Ammonium dihydrogen phosphate reacts with a certain 1:1 type metal oxide $\rm {\bf{M}\rm{O}}$ under humid conditions to obtain salt $\bf A$, whose crystal structure is shown in the figure below (hydrogen atoms on oxygen are omitted, each sphere represents an element, and the vertex is located on a certain atom). It is known that the crystal has $n$ glide planes or mirror planes in the $x/y$ direction and a twofold axis in the $z$ direction, where the coordinates of one $\bf M$ are (0.5000, 0.3639, 0.1286) and the coordinates of one $\rm P$ are (0.5000, 0.9773, 0.5000):

<MultiModal>![This is a schematic diagram of the unit cell structure of a crystal. The orientations of xyz are: to the right in the plane of the paper, perpendicular to the plane of the paper and out of the plane, and upwards in the plane of the paper, forming a left-handed helical coordinate system. Among them, chemical bonds and coordinate bonds are represented by bonds, and the coordination polyhedra of specific ions are represented by polyhedra. There are five different spheres in the figure, each representing a chemical element, which are red spheres, yellow spheres, blue spheres, pink spheres, and black spheres. The size relationship of their radii in the figure is: red sphere > yellow sphere > black sphere > blue sphere > pink sphere. The figure shows that they form various polyhedral ions with each other. 4 red spheres coordinate with 1 black sphere to form a 1 black 4 red tetrahedral ion I, 4 pink spheres coordinate with 1 blue sphere to form a 1 blue 4 pink tetrahedral ion II, and 6 red spheres coordinate with 1 yellow sphere to form a 1 yellow 6 red octahedral ion III. There are 2 of each of these three ions in the unit cell, and the orientation of each pair of the same type of polyhedra is mirrored with respect to the xz plane. All polyhedra have mirror planes parallel to the yz direction inside, and all polyhedra have a bottom surface that is almost parallel to the xy direction. Among them, the center of the tetrahedral ion I is located at the vertex of the unit cell, and the center of the xz face is slightly offset towards the negative y-axis at two positions; the tetrahedral ion II is located approximately at the center of the yz face offset towards the negative z-axis and the center of the unit cell offset towards the positive z-axis; the octahedral ion III is located approximately at the center of the yz face offset towards the positive z-axis and the center of the unit cell offset towards the negative z-axis. The a and b of the unit cell are close to or equal, and c is much larger than a/b](/media/uploads/880891a3-fe0b-425f-b90e-5e06f9f39bb5.png)</MultiModal>

Please observe and based on the above schematic diagram of the crystal unit cell, the following propositions are now given:

1. There are 4 kinds of ions with different compositions in the crystal
2. The molar mass of one structural unit of the crystal is $221.11 +M(\mathbf{M})~\mathrm{g\cdot mol^{-1}}$
3. When this substance is generated, for every $2.85~\mathrm{g~(NH_4)H_2PO_4} $ consumed, the volume of water vapor absorbed under standard conditions is $2.8~\mathrm{L}$
4. In the actual crystal, the polyhedral structure of at least one of the ions in the crystal is a regular polyhedron
5. The space group to which the crystal belongs may be $Pmm2$
6. There are two $n$ glide planes or mirror planes in the crystal that are parallel to the $xz$ plane, located at $y=0.0000/0.5000$
7. There are two $n$ glide planes or mirror planes in the crystal that are parallel to the $yz$ plane, located at $x=0.0000/0.5000$
8. One of the twofold axes of the crystal along the $z$ direction is located at $x=0.2500,y=0.5000$ in the $z$-axis projection
9. The coordinates of another $\bf M$ of the crystal are (0.0000, 0.6134, 0.6286)

Determine the correctness of the above propositions and find the value of $z$:

$$
z = \dfrac{\text{Sum of squares of correct propositions}}{(\text{Sum of incorrect propositions}+1)^2}
$$

