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