A crystal has monoclinic structure with a equals to 5.14 angstrom, b equals to 5.20 angstrom and strong c is equals to 5.30 angstrom and beta equals to 99 degree it undergoes phase transition to tetragonal structure with lattice parameter a = 5.09 angstrom , c = 5.27 angstrom the fractional change in volume of crystal due to transaction will be ?

Question

Answer

To calculate the fractional change in volume of the crystal due to the phase transition from monoclinic to tetragonal structure, we first need to determine the volumes of both crystal structures.

Monoclinic Volume Calculation
The volume $ V_m $ of the monoclinic unit cell can be calculated using the formula:
$$ V_m = a \cdot b \cdot c \cdot \sin(\beta) $$
Where:
- $ a = 5.14 $ Å
- $ b = 5.20 $ Å
- $ c = 5.30 $ Å
- $ \beta = 99^\circ $ (convert to radians for calculation)
Calculating $ V_m $:
$$ V_m = 5.14 \times 5.20 \times 5.30 \times \sin(99^\circ) $$
$$ V_m \approx 5.14 \times 5.20 \times 5.30 \times 0.9848 $$
$$ V_m \approx 140.89 \, \text{Å}^3 $$

Tetragonal Volume Calculation
The volume $ V_t $ of the tetragonal unit cell is given by:
$$ V_t = a^2 \cdot c $$
Where:
- $ a = 5.09 $ Å
- $ c = 5.27 $ Å
Calculating $ V_t $:
$$ V_t = (5.09)^2 \times 5.27 $$
$$ V_t \approx 25.9081 \times 5.27 $$
$$ V_t \approx 136.66 \, \text{Å}^3 $$

Fractional Change in Volume
Now, we can calculate the fractional change in volume:
$$ \text{Fractional Change} = \frac{V_t - V_m}{V_m} $$
$$ \text{Fractional Change} = \frac{136.66 - 140.89}{140.89} $$
$$ \text{Fractional Change} \approx \frac{-4.23}{140.89} $$
$$ \text{Fractional Change} \approx -0.030 $$
This indicates that there is a decrease in volume of approximately 3.0% during the phase transition from monoclinic to tetragonal structure.

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