In a cubic lattice of XYZ, X atoms are present at all corners except one corner which is occupied by Y atoms. Z atoms are present at face centres. What is the formula of compound? (Ans. X2YZ24)

My solution:

No. of X atoms in a unit cell = 7 * 1/8 = 7/8

No. of Y atoms in a unit cell = 1 * 1/8 = 1/8

No. of Z atoms in a unit cell = 6 * 1/2 = 3

Therefore, the ratio of X:Y:Z = 7/8:1/8:3 which is the same thing as 7:1:24

So, the formula = X7YZ24

What am I doing wrong?

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    $\begingroup$ Your solutions looks fine. May be the answer given is wrong, $\endgroup$ – Ayush Pateria Mar 22 '15 at 8:54
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    $\begingroup$ You need to take into account which atoms are shared with other unit cells. A face centre is shared with one, an edge centre with 4 other cells etc. $\endgroup$ – matt_black Mar 22 '15 at 10:45
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    $\begingroup$ The answer given is wrong and the question is poorly worded; a periodic cubic lattice requires that all the corners in a unit cell be of the same atom. If one of the corners of a cell is made of a different atom, then you're not actually looking at the unit cell. $\endgroup$ – Nicolau Saker Neto Mar 22 '15 at 13:01

In the spirit of "3 idiots", I would like to say your answer is right, but the question is wrong.

Remember that a lattice is "infinitely" repeated units in 3 dimensional space so you should be able to expand the unit cell in x, y and z axes, indefinitely.

Now imagine your cubic lattice with an Y atom in just 1 (out of 8) corners.

Expanding the unit cell in the x-axis will necessarily duplicate the Y atom in the x-axis because all expanded unit cells must be identical in composition. Similarly, expanding the unit cell in the y-axis will duplicate the Y atom in the y-axis, and expanding the unit cell in the z-axis will duplicate the Y atom in the z-axis.

In the end you will find the having one corner as Y atom will necessarily have ALL corners as Y atoms. In other words, X IS Y, and the formula is necessarily $\ce{XZ_3}$ or $\ce{YZ_3}$

Unfortunately, I don'k think the designer of the question saw his fatal contradiction because this very question is seen in the (mock) Joint Entrance Examination (JEE) in India.

cubic exam question

And $\ce{X_2YZ_24}$ is out of question even in the wrong question.


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