Question

A crystal has monoclinic structure, with lattice parameters \( a = 5.14 \, \text{Å} \), \( b = 5.20 \, \text{Å} \), \( c = 5.30 \, \text{Å} \) and angle \( \beta = 99^\circ \). It undergoes a phase transition to tetragonal structure with lattice parameters, \( a = 5.09 \, \text{Å} \) and \( c = 5.27 \, \text{Å} \). The fractional change in the volume \( \frac{\Delta V}{V} \) of the crystal due to this transition is ............... (Round off to two decimal places).

Hint:

When calculating the fractional change in volume, remember to use the volume formula for each crystal structure and compare the initial and final volumes.

Answer 1

Correct Answer: 0.02

Step 1: Calculate the initial volume of the crystal in the monoclinic phase.
The volume of a crystal with monoclinic structure is given by \[ V_{\text{monoclinic}} = a b c \sin \beta. \] Substituting the values, \[ V_{\text{monoclinic}} = 5.14 \times 5.20 \times 5.30 \times \sin(99^\circ) \approx 5.14 \times 5.20 \times 5.30 \times 0.9848 = 140.99 \, \text{Å}^3. \] Step 2: Calculate the volume of the crystal in the tetragonal phase.
The volume of a tetragonal crystal is given by \[ V_{\text{tetragonal}} = a^2 c. \] Substituting the values, \[ V_{\text{tetragonal}} = (5.09)^2 \times 5.27 \approx 25.91 \times 5.27 = 136.79 \, \text{Å}^3. \] Step 3: Calculate the fractional change in volume.
The fractional change in volume is \[ \frac{\Delta V}{V} = \frac{V_{\text{tetragonal}} - V_{\text{monoclinic}}}{V_{\text{monoclinic}}}. \] Substituting the values, \[ \frac{\Delta V}{V} = \frac{136.79 - 140.99}{140.99} = \frac{-4.20}{140.99} = -0.0298. \] Step 4: Final Answer.
Thus, the fractional change in the volume is \( -0.03 \).