Adding sodium hydroxide to an aqueous solution of a nitrate $\mathrm{\mathbf{M}(NO_3)_x}$ of a certain metal $\mathbf{M}$ yields a precipitate $\bf{X}$. Treating $\bf{X}$ and $\mathrm{\mathbf{M}(NO_3)_x}$ with a persulfate yields a precipitate $\bf{Y}$.
Hydrothermal synthesis of $\bf{Y}$ with M element under high pressure yields crystal $\bf{Z}$.

It is known that $\bf{X}$, $\bf{Y}$, and $\bf{Z}$ are all oxides of the metal $\mathbf{M}$, and the mass fraction of oxygen element in substance $\bf{X}$ is 6.9%, $\bf{Y}$ is 12.9%, and $\bf{Z}$ is 4.7%.

The crystal of $\bf{Z}$ can be approximately regarded as the stacking of different atomic layers in the manner of $\bf{……AcB□AcB□……}$. Where $\bf{A}$ and $\bf{B}$ are close-packed layers of $\mathbf{M}$ atoms, $\bf{c}$ is an octahedral interstitial layer filled with some $\mathrm{O}$ atoms, and □ is an empty octahedral interstitial layer.

The coordinates of all $\mathrm{O}$ atoms in the unit cell are:

$(\frac{2}{3},\,\frac{1}{3},\,\frac{1}{2})$

$(\frac{1}{3},\,\frac{2}{3},\,\frac{1}{2})$

And $\frac{2}{3}$ of the $\mathbf{M}$ atoms in the unit cell are located on the unit cell face

Which of the following statements are correct:

1.The occupancy rate of $\mathrm{O}$ atoms in the $\bf{c}$ layer is 1

2.The occupancy rate of $\mathrm{O}$ atoms in the $\bf{c}$ layer is $\frac{2}{3}$

3.The occupancy rate of $\mathrm{O}$ atoms in the $\bf{c}$ layer is $\frac{1}{3}$

4.The x, y coordinates of one of the $\mathbf{M}$ atoms are $(\frac{2}{3},\,\frac{2}{3})$

5.The x, y coordinates of one of the $\mathbf{M}$ atoms are $(\frac{1}{2},\,\frac{1}{2})$

6.The x, y coordinates of one of the $\mathbf{M}$ atoms are $(\frac{1}{3},\,\frac{2}{3})$
将氢氧化钠加入某金属 $\mathbf{M}$ 的硝酸盐 $\mathrm{\mathbf{M}(NO_3)_x}$ 的水溶液中可得到沉淀 $\bf{X}$。若将 $\bf{X}$ 和 $\mathrm{\mathbf{M}(NO_3)_x}$ 用过二硫酸盐处理则可得到沉淀 $\bf{Y}$。
将 $\bf{Y}$ 在高压下与 M 单质水热合成则可得到结晶 $\bf{Z}$。

已知 $\bf{X}$、$\bf{Y}$、$\bf{Z}$ 皆为金属 $\mathbf{M}$ 的氧化物，且物质 $\bf{X}$ 的氧元素质量分数为 6.9%，$\bf{Y}$ 为 12.9%，$\bf{Z}$ 为 4.7%

$\bf{Z}$ 的晶体可以近似看作不同原子层按照 $\bf{……AcB□AcB□……}$ 的方式堆积。其中 $\bf{A}$、$\bf{B}$ 为 $\mathbf{M}$ 原子的密堆积层，$\bf{c}$ 为填充有部分 $\mathrm{O}$ 原子的八面体空隙层，□ 为空的八面体空隙层。

