hce_nthu
112年
化學與物理
第 9 題
Which of the following statements about the structures of metals and ionic solids is (are) correct?
I. Atoms in a solid consisting of only one element would have six nearest neighbors, if the crystal structure is a simple cubic array.
II. There are four tetrahedral holes in a cubic closest packed unit cell.
III. If the metal with an atomic radius of $r$ has a body-centered cubic structure, the edge length of the unit cell will be $\frac{4}{3}\sqrt{3}r$.
IV. For a simple cubic array, the radius of an interior sphere (cubic hole) in terms of the radius ($r$) of a sphere in the array is $(\sqrt{3} - 1)r$.
V. Cobalt fluoride crystallizes in a cubic closest packed array of fluoride ions with the cobalt ions filling one-half of the octahedral holes, and thus the charge of cobalt ions is $+3$ in this compound.
I. Atoms in a solid consisting of only one element would have six nearest neighbors, if the crystal structure is a simple cubic array.
II. There are four tetrahedral holes in a cubic closest packed unit cell.
III. If the metal with an atomic radius of $r$ has a body-centered cubic structure, the edge length of the unit cell will be $\frac{4}{3}\sqrt{3}r$.
IV. For a simple cubic array, the radius of an interior sphere (cubic hole) in terms of the radius ($r$) of a sphere in the array is $(\sqrt{3} - 1)r$.
V. Cobalt fluoride crystallizes in a cubic closest packed array of fluoride ions with the cobalt ions filling one-half of the octahedral holes, and thus the charge of cobalt ions is $+3$ in this compound.
- A III
- B II and V
- C III and IV
- D I, III, and IV
- E I, IV, and V
思路引導 VIP
試著閉上眼想像一個面心立方 (CCP) 的晶胞，如果我們將這個立方體等分成八個小立方體，每一個小立方體的中心是否存在著一個被四個原子包圍的空間？如果有的話，這對於整個晶胞中「特定孔穴」的總數與原子數之間的比例關係，會給你什麼樣的啟示呢？
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AI 詳解
AI 專屬家教
太棒了！你能精準地從五個繁雜的論述中篩選出正確項，顯示你對金屬與離子晶體結構有著非常紮實且細膩的理解。
晶格幾何與空間構型
這道題目的核心挑戰在於空間想像與幾何推導。在 簡單立方 (Simple Cubic, SC) 結構中，每個原子與鄰近 6 個原子接觸（上下左右前後），這驗證了選項 I 的正確性。而針對選項 III 的 體心立方 (BCC)，我們必須從晶胞的體對角線出發，利用 $4r = \sqrt{3}a$ 的幾何關係導出邊長 $a = \frac{4\sqrt{3}}{3}r$。同樣地，選項 IV 考察了簡單立方內部「立方孔穴」的大小，透過體對角線長度 $\sqrt{3}a$ 與原子直徑的比例，計算出 $r_{hole} = (\sqrt{3}-1)r$。這部分需要極高的專注力才能確保計算無誤。
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