Retain 3 significant figures for the final result, and select the correct option from the following options, requiring the deviation from your calculated result to be less than 1%, otherwise select option **A**: All other options are incorrect.
磷酸二氢铵与某种 1:1 型金属氧化物 $\rm {\bf{M}\rm{O}}$ 在潮湿的状况下反应得到盐 $\bf A$，其晶体结构如下图所示（省略氧上的氢原子，每种球代表一种元素，顶点位于某种原子之上），已知该晶体在 $x/y$ 方向上均有 $n$ 滑移面或镜面、在 $z$ 方向上有二重轴，其中 1 个 $\bf M$ 的坐标为 (0.5000, 0.3639, 0.1286)、1 个 $\rm P$ 的坐标为 (0.5000, 0.9773, 0.5000)：

<MultiModal>![这是一个晶体的晶胞结构示意图，xyz的取向分别为：纸面内向正右方向、垂直于纸面向纸面外、纸面内向正上方向，呈左手螺旋坐标。其中，通过键连的方式表示了化学键与配位键、通过多面体表示出特定离子的配位多面体。图中共有五种不同的球，每种球代表一种化学元素，分别为红球、黄球、蓝球、粉球、黑球，它们的半径在图中的大小关系是：红球>黄球>黑球>蓝球>粉球，图中呈现出它们彼此之间形成各种多面体的离子，4个红球配位1个黑球形成1黑4红的四面体离子 I，4个粉球配位1个蓝球形成1蓝4粉的四面体离子 II，6个红球配位1个黄球形成1黄6红的八面体离子 III。这三种离子在晶胞中都有2个，且每对同种的多面体取向均相较于 xz 方向镜面发生了镜像，所有多面体内部都有平行于 yz 方向的镜面、所有多面体均有相较于 xy 方向几乎平行的底面。其中，四面体离子 I 的中心位于晶胞顶点、xz 面心略微向 y 轴负方向偏移两个位置；四面体离子 II 大致位于 yz 面心向 z 轴负方向偏移、晶胞体心向 z 轴正方向偏移的位置；八面体离子 III 大致位于位于 yz 面心向 z 轴正方向偏移、晶胞体心向 z 轴负方向偏移的位置。晶胞的 a 与 b 接近或相等，c 远大于 a/b](/media/uploads/880891a3-fe0b-425f-b90e-5e06f9f39bb5.png)</MultiModal>

请你观察并基于以上晶体晶胞示意图，现给出以下命题：

1. 晶体中存在 4 种不同组成的离子
2. 晶体一个结构基元的摩尔质量是 $221.11 +M(\mathbf{M})~\mathrm{g\cdot mol^{-1}}$
3. 生成该物质时，每消耗 $2.85~\mathrm{g~(NH_4)H_2PO_4} $ 的同时，吸收的水蒸气在标准状况下体积为 $2.8~\mathrm{L}$
4. 实际晶体中，晶体中离子的多面体结构至少有一种是正多面体
5. 该晶体所属的空间可能是 $Pmm2$
6. 该晶体中有两个平行于 $xz$ 面的 $n$ 滑移面或镜面，分别在 $y=0.0000/0.5000$ 的位置
7. 该晶体中有两个平行于 $yz$ 面的 $n$ 滑移面或镜面，分别在 $x=0.0000/0.5000$ 的位置
8. 该晶体的沿 $z$ 方向的二重轴，其中一个在 $z$ 轴投影中位于 $x=0.2500,y=0.5000$ 的位置
9. 该晶体的另一个  $\bf M$ 的坐标为 (0.0000, 0.6134, 0.6286)

判断以上命题的正误，并求以下 $z$ 的值：

$$
z = \dfrac{\text{正确命题的平方和}}{(\text{错误命题的和}+1)^2}
$$

最终结果保留3位有效数字，从以下选项中选择正确的选项，要求与你计算的结果偏差小于 1\%，否则选择选项 **A**：其他选项均不正确。
All other options are incorrect
0.00875
0.0138
0.0208
0.00907
0.0149
0.0300
0.0416
0.0744
0.0255
0.154
0.191
0.354
其他选项均不正确
0.00875
0.0138
0.0208
0.00907
0.0149
0.0300
0.0416
0.0744
0.0255
0.154
0.191
0.354
Observe the structure of the crystal. There are five different spheres, each representing a chemical element, which are red, yellow, blue, pink, and black spheres. Their radius relationship in the figure is: red sphere > yellow sphere > black sphere > blue sphere > pink sphere. Combined with the statement in the question: "Ammonium dihydrogen phosphate reacts with $\rm \bf{M}\rm{O}$ under humid conditions to obtain salt $\bf A$" and given the coordinates of some $\rm \bf{M}\rm{/P}$, the crystal should contain the five elements $\rm {O/\bf{M}\rm{/P/N/H}}$ in descending order of sphere radius in the figure.