晶胞中所有 $\mathrm{O}$ 原子的坐标为：  

$(\frac{2}{3},\,\frac{1}{3},\,\frac{1}{2})$  

$(\frac{1}{3},\,\frac{2}{3},\,\frac{1}{2})$

且晶胞内 $\frac{2}{3}$ 的 $\mathbf{M}$原子位于晶胞面上

下列说法正确的有：

1.$\mathrm{O}$ 原子在 $\bf{c}$ 层中的填隙率为 1

2.$\mathrm{O}$ 原子在 $\bf{c}$ 层中的填隙率为 $\frac{2}{3}$

3.$\mathrm{O}$ 原子在 $\bf{c}$ 层中的填隙率为 $\frac{1}{3}$

4.其中一个 $\mathbf{M}$原子的x、y坐标为$(\frac{2}{3},\,\frac{2}{3})$

5.其中一个 $\mathbf{M}$ 原子的x、y坐标为$(\frac{1}{2},\,\frac{1}{2})$

6.其中一个 $\mathbf{M}$原子的x、y坐标为$(\frac{1}{3},\,\frac{2}{3})$
1,4

1,5
1,6
2,4
2,5
2,6
3,4
3,5
3,6
All of the above options are incorrect or the answer is incomplete.
1,4
1,5
1,6
2,4
2,5
2,6
3,4
3,5
3,6
以上选项均不正确或答案不完全
$\bf{X}$ obtained by precipitating and drying with sodium hydroxide is one of the most common oxides of metal $\mathrm{M}$, which can be expressed as $\mathrm{M_pO_q}$, and its mass fraction of oxygen element $\omega_{\mathrm O}=\tfrac{q\times16}{p\,M+q\times16}=6.9\%$.

Assume the chemical formula of $\bf{X}$ to be $\mathrm{MO}$, $\mathrm{M_2O_3}$, $\mathrm{MO_2}$, etc., and substitute them into the equation. Finally, it is found that when it is $\mathrm{M_2O}$, the molar mass of $\mathrm{M}$ is 107.9, which is consistent with $\mathrm{Ag}$. Therefore, the metal $\mathrm{M}$ is $\mathrm{Ag}$, and $\bf{X}$ is $\mathrm{Ag_2O}$. Substituting the mass fraction, it can be further deduced that $\bf{Y}$ is $\mathrm{AgO}$, and $\bf{Z}$ is $\mathrm{Ag_3O}$.

Substitute into the question conditions for verification, which is consistent with the chemical properties of $\mathrm{Ag}$.

<Checkpoint>[$\bf{X}$ is $\mathrm{Ag_2O}$ , $\bf{Y}$ is $\mathrm{AgO}$ , $\bf{Z}$ is $\mathrm{Ag_3O}$ ](3)</Checkpoint>

There are 2 oxygen atoms in the unit cell, so according to the chemical formula, there are 6 silver atoms, of which layers $\bf{A}$ and $\bf{B}$ have three silver atoms respectively. According to the ratio of octahedral voids to the number of atoms in the closest packing being 1:1, 6 silver atoms have a total of 6 octahedral voids. Since the oxygen atom interstitial layer and the void layer are spaced apart, there are 3 octahedral voids in layer $\bf{c}$, which are filled with 2 oxygen atoms, so the filling rate is $\frac{2}{3}$.

<Checkpoint>[The filling rate of $\mathrm{O}$ atoms in layer $\bf{c}$ is $\frac{2}{3}$, option 2 is correct, options 1 and 3 are incorrect](2)</Checkpoint>

According to $\frac{2}{3}$ of the $\mathrm{M}$ atoms in the unit cell are located on the surface of the unit cell, there are 2 atoms on the surface of the unit cell and 1 atom inside the unit cell in each silver atom close-packed layer. According to the symmetry of the octahedron and the $\mathrm{C_3}$ axis in the z direction through the oxygen atom, it can be deduced that the x and y coordinates of the silver atoms in layer $\bf{A}$ are $(\frac{1}{3},0)$, $(0,\frac{1}{3})$, $(\frac{2}{3},\frac{2}{3})$, and the x and y coordinates of the silver atoms in layer $\bf{B}$ are $(\frac{2}{3},0)$, $(0,\frac{2}{3})$, $(\frac{2}{3},\frac{2}{3})$ (layers $\bf{A}$ and $\bf{B}$ are interchangeable). The crystal diagram is as follows

<MultiModal>![Z的晶胞参考图：$\bf{A}$ 层银原子的x、y坐标为$(\frac{1}{3},0)$、$(0,\frac{1}{3})$、$(\frac{2}{3},\frac{2}{3})$；$\bf{c}$ 层氧原子的坐标为 $(\frac{2}{3},\,\frac{1}{3},\,\frac{1}{2})$、  $(\frac{1}{3},\,\frac{2}{3},\,\frac{1}{2})$；$\bf{B}$ 层银原子的x、y坐标为$(\frac{2}{3},0)$、$(0,\frac{2}{3})$、$(\frac{2}{3},\frac{2}{3})$](/media/uploads/40716a74-3c6a-4b01-af4a-ed908d3d9f77.png)</MultiModal>

