Question

$\mathrm{N}\mathrm{a}\mathrm{H}$crystallizes with the same crystal structure as $\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}$. The edge length of the cubic unit cell of NaH is

(a) Calculate the ionic radius of ${\mathrm{H}}^{-}$. (The ionic radius of $\mathrm{L}{\mathrm{i}}^{+}$is

$0.95A$

(b) Calculate the density of $\mathrm{N}\mathrm{a}\mathrm{H}$.

Short answer

a. The radius of hydride ion is $1.630{\mathrm{A}}^{\circ }$

b. The density of $\mathrm{N}\mathrm{a}\mathrm{H}$ is $29.76\mathrm{g}/\mathrm{m}\mathrm{L}$

Step-by-step solution

Define the coordination number of a sphere

1. The relationship between edge length and radius for face cantered cubic unit cell is $r=\frac{a}{2\sqrt{2}}$.Here, "a" is edge length and " $r$ " is radius. Radius of a compound is average of cation and anion radius.
2. To calculate the density of tungsten, use the formula density $=\frac{Z\times M}{N_n\times a^3}$ Here, $Z$ is number of atoms present in the body centred cubic that is two. $\mathrm{M}$ is molar mass of metal; molar mass of $\mathrm{N}\mathrm{a}\mathrm{H}$ is $23\mathrm{g}/\mathrm{m}\mathrm{o}\mathrm{l}$ $N_o$ is a Avogadro's number that is $6.023\times {10}^{23}$

Find the radius of NaH  

To calculate the radius of $\mathrm{N}\mathrm{a}\mathrm{H}$ uses the formula $r=\frac{a}{2\sqrt{2}}$.

Here, $a$ is edge length and $r$ is radius.

Calculate the radius of $\mathrm{N}\mathrm{a}\mathrm{H}$ as follows:

$\begin{array}{rl}r& =\frac{a}{2\sqrt{2}}\\ r& =\frac{4.880{\mathrm{A}}^{\circ }}{2\sqrt{2}}\\ & =1.725{\mathrm{A}}^{\circ }\end{array}$

Calculate the radius of hydride ion is as follows:

$\begin{array}{rl}{r}_{N{a}^{+}}+r_{{\mathrm{H}}^{-}}& =1.725{\mathrm{A}}^{\circ }\\ r_{{\mathrm{H}}^{-}}& =1.725{\mathrm{A}}^{\circ }-r_{N{a}^{+}}\\ & =1.725{\mathrm{A}}^{\circ }-0.095{\mathrm{A}}^{\circ }\\ & =1.630{\mathrm{A}}^{\circ }\end{array}$

Find the density of NaH  

To calculate density of $\mathrm{N}\mathrm{a}\mathrm{H}$, substitute edge length, Avogadro's number, number of atoms present in the body centred cubic and its molar mass is as follows:

$\begin{array}{rl}\mathrm{D}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}& =\frac{Z\times M}{N_o\times a^3}\\ & =\frac{4\times 23\mathrm{g}}{6.023\times {10}^{23}\times {\left(1.725{\mathrm{A}}^{\circ }\times \frac{{10}^{-8}\mathrm{c}\mathrm{m}}{1{\mathrm{A}}^{\circ }}\right)}^3}\\ & =29.76\mathrm{g}/\mathrm{m}\mathrm{L}\end{array}$

Hence, density of $\mathrm{N}\mathrm{a}\mathrm{H}$ is $29.76\mathrm{g}/\mathrm{m}\mathrm{L}$.