<Checkpoint>[The unit cell diagram shows five elements $\rm O/\bf{M}\rm/P/N/H$ in descending order of sphere radius](0.5)</Checkpoint>

The figure shows three types of polyhedral ions formed between them, each with two. 4 red spheres coordinate with 1 black sphere to form a 1 black 4 red tetrahedral ion $\rm [PO_4]^{3-}$, 4 pink spheres coordinate with 1 blue sphere to form a 1 blue 4 pink tetrahedral ion $\rm [NH_4]^+$, and 6 red spheres coordinate with 1 yellow sphere to form a 1 yellow 6 red octahedral ion $[\bf{M}\rm{(H_2O)_6]^{2+}}$. Therefore, **Proposition 1 is incorrect**.

<Checkpoint>[The unit cell diagram has three types of ions: $\rm [PO_4]^{3-}$、$\rm [NH_4]^+$、$[\bf{M}\rm{(H_2O)_6]^{2+}}$, and there are two of each](0.5)</Checkpoint>

Therefore, the chemical formula of the crystal is $\rm (NH_4)\bf{M}\rm{(PO_4)\cdot 6H_2O}$.

<Checkpoint>[The chemical formula of the crystal is $\rm (NH_4)\bf{M}\rm{(PO_4)\cdot 6H_2O}$](1)</Checkpoint>

Observation reveals that each pair of the same polyhedron is mirrored with respect to the $xz$ direction. All polyhedra have a mirror plane parallel to the $yz$ direction inside. Therefore, each pair of identical polyhedra is not equivalent. One unit cell is one structural unit, and the composition of one unit cell is $\rm 2[(NH_4)\bf{M}\rm{(PO_4)\cdot 6H_2O}]$, and its molar mass is $2M_{\mathbf{A}}=2\times (221.11+M(\mathbf{M}))=442.22 +2M(\mathbf{M})~\mathrm{g\cdot mol^{-1}}$. Therefore, **Proposition 2 is incorrect**.

<Checkpoint>[The molar mass of one structural unit of the crystal is $442.22 +2M(\mathbf{M})~\mathrm{g\cdot mol^{-1}}$](1)</Checkpoint>

The chemical reaction equation for the formation of this crystal is:

$$
\rm \bf{M}\rm O + (NH_4)H_2PO_4+5H_2O\rightarrow (NH_4)\bf{M}\rm(PO_4)\cdot 6H_2O
$$

The stoichiometric ratio of $\mathrm{(NH_4)H_2PO_4}$ to $\rm H_2O$ is 1:5.

<Checkpoint>[The stoichiometric ratio of $\mathrm{(NH_4)H_2PO_4}$ to $\rm H_2O$ is 1:5](0.5)</Checkpoint>

According to the ideal gas state equation, for every $2.85~\mathrm{g~(NH_4)H_2PO_4}$ consumed, the volume of water vapor absorbed under standard conditions is:

$$
V_{\mathrm{H_2O}}=\dfrac{n_{\mathrm{H_2O}}RT}{p}=\dfrac{5n_{\mathrm{(NH_4)H_2PO_4}}RT}{p}=\dfrac{5m_{\mathrm{(NH_4)H_2PO_4}}RT}{M_{\mathrm{(NH_4)H_2PO_4}}p}=\dfrac{5\times 2.85\times 8.314\times 273}{115.03\times101}=2.81\approx2.8\mathrm{~L}
$$

Therefore, **Proposition 3 is correct**.

<Checkpoint>[For every $2.85~\mathrm{g~(NH_4)H_2PO_4}$ consumed, the volume of water vapor absorbed under standard conditions is $2.8\mathrm{~L}$](0.5)</Checkpoint>

According to the information in the question, "the crystal has $n$ glide planes or mirror planes in the $x/y$ directions and a twofold axis in the $z$ direction", and according to the orientation of the ions in the figure, and all polyhedra in the figure have a base almost parallel to the $xy$ direction. Therefore, the crystal should belong to the orthorhombic crystal system.