<Checkpoint>[The possible x and y coordinates of silver atoms are $(\frac{1}{3},0)$, $(0,\frac{1}{3})$, $(\frac{2}{3},\frac{2}{3})$, $(\frac{2}{3},0)$, $(0,\frac{2}{3})$, $(\frac{2}{3},\frac{2}{3})$, so option 4 is correct, options 5 and 6 are incorrect](3)</Checkpoint>

In summary, the answer is D

<Checkpoint>[The answer is D](1)</Checkpoint>
由氢氧化钠沉淀并干燥得到的 $\bf{X}$ 是金属 $\mathrm{M}$ 的一种最常见氧化物，可表示为 $\mathrm{M_pO_q}$，其氧元素质量分数 $\omega_{\mathrm O}=\tfrac{q\times16}{p\,M+q\times16}=6.9\%$。  
依次假设 $\bf{X}$ 的化学式为 $\mathrm{ \mathbf{M}O}$、$\mathrm{ \mathbf{M}_2O_3}$、$\mathrm{ \mathbf{M}O_2}$ 等，代入方程最终解得当为 $\mathrm{ \mathbf{M}_2O}$ 时，$ \mathbf{M}$ 的摩尔质量为107.9，与 $\mathrm{Ag}$ 符合，因此金属  $ \mathbf{M}$  是 $\mathrm{Ag}$，$\bf{X}$ 是 $\mathrm{Ag_2O}$。代入质量分数可进一步推出 $\bf{Y}$ 是 $\mathrm{AgO}$, $\bf{Z}$ 是 $\mathrm{Ag_3O}$。

代入题目条件检验，符合 $\mathrm{Ag}$ 的化学性质。

<Checkpoint>[$\bf{X}$ 是 $\mathrm{Ag_2O}$ , $\bf{Y}$ 是 $\mathrm{AgO}$ , $\bf{Z}$ 是 $\mathrm{Ag_3O}$ ](3)</Checkpoint>

晶胞内有2个氧原子，则根据化学式有6个银原子，其中 $\bf{A}$、$\bf{B}$ 层分别有三个银原子。根据最密堆积中八面体空隙与原子数之比为1:1，则6个银原子共有6个八面体空隙。由于氧原子填隙层与空隙层相互间隔，因此在 $\bf{c}$ 层中有3个八面体空隙，填充了2个氧原子，因此填隙率为 $\frac{2}{3}$。

<Checkpoint>[$\mathrm{O}$ 原子在 $\bf{c}$ 层中的填隙率为 $\frac{2}{3}$，选项2正确，选项1、3错误](2)</Checkpoint>

根据晶胞内 $\frac{2}{3}$ 的 $ \mathbf{M}$ 原子位于晶胞面上，则每个银原子密置层中有2个原子位于晶胞面上，1个位于晶胞内部。根据八面体的对称性，以及通过氧原子的z方向的 $\mathrm{C_3}$ 轴，则可推出 $\bf{A}$ 层银原子的x、y坐标为$(\frac{1}{3},0)$、$(0,\frac{1}{3})$、$(\frac{2}{3},\frac{2}{3})$，$\bf{B}$ 层银原子的x、y坐标为$(\frac{2}{3},0)$、$(0,\frac{2}{3})$、$(\frac{2}{3},\frac{2}{3})$（$\bf{A}$、$\bf{B}$ 层可互换）。晶体图如下

<MultiModal>![Z的晶胞参考图：$\bf{A}$ 层银原子的x、y坐标为$(\frac{1}{3},0)$、$(0,\frac{1}{3})$、$(\frac{2}{3},\frac{2}{3})$；$\bf{c}$ 层氧原子的坐标为 $(\frac{2}{3},\,\frac{1}{3},\,\frac{1}{2})$、  $(\frac{1}{3},\,\frac{2}{3},\,\frac{1}{2})$；$\bf{B}$ 层银原子的x、y坐标为$(\frac{2}{3},0)$、$(0,\frac{2}{3})$、$(\frac{2}{3},\frac{2}{3})$](/media/uploads/40716a74-3c6a-4b01-af4a-ed908d3d9f77.png)</MultiModal>

<Checkpoint>[银原子可能的x、y坐标为$(\frac{1}{3},0)$、$(0,\frac{1}{3})$、$(\frac{2}{3},\frac{2}{3})$、$(\frac{2}{3},0)$、$(0,\frac{2}{3})$、$(\frac{2}{3},\frac{2}{3})$，因此选项4正确，选项5、6错误](3)</Checkpoint>

综上，答案为D

<Checkpoint>[答案为D](1)</Checkpoint>