Question

derive the weight percentage crystallinity formula (see slide 25 of module 5.2):
%$_{c}=100\frac{\
ho_{c}(\
ho_{s}-\
ho_{a})}{\
ho_{s}(\
ho_{c}-\
ho_{a})}$
try to derive the equation with no exterior help.
if you need help, instead of asking your ai assistant to do it for you, try these hints:

• the definition of weight percentage of crystallinity is %$_{c}=\frac{m_{c}}{m_{s}}$ where $m_{c}$ and $m_{s}$ are the mass of crystalline regions and specimen, respectively.
• the definition of weight percentage of the amorphous regions is %$_{a}=\frac{m_{a}}{m_{s}} = 1 - $%$_{c}$
• the definition of density of the specimen is $\

ho_{s}=\frac{m_{s}}{v_{s}}=\frac{m_{a}+m_{c}}{v_{s}}$

• the definition of density of crystalline regions is $\

ho_{c}=\frac{m_{c}}{v_{c}}$ where $v_{c}$ is the volume of crystalline regions.

• the definition of density of amorphous regions is $\

ho_{a}=\frac{m_{a}}{v_{a}}$ where $v_{a}$ is the volume of amorphous regions.

• the volume of the specimen is $v_{s}=v_{c}+v_{a}$

Explanation:

Step1: Express mass in terms of density

We know $m =
ho V$. So, $m_c=
ho_cV_c$, $m_a =
ho_aV_a$ and $m_s=
ho_sV_s$. Since $V_s=V_c + V_a$, $m_s=
ho_s(V_c + V_a)$. Also, $\%_c=\frac{m_c}{m_s}$. Substituting $m_c$ and $m_s$ we get $\%_c=\frac{
ho_cV_c}{
ho_s(V_c + V_a)}$.

Step2: Express $V_a$ in terms of $V_c$

We know $\%_a=\frac{m_a}{m_s}=1 - \%_c$. Since $m_a=
ho_aV_a$ and $m_s=
ho_sV_s=
ho_s(V_c + V_a)$, $\frac{
ho_aV_a}{
ho_s(V_c + V_a)}=1-\frac{
ho_cV_c}{
ho_s(V_c + V_a)}$. Cross - multiplying gives $
ho_aV_a=
ho_s(V_c + V_a)-
ho_cV_c$. Rearranging for $V_a$: $V_a(
ho_s-
ho_a)=V_c(
ho_s -
ho_c)$, so $V_a=\frac{V_c(
ho_s -
ho_c)}{
ho_s-
ho_a}$.

Step3: Substitute $V_a$ into $\%_c$ formula

Substitute $V_a=\frac{V_c(
ho_s -
ho_c)}{
ho_s-
ho_a}$ into $\%_c=\frac{
ho_cV_c}{
ho_s(V_c + V_a)}$. First, $V_c + V_a=V_c+\frac{V_c(
ho_s -
ho_c)}{
ho_s-
ho_a}=V_c\frac{
ho_s-
ho_a+
ho_s -
ho_c}{
ho_s-
ho_a}$. Then $\%_c=\frac{
ho_cV_c}{
ho_sV_c\frac{2
ho_s-
ho_a -
ho_c}{
ho_s-
ho_a}}$. Canceling out $V_c$ and multiplying by 100 to get percentage: $\%_c = 100\frac{
ho_c(
ho_s-
ho_a)}{
ho_s(
ho_c-
ho_a)}$.

Answer:

$\%_c = 100\frac{
ho_c(
ho_s-
ho_a)}{
ho_s(
ho_c-
ho_a)}$