Question

An aqueous solution containing 1.00 g of bovine insulin (a protein, not ionized) per liter has an osmotic pressure of 3.1 mm Hg at 25 °C. How do you calculate the molar mass of bovine insulin?

Answer 1

The molar mass is `6.0 ×10^3color(white)(l)"g/mol"`.

Explanation:

The formula for osmotic pressure `Pi` is

`color(blue)(bar(ul(|color(white)(a/a)Pi = cRTcolor(white)(a/a)|)))" "`

where

`c =` the molar concentration of the solute
`R =` the universal gas constant
`T =` the Kelvin temperature of the solution

Since `c = "moles"/"litres" = n/V`, we can write

`Pi = (nRT)/V`

Since `n ="mass"/"molar mass"= m/M`, we can write

`Pi = (nRT)/(MV)`

We can solve this equation for the molar mass and get

`color(blue)(bar(ul(|color(white)(a/a)M=(mRT)/(PiV)color(white)(a/a)|)))" "`

In this problem

`m = "1.00 g"`
`R = "0.082 06 L·atm·K"^"-1""mol"^"-1"`
`T = "25 °C = 298.15 K"`
`Pi = 3.1 color(red)(cancel(color(black)("mmHg"))) × "1 atm"/(760 color(red)(cancel(color(black)("mmHg")))) = "0.004 08 atm"`
`V = "1 L"`

∴ `M = ("1.00 g" × "0.082 06" color(red)(cancel(color(black)("L·atm·K"^"-1")))"mol"^"-1" × 298.15 color(red)(cancel(color(black)("K"))))/("0.00 408"color(red)(cancel(color(black)("atm"))) × 1 color(red)(cancel(color(black)("L")))) = 6.0 × 10^3color(white)(l)"g/mol"`