where $x_{1}+x_{2}=1$.  
Substituting the known specific rotation data into $[\alpha]_{\mathrm{D}}=x_{1}{ }^{2}[\alpha]_{1}+x_{2}{ }^{2}[\alpha]_{2}$ yields: $-x_{1}{ }^{2}-203 x_{2}{ }^{2}=-6$.  
Solving these two equations simultaneously gives $x_{1}=0.838$ and $x_{2}=0.161$.  

<Checkpoint>[Mole fraction of $\mathbf{A}$=0.838, Mole fraction of $\mathbf{A_2}$=0.161](1)</Checkpoint>  

Given $c(\mathbf{A})+2 c\left(\mathbf{A}_{2}\right)=0.0080 \mathrm{~mol} / \mathrm{L}$, solving simultaneously yields $c(\mathbf{A})=5.78 \times 10^{-3} \mathrm{~mol} / \mathrm{L}$.  

Thus, $K_{c}{ }^{\theta}=c\left(\mathbf{A}_{2}\right) \times c^{\theta} /[c(\mathbf{A})]^{2}=\left(1.12 \times 10^{-3}\right) /\left(5.78 \times 10^{-3}\right)^{2}=33.5$.  

Statement 3 is correct, while statements 4 and 5 are incorrect, making option C the correct choice.  

<Checkpoint>[Equilibrium constant $K_{c}^\circ=33.5$](1)</Checkpoint>
观察化合物$\mathbf{A}$的结构，它含有：

羟基($\mathrm{-OH}$)
羰基($\mathrm{C=O}$)

这两个官能团可以通过氢键相互作用。二聚体$\mathrm{A_2}$最可能通过分子间氢键形成，即一个分子的($\mathrm{-OH}$)与另一个分子的($\mathrm{C=O}$)形成氢键，形成环状二聚体结构如下图，二聚过程未形成共价键，说法1错误。这两个单体由氢键连接，共形成两个$\mathrm{C=O…H-O}$氢键，该结构具有$C_2$对称性，因此2错误。（另外一种形式是发生分子间酯交换形成六元环，产物结构以SMILES表示为：CC(CO)([C@@H](OC([C@H](O1)C(CO)(C)C)=O)C1=O)C，但是该过程与分子内形成五元环酯键相比熵不利。）

<Checkpoint>[分子间酯交换形成六元环二聚是熵不利的](1)</Checkpoint>

<MultiModal>![图片为二聚体$\mathbf{A_2}$的结构，其中化合物$\mathbf{A}$单体不变，$\mathbf{A}$以SMILES描述为：CC1(C)COC(=O)[C@@H]1O，两个单体间通过羟基($\mathrm{-OH}$)
羰基($\mathrm{C=O}$)
这两个单体由氢键连接，共形成两个$\mathrm{C=O…H-O}$氢键。这张图片是一张化学结构示意图，其中包含两个互为镜像的分子结构，它们通过两条虚线相互作用。每个分子结构的核心是一个五元环，环内包含两个氧原子，以字母“O”表示。其中一个环内氧原子与一个羰基（碳氧双键）相邻，形成了内酯结构，该羰基的氧原子也用字母“O”表示。在每个五元环上，都有一个碳原子通过实心楔形键连接着一个羟基，该羟基由一个氧原子“O”和一个氢原子“H”组成。环上的另一个碳原子连接着两个甲基基团，由短线段表示。图像中心的相互作用由两条虚线表示：左侧分子的羟基氢原子“H”与右侧分子的羰基氧原子“O”之间有一条虚线；同时，右侧分子的羟基氢原子“H”与左侧分子的羰基氧原子“O”之间也有一条虚线。图中所有的化学键均由实线表示，除了表示分子间相互作用的虚线、表示立体化学的楔形键以及表示碳氧双键的双线。](/media/uploads/6a2cd921-58a2-45a4-b715-f877cae4ff38.png)</MultiModal>

<Checkpoint>[$\mathbf{A}$通过氢键二聚](1)</Checkpoint>

<Checkpoint>[二聚体具有$C_2$对称性。](1)</Checkpoint>

设 $\mathbf{A}$ 和 $\mathbf{A}_{2}$ 的摩尔分数分别为 $x_{1}$ 和 $x_{2}$, 其中 $x_{1}+x_{2}=1$。
将已知的比旋光度数据带入 $[\alpha]_{\mathrm{D}}=x_{1}{ }^{2}[\alpha]_{1}+x_{2}{ }^{2}[\alpha]_{2}$ 得: $-x_{1}{ }^{2}-203 x_{2}{ }^{2}=-6$
联立上述二式, 解得 $x_{1}=0.838, x_{2}=0.161$

<Checkpoint>[$\mathbf{A}$的摩尔分数=0.838, $\mathbf{A_2}$的摩尔分数=0.161](1)</Checkpoint>

又 $c(\mathbf{A})+2 c\left(\mathbf{A}_{2}\right)=0.0080 \mathrm{~mol} / \mathrm{L}$, 联立解得 $c(\mathbf{A})=5.78 \times 10^{-3} \mathrm{~mol} / \mathrm{L}$

故 $K_{c}{ }^{\theta}=c\left(\mathbf{A}_{2}\right) \times c^{\theta} /[c(\mathbf{A})]^{2}=\left(1.12 \times 10^{-3}\right) /\left(5.78 \times 10^{-3}\right)^{2}=33.5$

说法3正确，4，5均错误，从而选项C正确。

<Checkpoint>[平衡常数$K_{c}^\circ=33.5$](1)</Checkpoint>