Question

The osmotic pressure of solutions of PVC in cyclohexanone at $300 K$ are plotted on the graph The molar mass of $PVC$ is ____$g \,mol ^{-1}$ (Nearest integer)

( Given : R = 0.083 L atm K^-1 mol^-1 )

Answer 1

Correct Answer: 41500

The correct answer is 41500

$\pi =M^{\prime }RT=\left(\frac{W/M}{V}\right)RT$
$\Rightarrow \pi =\left(\frac{W}{V}\right)\left(\frac{1}{M}\right)RT=C\left(\frac{RT}{M}\right)$
$\Rightarrow \frac{\pi }{C}=\frac{RT}{M}\ne f(c)$
If we assume graph between $\frac{\pi }{C}$ and $C$
Assuming $\pi$ vs $C$ graph
Slope $=\frac{RT}{M}=\frac{0.083\times 300}{M}=6\times 10^{-4}$
$\therefore M=\frac{0.083\times 300}{6\times 10^{-4}}=\frac{830\times 300}{6}$
$=41,500gm/mole$

Answer 2

The van 't Hoff equation for osmotic pressure (\(\pi\)) is: 
\[ \pi = CRT \] 
Dividing both sides by concentration (\(C\)): 
\[ \frac{\pi}{C} = RT \times \frac{1}{M} \] 
From the graph, the slope (\(\frac{\pi}{C}\)) is determined to be 6.0 atm L g$^{-1}$. 
Using the relation: \[ M = \frac{RT}{\text{slope}} \] 
Substituting the values: 
\[ M = \frac{0.083 \times 300}{6.0} = 41500 \, \text{g mol$^{-1}$}. \] 
Thus, the molar mass of PVC is 41500 g \(mol^{-1}\). 

The molar mass of a polymer like PVC can be calculated from osmotic pressure data using the van 't Hoff equation. The slope of the \(\pi / C\) graph provides critical information for this calculation.