What is the molar mass of a protein if a solution containing 0.02 g of the protein in 25.0 mL of the solution has an osmotic pressure of 0.56 tor at 25 °C?

Osmotic pressure may be defined as the pressure that is exerted on a solvent to prevent it to pass through the osmosis. This is often used to express the concentration of the solution. It is given by

\({\bf{\Pi = iMRT}}\)

where,

π = Osmotic pressure

i= Van't Hoff factor

M = Molarity

R = Ideal gas constant, and

T= Temperature.

\({\bf{Molar mass = }}\frac{{{\bf{mass}}}}{{{\bf{mole}}}}{\bf{ = }}\frac{{\bf{g}}}{{{\bf{mol}}}}\)

Change osmotic pressure from tor to atm:

Osmotic pressure = 0.56 tor

\({\rm{Osmotic Pressure = 0}}{\rm{.56 tor \times }}\frac{{{\rm{1atm}}}}{{{\rm{760 tor}}}}{\rm{ = 7}}{\rm{.368 \times 1}}{{\rm{0}}^{{\rm{ - 4}}}}{\rm{atm}}\) \(\)

\({\rm{Temperature = 25 ^\circ C = }}\left( {{\rm{273 + 25}}} \right){\rm{ K = 298 K}}\)

Gas Constant, R =0.08206 atmL/Kmol

Π = MRT

\(M{\rm{ }} = {\rm{ }}\frac{{\rm{\Pi }}}{{RT}}\)

\(\begin{aligned}{{}{}}{{\rm{M = }}\frac{{{\rm{7}}{\rm{.368 \times 1}}{{\rm{0}}^{{\rm{ - 4}}}}{\rm{atm }}}}{{\left( {{\rm{0}}{\rm{.08206 atmL/Kmol}}} \right)\left( {{\rm{298K}}} \right)}}{\rm{ }}}\\{{\rm{M = 0}}{\rm{.3 \times 1}}{{\rm{0}}^{{\rm{ - 4}}}}{\rm{mole}}{{\rm{L}}^{{\rm{ - 1}}}}}\end{aligned}\)

\({\rm{Molarity = }}\frac{{{\rm{Number of Moles}}}}{{{\rm{Volume of Solvent }}\left( {{\rm{Litre}}} \right)}}\)

Number of Moles = Molarity × Volume of Solvent (Litre)

Volume of Solution = 25mL = 0.025L

\(\)

\(\begin{aligned}{{}{}}{{\rm{Number of Moles = 0}}{\rm{.3 \times 1}}{{\rm{0}}^{{\rm{ - 4}}}}{\rm{mol/L \times 0}}{\rm{.025L}}}\\{{\rm{Number of Moles = 0}}{\rm{.0075 \times 1}}{{\rm{0}}^{{\rm{ - 6}}}}{\rm{mole}}}\end{aligned}\)

\(Number{\rm{ }}of{\rm{ }}Moles{\rm{ }} = {\rm{ }}\frac{{Mass}}{{Molar{\rm{ }}Mass}}\) \(\)

\(\begin{aligned}{c}{\rm{Molar Mass = }}\frac{{{\rm{Mass }}}}{{{\rm{ Number of Moles}}}}\\{\rm{ = }}\frac{{{\rm{0}}{\rm{.02 }}}}{{{\rm{0}}{\rm{.0075 \times 1}}{{\rm{0}}^{{\rm{ - 4}}}}}}\\{\rm{Molar Mass = 2}}{\rm{.7 \times 1}}{{\rm{0}}^{\rm{4}}}{\rm{g/mole}}\end{aligned}\)