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$300 \mathrm{~mL}$ of an aqueous solution of a protein contains $2.52 \mathrm{~g}$ of the protein. If osmotic pressure of such a solution at $300 \mathrm{~K}$ is $5.04 \times 10^{-3}$ bar, the molar mass of the protein in $\mathrm{g} \mathrm{mol}^{-1}$ is
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Verified Answer
The correct answer is:
$41.5 \times 10^3$
$p=i \times M \times R \times T$
$$
\begin{aligned}
& p=\text { osmotic pressure }=5.04 \times 10^{-3} \text { bar }=5.04 \mathrm{~atm} \\
& i=\text { van't Hoff constant }=1 \text { for undissociating } \\
& \text { molecules like protein }
\end{aligned}
$$
molecules like protein
$$
\begin{aligned}
& M=\text { molarity }=\frac{\text { moles solute }}{\text { liter solution }} \\
& =\frac{\text { weight in } \mathrm{g}}{\text { molecular weight }} \times \frac{1}{\text { liter }}=\frac{2.52}{x} \times \frac{1000}{300}=\frac{25.2}{3 x}
\end{aligned}
$$
$$
\begin{aligned}
T & =\text { Temperature }=300 \mathrm{~K} \\
R & =\text { Gas constant }=0.08206 \mathrm{~L} \mathrm{~atm} / \mathrm{mol} \mathrm{K} .
\end{aligned}
$$
$$
\begin{aligned}
5.04 \times 10^{-3} & =1 \times \frac{25.2}{3 x} \times 0.08206 \times 300 \\
3 x & =\frac{25.2 \times 0.08206 \times 300}{5.04 \times 10^{-3}} \\
x & =\frac{25.2 \times 0.08206 \times 300}{5.04 \times 3 \times 10^{-3}}=\frac{6.2037 \times 100}{1512 \times 10^{-3}} \\
& =0.410 \times 100 \times 10^3=41.0 \times 10^3
\end{aligned}
$$
$$
\begin{aligned}
& p=\text { osmotic pressure }=5.04 \times 10^{-3} \text { bar }=5.04 \mathrm{~atm} \\
& i=\text { van't Hoff constant }=1 \text { for undissociating } \\
& \text { molecules like protein }
\end{aligned}
$$
molecules like protein
$$
\begin{aligned}
& M=\text { molarity }=\frac{\text { moles solute }}{\text { liter solution }} \\
& =\frac{\text { weight in } \mathrm{g}}{\text { molecular weight }} \times \frac{1}{\text { liter }}=\frac{2.52}{x} \times \frac{1000}{300}=\frac{25.2}{3 x}
\end{aligned}
$$
$$
\begin{aligned}
T & =\text { Temperature }=300 \mathrm{~K} \\
R & =\text { Gas constant }=0.08206 \mathrm{~L} \mathrm{~atm} / \mathrm{mol} \mathrm{K} .
\end{aligned}
$$
$$
\begin{aligned}
5.04 \times 10^{-3} & =1 \times \frac{25.2}{3 x} \times 0.08206 \times 300 \\
3 x & =\frac{25.2 \times 0.08206 \times 300}{5.04 \times 10^{-3}} \\
x & =\frac{25.2 \times 0.08206 \times 300}{5.04 \times 3 \times 10^{-3}}=\frac{6.2037 \times 100}{1512 \times 10^{-3}} \\
& =0.410 \times 100 \times 10^3=41.0 \times 10^3
\end{aligned}
$$
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