<Checkpoint>[The crystal belongs to the orthorhombic crystal system](0.5)</Checkpoint>

The symmetry of the crystal is lower than the symmetry of the coordination polyhedra (regular tetrahedron, regular octahedron). Therefore, due to the breaking of symmetry, there are no regular polyhedra in the actual crystal. Therefore, **Proposition 4 is incorrect**.

<Checkpoint>[In the actual crystal, none of the polyhedral structures of the ions in the crystal are regular polyhedra](1)</Checkpoint>

According to the question, the "vertices of the unit cell are located on some atom". Combined with the previous reasoning, we know that the vertices are located on the $\rm P$ atom. At the same time, the coordinates of one $\bf M$ are (0.5000, 0.3639, 0.1286), and the coordinates of one $\rm P$ are (0.5000, 0.9773, 0.5000). Further combined with the shape of the coordination polyhedra in the figure, it is obvious that: there are two mirror planes parallel to the $yz$ plane in the crystal, located at $x=0.0000/0.5000$. Therefore, **Proposition 7 is correct**.

<Checkpoint>[There are two mirror planes parallel to the $yz$ plane in the crystal, located at $x=0.0000/0.5000$](1)</Checkpoint>

Further combined with the information that the base of the polyhedron is almost parallel to the $xy$ plane and that there are mirror planes parallel to the $yz$ plane, it can be ruled out that there are two mirror planes parallel to the $xz$ plane in the crystal, which does not meet the requirements of the tetrahedral and octahedral shapes. Therefore, the crystal space group cannot be $Pmm2$. Therefore, **Proposition 5 is incorrect**.

<Checkpoint>[The crystal space group cannot be $Pmm2$](1)</Checkpoint>

Combined with the meaning of the question, there are two $n$ glide planes parallel to the $xz$ plane in the crystal. Combined with the coordinates of the two $\rm P$ (0.0000, 0.0000, 0.0000) and (0.5000, 0.9773, 0.5000), it is not difficult to find that the $n$ glide planes are not located at $y=0.0000/0.5000$, but should be located at $y=0.4886/0.9886$. Therefore, **Proposition 6 is incorrect**.

<Checkpoint>[There are two $n$ glide planes parallel to the $xz$ plane in the crystal, located at $y=0.4886/0.9886$](1)</Checkpoint>

Therefore, the space group to which the crystal belongs should be $Pmn\square$. At the same time, it is required to be an orthorhombic crystal system and $\square$ is a twofold axis. Therefore, among the 230 space groups, only $Pmn2_1$ meets the requirements. Therefore, there is a $2_1$ axis along the $z$ direction in the unit cell. Further verification of **Proposition 5 is incorrect**.

<Checkpoint>[The crystal space group is $Pmn2_1$](1)</Checkpoint>

Furthermore, examine the position of the $2_1$ axis, which should be located within the glide plane. Combined with the coordinates of the two $\rm P$ (0.0000, 0.0000, 0.0000) and (0.5000, 0.9773, 0.5000), its position in the $z$ axis projection is located at $x=0.2500,y=0.4886$. Similarly, it is deduced that other $2_1$ axes are located at $x=0.2500,y=0.9886$, $x=0.7500,y=0.4886$, $x=0.7500,y=0.9886$. There is no $2_1$ axis located at $x=0.2500,y=0.5000$. Therefore, **Proposition 8 is incorrect**.

<Checkpoint>[$2_1$ axes in $z$ axis projection are located at $x=0.2500,y=0.4886$、$x=0.2500,y=0.9886$、$x=0.7500,y=0.4886$、$x=0.7500,y=0.9886$](1)</Checkpoint>

Finally, combined with the symmetry of the crystal, the coordinates of the other $\bf M$ are indeed (0.0000, 0.6134, 0.6286). Therefore, **Proposition 9 is correct**.

