Question

Insulin is a hormone responsible for the regulation of glucose levels in the blood. An aqueous solution of insulin has an osmotic pressure of \(2.5 \mathrm{~mm} \mathrm{Hg}\) at \(25^{\circ} \mathrm{C}\). It is prepared by dissolving \(0.100 \mathrm{~g}\) of insulin in enough water to make \(125 \mathrm{~mL}\) of solution. What is the molar mass of insulin?

Short answer

Answer: The molar mass of insulin in the given aqueous solution is approximately 601 g/mol.

Step-by-step solution

Recall the osmotic pressure formula

The osmotic pressure (Π) formula is given by Π = MiRT, where M is the molarity, i is the van't Hoff factor (which is equal to 1 for non-electrolytes like insulin), R is the ideal gas constant, and T is the temperature in Kelvin.

Convert the temperature to Kelvin

We must convert the temperature from Celsius to Kelvin by using the formula K = °C + 273.15. In this case, we have 25°C: \(T = 25 + 273.15 = 298.15\,\mathrm{K}\)

Convert osmotic pressure to atm

Osmotic pressure is given in mm Hg, but we need it in atmospheres (atm) to use the ideal gas constant (R) in the formula. We use the conversion factor 1 atm = 760 mm Hg: \(2.5\,\mathrm{mm\,Hg} \times \frac{1\,\mathrm{atm}}{760\,\mathrm{mm\,Hg}} = 0.003289\,\mathrm{atm}\)

Rearrange the formula to solve for molarity (M)

We will rearrange the equation Π = MiRT to solve for M: \(M = \frac{Π}{iRT}\)

Plug in the values and find the molarity

Insert the values for osmotic pressure, gas constant, and temperature into the formula to find the molarity: \(M = \frac{0.003289\,\mathrm{atm}}{1 \times 0.0821\,\mathrm{\frac{L\,atm}{mol\,K}} \times 298.15\,\mathrm{K}} = 0.00133\,\mathrm{M}\)

Calculate moles of insulin

Use the molarity and solution volume to find the number of moles of insulin: \(n = M \times V = 0.00133\,\mathrm{M} \times 0.125\,\mathrm{L} = 0.0001662\,\mathrm{mol}\)

Calculate the molar mass of insulin

Finally, divide the mass of insulin by the number of moles to find the molar mass: \(Molar\,mass = \frac{mass}{moles} = \frac{0.1\,\mathrm{g}}{0.0001662\,\mathrm{mol}} = 601\,\mathrm{\frac{g}{mol}}\) The molar mass of insulin is approximately \(601\,\mathrm{\frac{g}{mol}}\).