lithium borohydride crystallizes in an orthorhombic system eith 4 molecules per unit cell .the unit cell dimensions are a=6.8A0,b=4.4A0 and c=7.2A0.if the cell molar mass is 21.76 then the density of crystal is ????

To find the density of lithium borohydride, we can use the formula for density, which is defined as mass per unit volume. In this case, we will calculate the density using the molar mass and the dimensions of the unit cell. Let's break this down step by step.

Understanding the Components

The density (\( \rho \)) of a crystal can be calculated using the formula:

Density (\( \rho \)) = \frac{Mass of the unit cell}{Volume of the unit cell}

Step 1: Calculate the Volume of the Unit Cell

The volume (\( V \)) of the unit cell in an orthorhombic system can be calculated using the formula:

Volume (\( V \)) = a \times b \times c

Given the dimensions:

• a = 6.8 Å
• b = 4.4 Å
• c = 7.2 Å

Now, substituting the values:

V = 6.8 \times 4.4 \times 7.2

Calculating this gives:

V = 6.8 \times 4.4 = 29.92

29.92 \times 7.2 = 215.424 ų

Step 2: Calculate the Mass of the Unit Cell

The mass of the unit cell can be calculated using the molar mass and Avogadro's number. The molar mass of lithium borohydride is given as 21.76 g/mol. Since there are 4 molecules per unit cell, the mass of the unit cell is:

Mass = \frac{Molar Mass}{Avogadro's Number} \times Number of Molecules

Using Avogadro's number, \( N_A = 6.022 \times 10^{23} \) mol⁻¹:

Mass = \frac{21.76 \text{ g/mol}}{6.022 \times 10^{23} \text{ mol}^{-1}} \times 4

Calculating the mass:

Mass = \frac{21.76}{6.022 \times 10^{23}} \approx 3.61 \times 10^{-23} \text{ g}

Mass of the unit cell = 3.61 \times 10^{-23} \times 4 \approx 1.44 \times 10^{-22} \text{ g}

Step 3: Calculate the Density

Now that we have both the mass and the volume, we can find the density:

Density (\( \rho \)) = \frac{Mass}{Volume}

Substituting the values:

Density = \frac{1.44 \times 10^{-22} \text{ g}}{215.424 \times 10^{-24} \text{ cm}^3}

Calculating this gives:

Density = \frac{1.44 \times 10^{-22}}{2.15424 \times 10^{-22}} \approx 0.667 \text{ g/cm}^3

Final Result

Thus, the density of lithium borohydride crystallizing in an orthorhombic system is approximately 0.667 g/cm³. This value reflects the compactness of the crystal structure and provides insight into its physical properties.