Question

For two isomorphous crystals A and B, the ratio of the density of A to that of B is 1.6, while the ratio of the edge length of B to that of A is 2. If the molar mass of crystal B is 200 g/mol, then that of crystal A is-
A. $240gmo{l^{ - 1}}$
B. $120gmo{l^{ - 1}}$
C. $80gmo{l^{ - 1}}$
D. $160gmo{l^{ - 1}}$
 E.$40gmo{l^{ - 1}}$

Answer 1

Hint: Density is a scientific concept that relates mass and volume of a substance. Volume refers to the measurement of the amount of three-dimensional space occupied by that substance. Try to recollect the formula for density, and you can easily solve this question.

Complete step by step answer:
For a crystal lattice, density is directly proportional to its molar mass ($M$)
$d\alpha M$ ……… (1)
Also, density is inversely proportional to volume ($V$), i.e., ${\left( {edgelength} \right)^3}$ = ${\left( a \right)^3}$
$d\alpha \dfrac{1}{{{a^3}}}$ ……. (2)
From equations (1) and (2), we can conclude that
$d = \dfrac{M}{{{a^3}}}$ …….. (3)
Let the density of crystal A be $dA$ and its molar mass be $MA$ .
Similarly, density for crystal B be $dB$ and its molar mass be $MB$ .
As per given question, $\dfrac{{{d_A}}}{{{d_B}}} = 1.6$ , $\dfrac{{{a_B}}}{{{a_A}}} = 2$ , ${M_B} = 200gmo{l^{ - 1}}$ , ${M_A} = ?$


On Combining all these conditions and substituting the values in equation (3), we will get-
$\dfrac{{{d_A}}}{{{d_B}}} = \dfrac{{{M_A}}}{{{M_B}}}x{\left( {\dfrac{{{a_B}}}{{{a_A}}}} \right)^3}$
$ \Rightarrow $ ${m_A} = \dfrac{{1.6x200}}{8}$
$ \Rightarrow $ ${m_A} = 40gmo{l^{ - 1}}$

So, the correct answer is option E.

Note:
Two crystals are said to be isomorphous if both of them have the same space group, same unit cell dimension, and the types and positions of all atoms are same in both except one or more atom replacement, that brings the difference. They have almost similar shapes. For e.g. Sodium nitrate and calcium sulfate are isomorphous crystals.