<Checkpoint>[The coordinates of the other $\bf M$ in the crystal are (0.0000, 0.6134, 0.6286)](1)</Checkpoint>

Finally, calculate

$$
z = \dfrac{\text{[The sum of squares of correct propositions]}}{(\text{[The sum of incorrect propositions]}+1)^2}=\frac{3^2+7^2+9^2}{(1+2+4+5+6+8+1)^2}\approx0.191
$$
观察该晶体的结构，共有五种不同的球，每种球代表一种化学元素，分别为红球、黄球、蓝球、粉球、黑球，它们的半径在图中的大小关系是：红球>黄球>黑球>蓝球>粉球，结合题目中所述：“磷酸二氢铵与 $\rm \bf{M}\rm{O}$ 在潮湿的状况下反应得到盐 $\bf A$”并告知了部分 $\rm \bf{M}\rm{/P}$ 的坐标，晶体中应该存在有按照图中小球半径从大到小 $\rm {O/\bf{M}\rm{/P/N/H}}$ 这五种元素。

<Checkpoint>[晶胞图中小球半径从大到小存在 $\rm O/\bf M\rm/P/N/H$ 五种元素](0.5)</Checkpoint>

图中呈现出它们彼此之间形成的三种多面体的离子，它们均有两个，4个红球配位1个黑球形成1黑4红的四面体离子 $\rm [PO_4]^{3-}$，4个粉球配位1个蓝球形成1蓝4粉的四面体离子 $\rm [NH_4]^+$，6个红球配位1个黄球形成1黄6红的八面体离子  $[\bf{M}\rm{(H_2O)_6]^{2+}}$。故**命题 1 错误**。

<Checkpoint>[晶胞图中有 $\rm [PO_4]^{3-}$、$\rm [NH_4]^+$、$[\bf{M}\rm{(H_2O)_6]^{2+}}$ 三种离子，且均有两个](0.5)</Checkpoint>

因此，晶体的化学式为 $\rm (NH_4)\bf{M}\rm{(PO_4)\cdot 6H_2O}$。

<Checkpoint>[晶体的化学式为 $\rm (NH_4)\bf{M}\rm{(PO_4)\cdot 6H_2O}$](1)</Checkpoint>

观察得到，每对同种的多面体取向均相较于 $xz$ 方向镜面发生了镜像，所有多面体内部都有平行于 $yz$ 方向的镜面，因此，每对相同的多面体不等价，一个晶胞即为一个结构基元，而一个晶胞的组成为 $\rm 2[(NH_4)\bf{M}\rm{(PO_4)\cdot 6H_2O}]$，其摩尔质量为 $2M_{\mathbf{A}}=2\times (221.11+M(\mathbf{M}))=442.22 +2M(\mathbf{M})~\mathrm{g\cdot mol^{-1}}$，因此，**命题 2 错误**。

<Checkpoint>[晶体一个结构基元的摩尔质量为 $442.22 +2M(\mathbf{M})~\mathrm{g\cdot mol^{-1}}$](1)</Checkpoint>

生成该晶体的化学反应方程式为：

$$
\rm \bf{M}\rm O + (NH_4)H_2PO_4+5H_2O\rightarrow (NH_4)\bf{M}\rm(PO_4)\cdot 6H_2O
$$

$\mathrm{(NH_4)H_2PO_4}$ 与 $\rm H_2O$ 的反应计量比为 1:5。

<Checkpoint>[$\mathrm{(NH_4)H_2PO_4}$ 与 $\rm H_2O$ 的反应计量比为 1:5](0.5)</Checkpoint>

根据理想气体状态方程，每消耗 $2.85~\mathrm{g~(NH_4)H_2PO_4}$ 的同时，吸收的水蒸气在标准状况下体积为：

$$
V_{\mathrm{H_2O}}=\dfrac{n_{\mathrm{H_2O}}RT}{p}=\dfrac{5n_{\mathrm{(NH_4)H_2PO_4}}RT}{p}=\dfrac{5m_{\mathrm{(NH_4)H_2PO_4}}RT}{M_{\mathrm{(NH_4)H_2PO_4}}p}=\dfrac{5\times 2.85\times 8.314\times 273}{115.03\times101}=2.81\approx2.8\mathrm{~L}
$$

因此**命题 3 正确**。

<Checkpoint>[每消耗 $2.85~\mathrm{g~(NH_4)H_2PO_4}$ 的同时，吸收的水蒸气在标准状况下体积为 $2.8\mathrm{~L}$](0.5)</Checkpoint>

