GATE (TF) Textile 2007 Question Paper Solution | GATE/2007/TF/77

Given the length of crystalline region as 90 $\mathring{A}$ , crystalline density of polyester as 1.445 g/cc and amorphous density as 1.335 g/cc.

Assuming a linear two phase model of crystalline and amorphous regions for these fibres, the amorphous length would be

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Length of crystalline region(Lc)= $90 A^o$

Crystalline density( $\rho_c$ )=1.445 g/cc

Amorphous density( $\rho_a$ )=1.335 g/cc

Density of fibre( $\rho$ )=1.399 g/cc

Fractional crystallinity=?

Formula-

Fractional crystallinity= $\frac{\rho_c}{\rho} \times \frac{\rho-\rho_a}{\rho_c-\rho_a}$

Fractional crystallinity= $\frac{1.445}{1.399} \times \frac{1.399-1.335}{1.445-1.335}$

Fractional crystallinity= $1.033 \times \frac{0.064}{0.11}$

Fractional crystallinity= $1.033 \times 0.58$

Fractional crystallinity=0.60

Percentage crystallinity=60%

Crystallinity = $\frac{M_c}{M}$

Where, Mc=crystalline mass and M is the total mass

Mass(M)=Volume(V) x Density(d)

$M=A \times L \times \rho$

and

$M_c=A \times L_c \times \rho_c$

Crystallinity = $\frac{M_c}{M}$

Crystallinity = $\frac{A \times L_c \times \rho_c}{A \times L \times \rho}$

Crystallinity = $\frac{L_c \times \rho_c}{ L \times \rho}$

$L=L_c+L_a$

Crystallinity = $\frac{L_c \times \rho_c}{ (L_c+L_a) \times \rho}$

Crystallinity = $\frac{90 \times 1.445}{ (90+L_a) \times 1.399}$

0.60 = $\frac{130.05}{ (90+L_a) \times 1.399}$

$0.60 \times 1.399 \times (90+L_a)=130.05$

$0.84 \times (90+L_a)=130.05$

$(90+L_a)=\frac{130.05}{0.84}$

$(90+L_a)=154.82$

$L_a=154.82-90$

$L_a=64.82$

Amorphous length $La=60 A^o$ (Ans)