根据题目中的信息，“该晶体在 $x/y$ 方向上均有 $n$ 滑移面或镜面、在 $z$ 方向上有二重轴”，且根据图中离子的取向，以及图中所有多面体均有相较于 $xy$ 方向几乎平行的底面。因此，该晶体应该属于正交晶系。

<Checkpoint>[该晶体属于正交晶系](0.5)</Checkpoint>

晶体的对称性低于配位多面体（正四面体、正八面体）的对称性，因此，由于对称性的破坏，在实际情况时，晶体中没有正多面体存在。故**命题 4 错误**。

<Checkpoint>[实际晶体中，晶体中离子的多面体结构没有一种是正多面体](1)</Checkpoint>

根据题目中晶胞的“顶点位于某种原子之上”，结合之前推理，我们知道顶点位于 $\rm P$ 原子之上，同时 1 个 $\bf M$ 的坐标为 (0.5000, 0.3639, 0.1286)、1 个 $\rm P$ 的坐标为 (0.5000, 0.9773, 0.5000)，进一步结合图中配位多面体的形状，显然：该晶体中有两个平行于 $yz$ 面的镜面，分别在 $x=0.0000/0.5000$ 的位置。故**命题 7 正确**。

<Checkpoint>[该晶体中有两个平行于 $yz$ 面的镜面，分别在 $x=0.0000/0.5000$ 的位置](1)</Checkpoint>

进一步结合多面体底面与 $xy$ 面几乎平行、存在平行于 $yz$ 面的镜面的信息，可以排除该晶体中有两个平行于 $xz$ 面的镜面的情况，这不符合四面体和八面体形状要求。因此，晶体空间群不可能为 $Pmm2$。故**命题 5 错误**。

<Checkpoint>[晶体空间群不可能为 $Pmm2$](1)</Checkpoint>

结合题意，晶体中有两个平行于 $xz$ 面的 $n$ 滑移面，结合两个$\rm P$ 的坐标(0.0000, 0.0000, 0.0000)与(0.5000, 0.9773, 0.5000)，不难发现 $n$ 滑移面不在 $y=0.0000/0.5000$ 的位置，而应该在 $y=0.4886/0.9886$ 的位置。故**命题 6 错误**。

<Checkpoint>[该晶体中有两个平行于 $xz$ 面的 $n$ 滑移面，分别在 $y=0.4886/0.9886$ 的位置](1)</Checkpoint>

因此，该晶体所属的空间群应该为 $Pmn\square$，同时要求其为正交晶系、$\square$ 为二重轴，因此，在 230 个空间群中，只有 $Pmn2_1$ 符合要求，因此，晶胞中沿 $z$ 方向有 $2_1$ 轴。进一步验证了**命题 5 错误**。

<Checkpoint>[晶体空间群为 $Pmn2_1$](1)</Checkpoint>

进一步，考察 $2_1$ 轴的位置，其应该位于滑移面内，结合两个$\rm P$ 的坐标(0.0000, 0.0000, 0.0000)与(0.5000, 0.9773, 0.5000)，其位置在在 $z$ 轴投影中位于 $x=0.2500,y=0.4886$，同理推得其他 $2_1$ 轴位于 $x=0.2500,y=0.9886$、$x=0.7500,y=0.4886$、$x=0.7500,y=0.9886$。不存在位于 $x=0.2500,y=0.5000$ 的 $2_1$ 轴，故**命题 8 错误**。

<Checkpoint>[$2_1$ 轴在 $z$ 轴投影中位于 $x=0.2500,y=0.4886$、$x=0.2500,y=0.9886$、$x=0.7500,y=0.4886$、$x=0.7500,y=0.9886$](1)</Checkpoint>

最终，结合晶体的对称性，另一个 $\bf M$ 的坐标确为 (0.0000, 0.6134, 0.6286)。故**命题 9 正确**。

<Checkpoint>[晶体中另一个 $\bf M$ 的坐标为 (0.0000, 0.6134, 0.6286)](1)</Checkpoint>

最终，计算

$$
z = \dfrac{\text{正确命题的平方和}}{(\text{错误命题的和}+1)^2}=\frac{3^2+7^2+9^2}{(1+2+4+5+6+8+1)^2}\approx0.191
$$